# HG changeset patch
# User hoelzl
# Date 1400493885 -7200
# Node ID e5366291d6aaa0cf08f1e2ec23f0b9e9ae32889f
# Parent  d0e5225601d3c6daee52537fa12e8f7e65b4bbe7
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces

diff -r d0e5225601d3 -r e5366291d6aa NEWS
--- a/NEWS	Mon May 19 11:27:02 2014 +0200
+++ b/NEWS	Mon May 19 12:04:45 2014 +0200
@@ -670,6 +670,68 @@
     derivative_is_linear      ~>  has_derivative_linear
     bounded_linear_imp_linear ~>  bounded_linear.linear
 
+* HOL-Probability:
+  - replaced the Lebesgue integral on real numbers by the more general Bochner
+    integral for functions into a real-normed vector space.
+
+    integral_zero               ~>  integral_zero / integrable_zero
+    integral_minus              ~>  integral_minus / integrable_minus
+    integral_add                ~>  integral_add / integrable_add
+    integral_diff               ~>  integral_diff / integrable_diff
+    integral_setsum             ~>  integral_setsum / integrable_setsum
+    integral_multc              ~>  integral_mult_left / integrable_mult_left
+    integral_cmult              ~>  integral_mult_right / integrable_mult_right
+    integral_triangle_inequality~>  integral_norm_bound
+    integrable_nonneg           ~>  integrableI_nonneg
+    integral_positive           ~>  integral_nonneg_AE
+    integrable_abs_iff          ~>  integrable_abs_cancel
+    positive_integral_lim_INF   ~>  positive_integral_liminf
+    lebesgue_real_affine        ~>  lborel_real_affine
+    borel_integral_has_integral ~>  has_integral_lebesgue_integral
+    integral_indicator          ~>
+         integral_real_indicator / integrable_real_indicator
+    positive_integral_fst       ~>  positive_integral_fst'
+    positive_integral_fst_measurable ~> positive_integral_fst
+    positive_integral_snd_measurable ~> positive_integral_snd
+
+    integrable_fst_measurable   ~>
+         integral_fst / integrable_fst / AE_integrable_fst
+
+    integrable_snd_measurable   ~>
+         integral_snd / integrable_snd / AE_integrable_snd
+
+    integral_monotone_convergence  ~>
+         integral_monotone_convergence / integrable_monotone_convergence
+
+    integral_monotone_convergence_at_top  ~>
+         integral_monotone_convergence_at_top /
+         integrable_monotone_convergence_at_top
+
+    has_integral_iff_positive_integral_lebesgue  ~>
+         has_integral_iff_has_bochner_integral_lebesgue_nonneg
+
+    lebesgue_integral_has_integral  ~>
+         has_integral_integrable_lebesgue_nonneg
+
+    positive_integral_lebesgue_has_integral  ~>
+         integral_has_integral_lebesgue_nonneg /
+         integrable_has_integral_lebesgue_nonneg
+
+    lebesgue_integral_real_affine  ~>
+         positive_integral_real_affine
+
+    has_integral_iff_positive_integral_lborel  ~>
+         integral_has_integral_nonneg / integrable_has_integral_nonneg
+
+    The following theorems where removed:
+
+    lebesgue_integral_nonneg
+    lebesgue_integral_uminus
+    lebesgue_integral_cmult
+    lebesgue_integral_multc
+    lebesgue_integral_cmult_nonneg
+    integral_cmul_indicator
+    integral_real
 
 *** Scala ***
 
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Lattices_Big.thy
--- a/src/HOL/Lattices_Big.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Lattices_Big.thy	Mon May 19 12:04:45 2014 +0200
@@ -125,6 +125,9 @@
   finally show ?case .
 qed
 
+lemma infinite: "\<not> finite A \<Longrightarrow> F A = the None"
+  unfolding eq_fold' by (cases "finite (UNIV::'a set)") (auto intro: finite_subset fold_infinite)
+
 end
 
 locale semilattice_order_set = binary?: semilattice_order + semilattice_set
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Library/Extended_Real.thy
--- a/src/HOL/Library/Extended_Real.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Library/Extended_Real.thy	Mon May 19 12:04:45 2014 +0200
@@ -453,7 +453,7 @@
 
 lemma ereal_add_strict_mono:
   fixes a b c d :: ereal
-  assumes "a = b"
+  assumes "a \<le> b"
     and "0 \<le> a"
     and "a \<noteq> \<infinity>"
     and "c < d"
@@ -2022,6 +2022,22 @@
      by auto
 qed (auto simp add: image_Union image_Int)
 
+
+lemma eventually_finite:
+  fixes x :: ereal
+  assumes "\<bar>x\<bar> \<noteq> \<infinity>" "(f ---> x) F"
+  shows "eventually (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>) F"
+proof -
+  have "(f ---> ereal (real x)) F"
+    using assms by (cases x) auto
+  then have "eventually (\<lambda>x. f x \<in> ereal ` UNIV) F"
+    by (rule topological_tendstoD) (auto intro: open_ereal)
+  also have "(\<lambda>x. f x \<in> ereal ` UNIV) = (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>)"
+    by auto
+  finally show ?thesis .
+qed
+
+
 lemma open_ereal_def:
   "open A \<longleftrightarrow> open (ereal -` A) \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {ereal x <..} \<subseteq> A)) \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A))"
   (is "open A \<longleftrightarrow> ?rhs")
@@ -2265,6 +2281,27 @@
   shows "a * b = a * c \<longleftrightarrow> (\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c"
   by (cases rule: ereal3_cases[of a b c]) (simp_all add: zero_less_mult_iff)
 
+lemma tendsto_add_ereal:
+  fixes x y :: ereal
+  assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and y: "\<bar>y\<bar> \<noteq> \<infinity>"
+  assumes f: "(f ---> x) F" and g: "(g ---> y) F"
+  shows "((\<lambda>x. f x + g x) ---> x + y) F"
+proof -
+  from x obtain r where x': "x = ereal r" by (cases x) auto
+  with f have "((\<lambda>i. real (f i)) ---> r) F" by simp
+  moreover
+  from y obtain p where y': "y = ereal p" by (cases y) auto
+  with g have "((\<lambda>i. real (g i)) ---> p) F" by simp
+  ultimately have "((\<lambda>i. real (f i) + real (g i)) ---> r + p) F"
+    by (rule tendsto_add)
+  moreover
+  from eventually_finite[OF x f] eventually_finite[OF y g]
+  have "eventually (\<lambda>x. f x + g x = ereal (real (f x) + real (g x))) F"
+    by eventually_elim auto
+  ultimately show ?thesis
+    by (simp add: x' y' cong: filterlim_cong)
+qed
+
 lemma ereal_inj_affinity:
   fixes m t :: ereal
   assumes "\<bar>m\<bar> \<noteq> \<infinity>"
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Library/Indicator_Function.thy
--- a/src/HOL/Library/Indicator_Function.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Library/Indicator_Function.thy	Mon May 19 12:04:45 2014 +0200
@@ -5,7 +5,7 @@
 header {* Indicator Function *}
 
 theory Indicator_Function
-imports Main
+imports Complex_Main
 begin
 
 definition "indicator S x = (if x \<in> S then 1 else 0)"
@@ -65,4 +65,9 @@
   using setsum_mult_indicator[OF assms, of "%x. 1::nat"]
   unfolding card_eq_setsum by simp
 
+lemma setsum_indicator_scaleR[simp]:
+  "finite A \<Longrightarrow>
+    (\<Sum>x \<in> A. indicator (B x) (g x) *\<^sub>R f x) = (\<Sum>x \<in> {x\<in>A. g x \<in> B x}. f x::'a::real_vector)"
+  using assms by (auto intro!: setsum_mono_zero_cong_right split: split_if_asm simp: indicator_def)
+
 end
\ No newline at end of file
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Binary_Product_Measure.thy
--- a/src/HOL/Probability/Binary_Product_Measure.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Binary_Product_Measure.thy	Mon May 19 12:04:45 2014 +0200
@@ -5,7 +5,7 @@
 header {*Binary product measures*}
 
 theory Binary_Product_Measure
-imports Lebesgue_Integration
+imports Nonnegative_Lebesgue_Integration
 begin
 
 lemma Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
@@ -510,7 +510,7 @@
                    positive_integral_monotone_convergence_SUP incseq_def le_fun_def
               cong: measurable_cong)
 
-lemma (in sigma_finite_measure) positive_integral_fst:
+lemma (in sigma_finite_measure) positive_integral_fst':
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" "\<And>x. 0 \<le> f x"
   shows "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M \<partial>M1) = integral\<^sup>P (M1 \<Otimes>\<^sub>M M) f" (is "?I f = _")
 using f proof induct
@@ -525,30 +525,25 @@
                    borel_measurable_positive_integral_fst' positive_integral_mono incseq_def le_fun_def
               cong: positive_integral_cong)
 
-lemma (in sigma_finite_measure) positive_integral_fst_measurable:
+lemma (in sigma_finite_measure) positive_integral_fst:
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)"
-  shows "(\<lambda>x. \<integral>\<^sup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1"
-      (is "?C f \<in> borel_measurable M1")
-    and "(\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M) \<partial>M1) = integral\<^sup>P (M1 \<Otimes>\<^sub>M M) f"
+  shows "(\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M) \<partial>M1) = integral\<^sup>P (M1 \<Otimes>\<^sub>M M) f"
   using f
     borel_measurable_positive_integral_fst'[of "\<lambda>x. max 0 (f x)"]
-    positive_integral_fst[of "\<lambda>x. max 0 (f x)"]
+    positive_integral_fst'[of "\<lambda>x. max 0 (f x)"]
   unfolding positive_integral_max_0 by auto
 
 lemma (in sigma_finite_measure) borel_measurable_positive_integral[measurable (raw)]:
   "split f \<in> borel_measurable (N \<Otimes>\<^sub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^sup>+ y. f x y \<partial>M) \<in> borel_measurable N"
-  using positive_integral_fst_measurable(1)[of "split f" N] by simp
+  using borel_measurable_positive_integral_fst'[of "\<lambda>x. max 0 (split f x)" N]
+  by (simp add: positive_integral_max_0)
 
-lemma (in sigma_finite_measure) borel_measurable_lebesgue_integral[measurable (raw)]:
-  "split f \<in> borel_measurable (N \<Otimes>\<^sub>M M) \<Longrightarrow> (\<lambda>x. \<integral> y. f x y \<partial>M) \<in> borel_measurable N"
-  by (simp add: lebesgue_integral_def)
-
-lemma (in pair_sigma_finite) positive_integral_snd_measurable:
+lemma (in pair_sigma_finite) positive_integral_snd:
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>P (M1 \<Otimes>\<^sub>M M2) f"
 proof -
   note measurable_pair_swap[OF f]
-  from M1.positive_integral_fst_measurable[OF this]
+  from M1.positive_integral_fst[OF this]
   have "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1))"
     by simp
   also have "(\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>P (M1 \<Otimes>\<^sub>M M2) f"
@@ -560,113 +555,7 @@
 lemma (in pair_sigma_finite) Fubini:
   assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
   shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M2) \<partial>M1)"
-  unfolding positive_integral_snd_measurable[OF assms]
-  unfolding M2.positive_integral_fst_measurable[OF assms] ..
-
-lemma (in pair_sigma_finite) integrable_product_swap:
-  assumes "integrable (M1 \<Otimes>\<^sub>M M2) f"
-  shows "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x,y). f (y,x))"
-proof -
-  interpret Q: pair_sigma_finite M2 M1 by default
-  have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
-  show ?thesis unfolding *
-    by (rule integrable_distr[OF measurable_pair_swap'])
-       (simp add: distr_pair_swap[symmetric] assms)
-qed
-
-lemma (in pair_sigma_finite) integrable_product_swap_iff:
-  "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable (M1 \<Otimes>\<^sub>M M2) f"
-proof -
-  interpret Q: pair_sigma_finite M2 M1 by default
-  from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
-  show ?thesis by auto
-qed
-
-lemma (in pair_sigma_finite) integral_product_swap:
-  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
-  shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
-proof -
-  have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
-  show ?thesis unfolding *
-    by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric])
-qed
-
-lemma (in pair_sigma_finite) integrable_fst_measurable:
-  assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) f"
-  shows "AE x in M1. integrable M2 (\<lambda> y. f (x, y))" (is "?AE")
-    and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f" (is "?INT")
-proof -
-  have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
-    using f by auto
-  let ?pf = "\<lambda>x. ereal (f x)" and ?nf = "\<lambda>x. ereal (- f x)"
-  have
-    borel: "?nf \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)""?pf \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" and
-    int: "integral\<^sup>P (M1 \<Otimes>\<^sub>M M2) ?nf \<noteq> \<infinity>" "integral\<^sup>P (M1 \<Otimes>\<^sub>M M2) ?pf \<noteq> \<infinity>"
-    using assms by auto
-  have "(\<integral>\<^sup>+x. (\<integral>\<^sup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
-     "(\<integral>\<^sup>+x. (\<integral>\<^sup>+y. ereal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
-    using borel[THEN M2.positive_integral_fst_measurable(1)] int
-    unfolding borel[THEN M2.positive_integral_fst_measurable(2)] by simp_all
-  with borel[THEN M2.positive_integral_fst_measurable(1)]
-  have AE_pos: "AE x in M1. (\<integral>\<^sup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
-    "AE x in M1. (\<integral>\<^sup>+y. ereal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
-    by (auto intro!: positive_integral_PInf_AE )
-  then have AE: "AE x in M1. \<bar>\<integral>\<^sup>+y. ereal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
-    "AE x in M1. \<bar>\<integral>\<^sup>+y. ereal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
-    by (auto simp: positive_integral_positive)
-  from AE_pos show ?AE using assms
-    by (simp add: measurable_Pair2[OF f_borel] integrable_def)
-  { fix f have "(\<integral>\<^sup>+ x. - \<integral>\<^sup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^sup>+x. 0 \<partial>M1)"
-      using positive_integral_positive
-      by (intro positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder)
-    then have "(\<integral>\<^sup>+ x. - \<integral>\<^sup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
-  note this[simp]
-  { fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
-      and int: "integral\<^sup>P (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. ereal (f x)) \<noteq> \<infinity>"
-      and AE: "AE x in M1. (\<integral>\<^sup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
-    have "integrable M1 (\<lambda>x. real (\<integral>\<^sup>+y. ereal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
-    proof (intro integrable_def[THEN iffD2] conjI)
-      show "?f \<in> borel_measurable M1"
-        using borel by (auto intro!: M2.positive_integral_fst_measurable)
-      have "(\<integral>\<^sup>+x. ereal (?f x) \<partial>M1) = (\<integral>\<^sup>+x. (\<integral>\<^sup>+y. ereal (f (x, y))  \<partial>M2) \<partial>M1)"
-        using AE positive_integral_positive[of M2]
-        by (auto intro!: positive_integral_cong_AE simp: ereal_real)
-      then show "(\<integral>\<^sup>+x. ereal (?f x) \<partial>M1) \<noteq> \<infinity>"
-        using M2.positive_integral_fst_measurable[OF borel] int by simp
-      have "(\<integral>\<^sup>+x. ereal (- ?f x) \<partial>M1) = (\<integral>\<^sup>+x. 0 \<partial>M1)"
-        by (intro positive_integral_cong_pos)
-           (simp add: positive_integral_positive real_of_ereal_pos)
-      then show "(\<integral>\<^sup>+x. ereal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
-    qed }
-  with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
-  show ?INT
-    unfolding lebesgue_integral_def[of "M1 \<Otimes>\<^sub>M M2"] lebesgue_integral_def[of M2]
-      borel[THEN M2.positive_integral_fst_measurable(2), symmetric]
-    using AE[THEN integral_real]
-    by simp
-qed
-
-lemma (in pair_sigma_finite) integrable_snd_measurable:
-  assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) f"
-  shows "AE y in M2. integrable M1 (\<lambda>x. f (x, y))" (is "?AE")
-    and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f" (is "?INT")
-proof -
-  interpret Q: pair_sigma_finite M2 M1 by default
-  have Q_int: "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x, y). f (y, x))"
-    using f unfolding integrable_product_swap_iff .
-  show ?INT
-    using Q.integrable_fst_measurable(2)[OF Q_int]
-    using integral_product_swap[of f] f by auto
-  show ?AE
-    using Q.integrable_fst_measurable(1)[OF Q_int]
-    by simp
-qed
-
-lemma (in pair_sigma_finite) Fubini_integral:
-  assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) f"
-  shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"
-  unfolding integrable_snd_measurable[OF assms]
-  unfolding integrable_fst_measurable[OF assms] ..
+  unfolding positive_integral_snd[OF assms] M2.positive_integral_fst[OF assms] ..
 
 section {* Products on counting spaces, densities and distributions *}
 
@@ -741,7 +630,7 @@
        (auto simp add: positive_integral_cmult[symmetric] ac_simps)
   with A f g show "emeasure ?L A = emeasure ?R A"
     by (simp add: D2.emeasure_pair_measure emeasure_density positive_integral_density
-                  M2.positive_integral_fst_measurable(2)[symmetric]
+                  M2.positive_integral_fst[symmetric]
              cong: positive_integral_cong)
 qed simp
 
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Bochner_Integration.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Bochner_Integration.thy	Mon May 19 12:04:45 2014 +0200
@@ -0,0 +1,2615 @@
+(*  Title:      HOL/Probability/Bochner_Integration.thy
+    Author:     Johannes Hölzl, TU München
+*)
+
+header {* Bochner Integration for Vector-Valued Functions *}
+
+theory Bochner_Integration
+  imports Finite_Product_Measure
+begin
+
+text {*
+
+In the following development of the Bochner integral we use second countable topologies instead
+of separable spaces. A second countable topology is also separable.
+
+*}
+
+lemma borel_measurable_implies_sequence_metric:
+  fixes f :: "'a \<Rightarrow> 'b :: {metric_space, second_countable_topology}"
+  assumes [measurable]: "f \<in> borel_measurable M"
+  shows "\<exists>F. (\<forall>i. simple_function M (F i)) \<and> (\<forall>x\<in>space M. (\<lambda>i. F i x) ----> f x) \<and>
+    (\<forall>i. \<forall>x\<in>space M. dist (F i x) z \<le> 2 * dist (f x) z)"
+proof -
+  obtain D :: "'b set" where "countable D" and D: "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d\<in>D. d \<in> X"
+    by (erule countable_dense_setE)
+
+  def e \<equiv> "from_nat_into D"
+  { fix n x
+    obtain d where "d \<in> D" and d: "d \<in> ball x (1 / Suc n)"
+      using D[of "ball x (1 / Suc n)"] by auto
+    from `d \<in> D` D[of UNIV] `countable D` obtain i where "d = e i"
+      unfolding e_def by (auto dest: from_nat_into_surj)
+    with d have "\<exists>i. dist x (e i) < 1 / Suc n"
+      by auto }
+  note e = this
+
+  def A \<equiv> "\<lambda>m n. {x\<in>space M. dist (f x) (e n) < 1 / (Suc m) \<and> 1 / (Suc m) \<le> dist (f x) z}"
+  def B \<equiv> "\<lambda>m. disjointed (A m)"
+  
+  def m \<equiv> "\<lambda>N x. Max {m::nat. m \<le> N \<and> x \<in> (\<Union>n\<le>N. B m n)}"
+  def F \<equiv> "\<lambda>N::nat. \<lambda>x. if (\<exists>m\<le>N. x \<in> (\<Union>n\<le>N. B m n)) \<and> (\<exists>n\<le>N. x \<in> B (m N x) n) 
+    then e (LEAST n. x \<in> B (m N x) n) else z"
+
+  have B_imp_A[intro, simp]: "\<And>x m n. x \<in> B m n \<Longrightarrow> x \<in> A m n"
+    using disjointed_subset[of "A m" for m] unfolding B_def by auto
+
+  { fix m
+    have "\<And>n. A m n \<in> sets M"
+      by (auto simp: A_def)
+    then have "\<And>n. B m n \<in> sets M"
+      using sets.range_disjointed_sets[of "A m" M] by (auto simp: B_def) }
+  note this[measurable]
+
+  { fix N i x assume "\<exists>m\<le>N. x \<in> (\<Union>n\<le>N. B m n)"
+    then have "m N x \<in> {m::nat. m \<le> N \<and> x \<in> (\<Union>n\<le>N. B m n)}"
+      unfolding m_def by (intro Max_in) auto
+    then have "m N x \<le> N" "\<exists>n\<le>N. x \<in> B (m N x) n"
+      by auto }
+  note m = this
+
+  { fix j N i x assume "j \<le> N" "i \<le> N" "x \<in> B j i"
+    then have "j \<le> m N x"
+      unfolding m_def by (intro Max_ge) auto }
+  note m_upper = this
+
+  show ?thesis
+    unfolding simple_function_def
+  proof (safe intro!: exI[of _ F])
+    have [measurable]: "\<And>i. F i \<in> borel_measurable M"
+      unfolding F_def m_def by measurable
+    show "\<And>x i. F i -` {x} \<inter> space M \<in> sets M"
+      by measurable
+
+    { fix i
+      { fix n x assume "x \<in> B (m i x) n"
+        then have "(LEAST n. x \<in> B (m i x) n) \<le> n"
+          by (intro Least_le)
+        also assume "n \<le> i" 
+        finally have "(LEAST n. x \<in> B (m i x) n) \<le> i" . }
+      then have "F i ` space M \<subseteq> {z} \<union> e ` {.. i}"
+        by (auto simp: F_def)
+      then show "finite (F i ` space M)"
+        by (rule finite_subset) auto }
+    
+    { fix N i n x assume "i \<le> N" "n \<le> N" "x \<in> B i n"
+      then have 1: "\<exists>m\<le>N. x \<in> (\<Union> n\<le>N. B m n)" by auto
+      from m[OF this] obtain n where n: "m N x \<le> N" "n \<le> N" "x \<in> B (m N x) n" by auto
+      moreover
+      def L \<equiv> "LEAST n. x \<in> B (m N x) n"
+      have "dist (f x) (e L) < 1 / Suc (m N x)"
+      proof -
+        have "x \<in> B (m N x) L"
+          using n(3) unfolding L_def by (rule LeastI)
+        then have "x \<in> A (m N x) L"
+          by auto
+        then show ?thesis
+          unfolding A_def by simp
+      qed
+      ultimately have "dist (f x) (F N x) < 1 / Suc (m N x)"
+        by (auto simp add: F_def L_def) }
+    note * = this
+
+    fix x assume "x \<in> space M"
+    show "(\<lambda>i. F i x) ----> f x"
+    proof cases
+      assume "f x = z"
+      then have "\<And>i n. x \<notin> A i n"
+        unfolding A_def by auto
+      then have "\<And>i. F i x = z"
+        by (auto simp: F_def)
+      then show ?thesis
+        using `f x = z` by auto
+    next
+      assume "f x \<noteq> z"
+
+      show ?thesis
+      proof (rule tendstoI)
+        fix e :: real assume "0 < e"
+        with `f x \<noteq> z` obtain n where "1 / Suc n < e" "1 / Suc n < dist (f x) z"
+          by (metis dist_nz order_less_trans neq_iff nat_approx_posE)
+        with `x\<in>space M` `f x \<noteq> z` have "x \<in> (\<Union>i. B n i)"
+          unfolding A_def B_def UN_disjointed_eq using e by auto
+        then obtain i where i: "x \<in> B n i" by auto
+
+        show "eventually (\<lambda>i. dist (F i x) (f x) < e) sequentially"
+          using eventually_ge_at_top[of "max n i"]
+        proof eventually_elim
+          fix j assume j: "max n i \<le> j"
+          with i have "dist (f x) (F j x) < 1 / Suc (m j x)"
+            by (intro *[OF _ _ i]) auto
+          also have "\<dots> \<le> 1 / Suc n"
+            using j m_upper[OF _ _ i]
+            by (auto simp: field_simps)
+          also note `1 / Suc n < e`
+          finally show "dist (F j x) (f x) < e"
+            by (simp add: less_imp_le dist_commute)
+        qed
+      qed
+    qed
+    fix i 
+    { fix n m assume "x \<in> A n m"
+      then have "dist (e m) (f x) + dist (f x) z \<le> 2 * dist (f x) z"
+        unfolding A_def by (auto simp: dist_commute)
+      also have "dist (e m) z \<le> dist (e m) (f x) + dist (f x) z"
+        by (rule dist_triangle)
+      finally (xtrans) have "dist (e m) z \<le> 2 * dist (f x) z" . }
+    then show "dist (F i x) z \<le> 2 * dist (f x) z"
+      unfolding F_def
+      apply auto
+      apply (rule LeastI2)
+      apply auto
+      done
+  qed
+qed
+
+lemma real_indicator: "real (indicator A x :: ereal) = indicator A x"
+  unfolding indicator_def by auto
+
+lemma split_indicator_asm:
+  "P (indicator S x) \<longleftrightarrow> \<not> ((x \<in> S \<and> \<not> P 1) \<or> (x \<notin> S \<and> \<not> P 0))"
+  unfolding indicator_def by auto
+
+lemma
+  fixes f :: "'a \<Rightarrow> 'b::semiring_1" assumes "finite A"
+  shows setsum_mult_indicator[simp]: "(\<Sum>x \<in> A. f x * indicator (B x) (g x)) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
+  and setsum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator (B x) (g x) * f x) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
+  unfolding indicator_def
+  using assms by (auto intro!: setsum_mono_zero_cong_right split: split_if_asm)
+
+lemma borel_measurable_induct_real[consumes 2, case_names set mult add seq]:
+  fixes P :: "('a \<Rightarrow> real) \<Rightarrow> bool"
+  assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x"
+  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
+  assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
+  assumes add: "\<And>u v. u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
+  assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow> (\<And>i x. 0 \<le> U i x) \<Longrightarrow> (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. U i x) ----> u x) \<Longrightarrow> P u"
+  shows "P u"
+proof -
+  have "(\<lambda>x. ereal (u x)) \<in> borel_measurable M" using u by auto
+  from borel_measurable_implies_simple_function_sequence'[OF this]
+  obtain U where U: "\<And>i. simple_function M (U i)" "incseq U" "\<And>i. \<infinity> \<notin> range (U i)" and
+    sup: "\<And>x. (SUP i. U i x) = max 0 (ereal (u x))" and nn: "\<And>i x. 0 \<le> U i x"
+    by blast
+
+  def U' \<equiv> "\<lambda>i x. indicator (space M) x * real (U i x)"
+  then have U'_sf[measurable]: "\<And>i. simple_function M (U' i)"
+    using U by (auto intro!: simple_function_compose1[where g=real])
+
+  show "P u"
+  proof (rule seq)
+    fix i show "U' i \<in> borel_measurable M" "\<And>x. 0 \<le> U' i x"
+      using U nn by (auto
+          intro: borel_measurable_simple_function 
+          intro!: borel_measurable_real_of_ereal real_of_ereal_pos borel_measurable_times
+          simp: U'_def zero_le_mult_iff)
+    show "incseq U'"
+      using U(2,3) nn
+      by (auto simp: incseq_def le_fun_def image_iff eq_commute U'_def indicator_def
+               intro!: real_of_ereal_positive_mono)
+  next
+    fix x assume x: "x \<in> space M"
+    have "(\<lambda>i. U i x) ----> (SUP i. U i x)"
+      using U(2) by (intro LIMSEQ_SUP) (auto simp: incseq_def le_fun_def)
+    moreover have "(\<lambda>i. U i x) = (\<lambda>i. ereal (U' i x))"
+      using x nn U(3) by (auto simp: fun_eq_iff U'_def ereal_real image_iff eq_commute)
+    moreover have "(SUP i. U i x) = ereal (u x)"
+      using sup u(2) by (simp add: max_def)
+    ultimately show "(\<lambda>i. U' i x) ----> u x" 
+      by simp
+  next
+    fix i
+    have "U' i ` space M \<subseteq> real ` (U i ` space M)" "finite (U i ` space M)"
+      unfolding U'_def using U(1) by (auto dest: simple_functionD)
+    then have fin: "finite (U' i ` space M)"
+      by (metis finite_subset finite_imageI)
+    moreover have "\<And>z. {y. U' i z = y \<and> y \<in> U' i ` space M \<and> z \<in> space M} = (if z \<in> space M then {U' i z} else {})"
+      by auto
+    ultimately have U': "(\<lambda>z. \<Sum>y\<in>U' i`space M. y * indicator {x\<in>space M. U' i x = y} z) = U' i"
+      by (simp add: U'_def fun_eq_iff)
+    have "\<And>x. x \<in> U' i ` space M \<Longrightarrow> 0 \<le> x"
+      using nn by (auto simp: U'_def real_of_ereal_pos)
+    with fin have "P (\<lambda>z. \<Sum>y\<in>U' i`space M. y * indicator {x\<in>space M. U' i x = y} z)"
+    proof induct
+      case empty from set[of "{}"] show ?case
+        by (simp add: indicator_def[abs_def])
+    next
+      case (insert x F)
+      then show ?case
+        by (auto intro!: add mult set setsum_nonneg split: split_indicator split_indicator_asm
+                 simp del: setsum_mult_indicator simp: setsum_nonneg_eq_0_iff )
+    qed
+    with U' show "P (U' i)" by simp
+  qed
+qed
+
+lemma scaleR_cong_right:
+  fixes x :: "'a :: real_vector"
+  shows "(x \<noteq> 0 \<Longrightarrow> r = p) \<Longrightarrow> r *\<^sub>R x = p *\<^sub>R x"
+  by (cases "x = 0") auto
+
+inductive simple_bochner_integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b::real_vector) \<Rightarrow> bool" for M f where
+  "simple_function M f \<Longrightarrow> emeasure M {y\<in>space M. f y \<noteq> 0} \<noteq> \<infinity> \<Longrightarrow>
+    simple_bochner_integrable M f"
+
+lemma simple_bochner_integrable_compose2:
+  assumes p_0: "p 0 0 = 0"
+  shows "simple_bochner_integrable M f \<Longrightarrow> simple_bochner_integrable M g \<Longrightarrow>
+    simple_bochner_integrable M (\<lambda>x. p (f x) (g x))"
+proof (safe intro!: simple_bochner_integrable.intros elim!: simple_bochner_integrable.cases del: notI)
+  assume sf: "simple_function M f" "simple_function M g"
+  then show "simple_function M (\<lambda>x. p (f x) (g x))"
+    by (rule simple_function_compose2)
+
+  from sf have [measurable]:
+      "f \<in> measurable M (count_space UNIV)"
+      "g \<in> measurable M (count_space UNIV)"
+    by (auto intro: measurable_simple_function)
+
+  assume fin: "emeasure M {y \<in> space M. f y \<noteq> 0} \<noteq> \<infinity>" "emeasure M {y \<in> space M. g y \<noteq> 0} \<noteq> \<infinity>"
+   
+  have "emeasure M {x\<in>space M. p (f x) (g x) \<noteq> 0} \<le>
+      emeasure M ({x\<in>space M. f x \<noteq> 0} \<union> {x\<in>space M. g x \<noteq> 0})"
+    by (intro emeasure_mono) (auto simp: p_0)
+  also have "\<dots> \<le> emeasure M {x\<in>space M. f x \<noteq> 0} + emeasure M {x\<in>space M. g x \<noteq> 0}"
+    by (intro emeasure_subadditive) auto
+  finally show "emeasure M {y \<in> space M. p (f y) (g y) \<noteq> 0} \<noteq> \<infinity>"
+    using fin by auto
+qed
+
+lemma simple_function_finite_support:
+  assumes f: "simple_function M f" and fin: "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>" and nn: "\<And>x. 0 \<le> f x"
+  shows "emeasure M {x\<in>space M. f x \<noteq> 0} \<noteq> \<infinity>"
+proof cases
+  from f have meas[measurable]: "f \<in> borel_measurable M"
+    by (rule borel_measurable_simple_function)
+
+  assume non_empty: "\<exists>x\<in>space M. f x \<noteq> 0"
+
+  def m \<equiv> "Min (f`space M - {0})"
+  have "m \<in> f`space M - {0}"
+    unfolding m_def using f non_empty by (intro Min_in) (auto simp: simple_function_def)
+  then have m: "0 < m"
+    using nn by (auto simp: less_le)
+
+  from m have "m * emeasure M {x\<in>space M. 0 \<noteq> f x} = 
+    (\<integral>\<^sup>+x. m * indicator {x\<in>space M. 0 \<noteq> f x} x \<partial>M)"
+    using f by (intro positive_integral_cmult_indicator[symmetric]) auto
+  also have "\<dots> \<le> (\<integral>\<^sup>+x. f x \<partial>M)"
+    using AE_space
+  proof (intro positive_integral_mono_AE, eventually_elim)
+    fix x assume "x \<in> space M"
+    with nn show "m * indicator {x \<in> space M. 0 \<noteq> f x} x \<le> f x"
+      using f by (auto split: split_indicator simp: simple_function_def m_def)
+  qed
+  also note `\<dots> < \<infinity>`
+  finally show ?thesis
+    using m by auto 
+next
+  assume "\<not> (\<exists>x\<in>space M. f x \<noteq> 0)"
+  with nn have *: "{x\<in>space M. f x \<noteq> 0} = {}"
+    by auto
+  show ?thesis unfolding * by simp
+qed
+
+lemma simple_bochner_integrableI_bounded:
+  assumes f: "simple_function M f" and fin: "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
+  shows "simple_bochner_integrable M f"
+proof
+  have "emeasure M {y \<in> space M. ereal (norm (f y)) \<noteq> 0} \<noteq> \<infinity>"
+  proof (rule simple_function_finite_support)
+    show "simple_function M (\<lambda>x. ereal (norm (f x)))"
+      using f by (rule simple_function_compose1)
+    show "(\<integral>\<^sup>+ y. ereal (norm (f y)) \<partial>M) < \<infinity>" by fact
+  qed simp
+  then show "emeasure M {y \<in> space M. f y \<noteq> 0} \<noteq> \<infinity>" by simp
+qed fact
+
+definition simple_bochner_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b::real_vector) \<Rightarrow> 'b" where
+  "simple_bochner_integral M f = (\<Sum>y\<in>f`space M. measure M {x\<in>space M. f x = y} *\<^sub>R y)"
+
+lemma simple_bochner_integral_partition:
+  assumes f: "simple_bochner_integrable M f" and g: "simple_function M g"
+  assumes sub: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow> g x = g y \<Longrightarrow> f x = f y"
+  assumes v: "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)"
+  shows "simple_bochner_integral M f = (\<Sum>y\<in>g ` space M. measure M {x\<in>space M. g x = y} *\<^sub>R v y)"
+    (is "_ = ?r")
+proof -
+  from f g have [simp]: "finite (f`space M)" "finite (g`space M)"
+    by (auto simp: simple_function_def elim: simple_bochner_integrable.cases)
+
+  from f have [measurable]: "f \<in> measurable M (count_space UNIV)"
+    by (auto intro: measurable_simple_function elim: simple_bochner_integrable.cases)
+
+  from g have [measurable]: "g \<in> measurable M (count_space UNIV)"
+    by (auto intro: measurable_simple_function elim: simple_bochner_integrable.cases)
+
+  { fix y assume "y \<in> space M"
+    then have "f ` space M \<inter> {i. \<exists>x\<in>space M. i = f x \<and> g y = g x} = {v (g y)}"
+      by (auto cong: sub simp: v[symmetric]) }
+  note eq = this
+
+  have "simple_bochner_integral M f =
+    (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M. 
+      if \<exists>x\<in>space M. y = f x \<and> z = g x then measure M {x\<in>space M. g x = z} else 0) *\<^sub>R y)"
+    unfolding simple_bochner_integral_def
+  proof (safe intro!: setsum_cong scaleR_cong_right)
+    fix y assume y: "y \<in> space M" "f y \<noteq> 0"
+    have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} = 
+        {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
+      by auto
+    have eq:"{x \<in> space M. f x = f y} =
+        (\<Union>i\<in>{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}. {x \<in> space M. g x = i})"
+      by (auto simp: eq_commute cong: sub rev_conj_cong)
+    have "finite (g`space M)" by simp
+    then have "finite {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
+      by (rule rev_finite_subset) auto
+    moreover
+    { fix x assume "x \<in> space M" "f x = f y"
+      then have "x \<in> space M" "f x \<noteq> 0"
+        using y by auto
+      then have "emeasure M {y \<in> space M. g y = g x} \<le> emeasure M {y \<in> space M. f y \<noteq> 0}"
+        by (auto intro!: emeasure_mono cong: sub)
+      then have "emeasure M {xa \<in> space M. g xa = g x} < \<infinity>"
+        using f by (auto simp: simple_bochner_integrable.simps) }
+    ultimately
+    show "measure M {x \<in> space M. f x = f y} =
+      (\<Sum>z\<in>g ` space M. if \<exists>x\<in>space M. f y = f x \<and> z = g x then measure M {x \<in> space M. g x = z} else 0)"
+      apply (simp add: setsum_cases eq)
+      apply (subst measure_finite_Union[symmetric])
+      apply (auto simp: disjoint_family_on_def)
+      done
+  qed
+  also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M. 
+      if \<exists>x\<in>space M. y = f x \<and> z = g x then measure M {x\<in>space M. g x = z} *\<^sub>R y else 0))"
+    by (auto intro!: setsum_cong simp: scaleR_setsum_left)
+  also have "\<dots> = ?r"
+    by (subst setsum_commute)
+       (auto intro!: setsum_cong simp: setsum_cases scaleR_setsum_right[symmetric] eq)
+  finally show "simple_bochner_integral M f = ?r" .
+qed
+
+lemma simple_bochner_integral_add:
+  assumes f: "simple_bochner_integrable M f" and g: "simple_bochner_integrable M g"
+  shows "simple_bochner_integral M (\<lambda>x. f x + g x) =
+    simple_bochner_integral M f + simple_bochner_integral M g"
+proof -
+  from f g have "simple_bochner_integral M (\<lambda>x. f x + g x) =
+    (\<Sum>y\<in>(\<lambda>x. (f x, g x)) ` space M. measure M {x \<in> space M. (f x, g x) = y} *\<^sub>R (fst y + snd y))"
+    by (intro simple_bochner_integral_partition)
+       (auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
+  moreover from f g have "simple_bochner_integral M f =
+    (\<Sum>y\<in>(\<lambda>x. (f x, g x)) ` space M. measure M {x \<in> space M. (f x, g x) = y} *\<^sub>R fst y)"
+    by (intro simple_bochner_integral_partition)
+       (auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
+  moreover from f g have "simple_bochner_integral M g =
+    (\<Sum>y\<in>(\<lambda>x. (f x, g x)) ` space M. measure M {x \<in> space M. (f x, g x) = y} *\<^sub>R snd y)"
+    by (intro simple_bochner_integral_partition)
+       (auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
+  ultimately show ?thesis
+    by (simp add: setsum_addf[symmetric] scaleR_add_right)
+qed
+
+lemma (in linear) simple_bochner_integral_linear:
+  assumes g: "simple_bochner_integrable M g"
+  shows "simple_bochner_integral M (\<lambda>x. f (g x)) = f (simple_bochner_integral M g)"
+proof -
+  from g have "simple_bochner_integral M (\<lambda>x. f (g x)) =
+    (\<Sum>y\<in>g ` space M. measure M {x \<in> space M. g x = y} *\<^sub>R f y)"
+    by (intro simple_bochner_integral_partition)
+       (auto simp: simple_bochner_integrable_compose2[where p="\<lambda>x y. f x"] zero
+             elim: simple_bochner_integrable.cases)
+  also have "\<dots> = f (simple_bochner_integral M g)"
+    by (simp add: simple_bochner_integral_def setsum scaleR)
+  finally show ?thesis .
+qed
+
+lemma simple_bochner_integral_minus:
+  assumes f: "simple_bochner_integrable M f"
+  shows "simple_bochner_integral M (\<lambda>x. - f x) = - simple_bochner_integral M f"
+proof -
+  interpret linear uminus by unfold_locales auto
+  from f show ?thesis
+    by (rule simple_bochner_integral_linear)
+qed
+
+lemma simple_bochner_integral_diff:
+  assumes f: "simple_bochner_integrable M f" and g: "simple_bochner_integrable M g"
+  shows "simple_bochner_integral M (\<lambda>x. f x - g x) =
+    simple_bochner_integral M f - simple_bochner_integral M g"
+  unfolding diff_conv_add_uminus using f g
+  by (subst simple_bochner_integral_add)
+     (auto simp: simple_bochner_integral_minus simple_bochner_integrable_compose2[where p="\<lambda>x y. - y"])
+
+lemma simple_bochner_integral_norm_bound:
+  assumes f: "simple_bochner_integrable M f"
+  shows "norm (simple_bochner_integral M f) \<le> simple_bochner_integral M (\<lambda>x. norm (f x))"
+proof -
+  have "norm (simple_bochner_integral M f) \<le> 
+    (\<Sum>y\<in>f ` space M. norm (measure M {x \<in> space M. f x = y} *\<^sub>R y))"
+    unfolding simple_bochner_integral_def by (rule norm_setsum)
+  also have "\<dots> = (\<Sum>y\<in>f ` space M. measure M {x \<in> space M. f x = y} *\<^sub>R norm y)"
+    by (simp add: measure_nonneg)
+  also have "\<dots> = simple_bochner_integral M (\<lambda>x. norm (f x))"
+    using f
+    by (intro simple_bochner_integral_partition[symmetric])
+       (auto intro: f simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
+  finally show ?thesis .
+qed
+
+lemma simple_bochner_integral_eq_positive_integral:
+  assumes f: "simple_bochner_integrable M f" "\<And>x. 0 \<le> f x"
+  shows "simple_bochner_integral M f = (\<integral>\<^sup>+x. f x \<partial>M)"
+proof -
+  { fix x y z have "(x \<noteq> 0 \<Longrightarrow> y = z) \<Longrightarrow> ereal x * y = ereal x * z"
+      by (cases "x = 0") (auto simp: zero_ereal_def[symmetric]) }
+  note ereal_cong_mult = this
+
+  have [measurable]: "f \<in> borel_measurable M"
+    using f(1) by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
+
+  { fix y assume y: "y \<in> space M" "f y \<noteq> 0"
+    have "ereal (measure M {x \<in> space M. f x = f y}) = emeasure M {x \<in> space M. f x = f y}"
+    proof (rule emeasure_eq_ereal_measure[symmetric])
+      have "emeasure M {x \<in> space M. f x = f y} \<le> emeasure M {x \<in> space M. f x \<noteq> 0}"
+        using y by (intro emeasure_mono) auto
+      with f show "emeasure M {x \<in> space M. f x = f y} \<noteq> \<infinity>"
+        by (auto simp: simple_bochner_integrable.simps)
+    qed
+    moreover have "{x \<in> space M. f x = f y} = (\<lambda>x. ereal (f x)) -` {ereal (f y)} \<inter> space M"
+      by auto
+    ultimately have "ereal (measure M {x \<in> space M. f x = f y}) =
+          emeasure M ((\<lambda>x. ereal (f x)) -` {ereal (f y)} \<inter> space M)" by simp }
+  with f have "simple_bochner_integral M f = (\<integral>\<^sup>Sx. f x \<partial>M)"
+    unfolding simple_integral_def
+    by (subst simple_bochner_integral_partition[OF f(1), where g="\<lambda>x. ereal (f x)" and v=real])
+       (auto intro: f simple_function_compose1 elim: simple_bochner_integrable.cases
+             intro!: setsum_cong ereal_cong_mult
+             simp: setsum_ereal[symmetric] times_ereal.simps(1)[symmetric] mult_ac
+             simp del: setsum_ereal times_ereal.simps(1))
+  also have "\<dots> = (\<integral>\<^sup>+x. f x \<partial>M)"
+    using f
+    by (intro positive_integral_eq_simple_integral[symmetric])
+       (auto simp: simple_function_compose1 simple_bochner_integrable.simps)
+  finally show ?thesis .
+qed
+
+lemma simple_bochner_integral_bounded:
+  fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector, second_countable_topology}"
+  assumes f[measurable]: "f \<in> borel_measurable M"
+  assumes s: "simple_bochner_integrable M s" and t: "simple_bochner_integrable M t"
+  shows "ereal (norm (simple_bochner_integral M s - simple_bochner_integral M t)) \<le>
+    (\<integral>\<^sup>+ x. norm (f x - s x) \<partial>M) + (\<integral>\<^sup>+ x. norm (f x - t x) \<partial>M)"
+    (is "ereal (norm (?s - ?t)) \<le> ?S + ?T")
+proof -
+  have [measurable]: "s \<in> borel_measurable M" "t \<in> borel_measurable M"
+    using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
+
+  have "ereal (norm (?s - ?t)) = norm (simple_bochner_integral M (\<lambda>x. s x - t x))"
+    using s t by (subst simple_bochner_integral_diff) auto
+  also have "\<dots> \<le> simple_bochner_integral M (\<lambda>x. norm (s x - t x))"
+    using simple_bochner_integrable_compose2[of "op -" M "s" "t"] s t
+    by (auto intro!: simple_bochner_integral_norm_bound)
+  also have "\<dots> = (\<integral>\<^sup>+x. norm (s x - t x) \<partial>M)"
+    using simple_bochner_integrable_compose2[of "\<lambda>x y. norm (x - y)" M "s" "t"] s t
+    by (auto intro!: simple_bochner_integral_eq_positive_integral)
+  also have "\<dots> \<le> (\<integral>\<^sup>+x. ereal (norm (f x - s x)) + ereal (norm (f x - t x)) \<partial>M)"
+    by (auto intro!: positive_integral_mono)
+       (metis (erased, hide_lams) add_diff_cancel_left add_diff_eq diff_add_eq order_trans
+              norm_minus_commute norm_triangle_ineq4 order_refl)
+  also have "\<dots> = ?S + ?T"
+   by (rule positive_integral_add) auto
+  finally show ?thesis .
+qed
+
+inductive has_bochner_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b::{real_normed_vector, second_countable_topology} \<Rightarrow> bool"
+  for M f x where
+  "f \<in> borel_measurable M \<Longrightarrow>
+    (\<And>i. simple_bochner_integrable M (s i)) \<Longrightarrow>
+    (\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) ----> 0 \<Longrightarrow>
+    (\<lambda>i. simple_bochner_integral M (s i)) ----> x \<Longrightarrow>
+    has_bochner_integral M f x"
+
+lemma has_bochner_integral_cong:
+  assumes "M = N" "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" "x = y"
+  shows "has_bochner_integral M f x \<longleftrightarrow> has_bochner_integral N g y"
+  unfolding has_bochner_integral.simps assms(1,3)
+  using assms(2) by (simp cong: measurable_cong_strong positive_integral_cong_strong)
+
+lemma has_bochner_integral_cong_AE:
+  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow>
+    has_bochner_integral M f x \<longleftrightarrow> has_bochner_integral M g x"
+  unfolding has_bochner_integral.simps
+  by (intro arg_cong[where f=Ex] ext conj_cong rev_conj_cong refl arg_cong[where f="\<lambda>x. x ----> 0"]
+            positive_integral_cong_AE)
+     auto
+
+lemma borel_measurable_has_bochner_integral[measurable_dest]:
+  "has_bochner_integral M f x \<Longrightarrow> f \<in> borel_measurable M"
+  by (auto elim: has_bochner_integral.cases)
+
+lemma has_bochner_integral_simple_bochner_integrable:
+  "simple_bochner_integrable M f \<Longrightarrow> has_bochner_integral M f (simple_bochner_integral M f)"
+  by (rule has_bochner_integral.intros[where s="\<lambda>_. f"])
+     (auto intro: borel_measurable_simple_function 
+           elim: simple_bochner_integrable.cases
+           simp: zero_ereal_def[symmetric])
+
+lemma has_bochner_integral_real_indicator:
+  assumes [measurable]: "A \<in> sets M" and A: "emeasure M A < \<infinity>"
+  shows "has_bochner_integral M (indicator A) (measure M A)"
+proof -
+  have sbi: "simple_bochner_integrable M (indicator A::'a \<Rightarrow> real)"
+  proof
+    have "{y \<in> space M. (indicator A y::real) \<noteq> 0} = A"
+      using sets.sets_into_space[OF `A\<in>sets M`] by (auto split: split_indicator)
+    then show "emeasure M {y \<in> space M. (indicator A y::real) \<noteq> 0} \<noteq> \<infinity>"
+      using A by auto
+  qed (rule simple_function_indicator assms)+
+  moreover have "simple_bochner_integral M (indicator A) = measure M A"
+    using simple_bochner_integral_eq_positive_integral[OF sbi] A
+    by (simp add: ereal_indicator emeasure_eq_ereal_measure)
+  ultimately show ?thesis
+    by (metis has_bochner_integral_simple_bochner_integrable)
+qed
+
+lemma has_bochner_integral_add:
+  "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M g y \<Longrightarrow>
+    has_bochner_integral M (\<lambda>x. f x + g x) (x + y)"
+proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
+  fix sf sg
+  assume f_sf: "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - sf i x) \<partial>M) ----> 0"
+  assume g_sg: "(\<lambda>i. \<integral>\<^sup>+ x. norm (g x - sg i x) \<partial>M) ----> 0"
+
+  assume sf: "\<forall>i. simple_bochner_integrable M (sf i)"
+    and sg: "\<forall>i. simple_bochner_integrable M (sg i)"
+  then have [measurable]: "\<And>i. sf i \<in> borel_measurable M" "\<And>i. sg i \<in> borel_measurable M"
+    by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
+  assume [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
+
+  show "\<And>i. simple_bochner_integrable M (\<lambda>x. sf i x + sg i x)"
+    using sf sg by (simp add: simple_bochner_integrable_compose2)
+
+  show "(\<lambda>i. \<integral>\<^sup>+ x. (norm (f x + g x - (sf i x + sg i x))) \<partial>M) ----> 0"
+    (is "?f ----> 0")
+  proof (rule tendsto_sandwich)
+    show "eventually (\<lambda>n. 0 \<le> ?f n) sequentially" "(\<lambda>_. 0) ----> 0"
+      by (auto simp: positive_integral_positive)
+    show "eventually (\<lambda>i. ?f i \<le> (\<integral>\<^sup>+ x. (norm (f x - sf i x)) \<partial>M) + \<integral>\<^sup>+ x. (norm (g x - sg i x)) \<partial>M) sequentially"
+      (is "eventually (\<lambda>i. ?f i \<le> ?g i) sequentially")
+    proof (intro always_eventually allI)
+      fix i have "?f i \<le> (\<integral>\<^sup>+ x. (norm (f x - sf i x)) + ereal (norm (g x - sg i x)) \<partial>M)"
+        by (auto intro!: positive_integral_mono norm_diff_triangle_ineq)
+      also have "\<dots> = ?g i"
+        by (intro positive_integral_add) auto
+      finally show "?f i \<le> ?g i" .
+    qed
+    show "?g ----> 0"
+      using tendsto_add_ereal[OF _ _ f_sf g_sg] by simp
+  qed
+qed (auto simp: simple_bochner_integral_add tendsto_add)
+
+lemma has_bochner_integral_bounded_linear:
+  assumes "bounded_linear T"
+  shows "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M (\<lambda>x. T (f x)) (T x)"
+proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
+  interpret T: bounded_linear T by fact
+  have [measurable]: "T \<in> borel_measurable borel"
+    by (intro borel_measurable_continuous_on1 T.continuous_on continuous_on_id)
+  assume [measurable]: "f \<in> borel_measurable M"
+  then show "(\<lambda>x. T (f x)) \<in> borel_measurable M"
+    by auto
+
+  fix s assume f_s: "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M) ----> 0"
+  assume s: "\<forall>i. simple_bochner_integrable M (s i)"
+  then show "\<And>i. simple_bochner_integrable M (\<lambda>x. T (s i x))"
+    by (auto intro: simple_bochner_integrable_compose2 T.zero)
+
+  have [measurable]: "\<And>i. s i \<in> borel_measurable M"
+    using s by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
+
+  obtain K where K: "K > 0" "\<And>x i. norm (T (f x) - T (s i x)) \<le> norm (f x - s i x) * K"
+    using T.pos_bounded by (auto simp: T.diff[symmetric])
+
+  show "(\<lambda>i. \<integral>\<^sup>+ x. norm (T (f x) - T (s i x)) \<partial>M) ----> 0"
+    (is "?f ----> 0")
+  proof (rule tendsto_sandwich)
+    show "eventually (\<lambda>n. 0 \<le> ?f n) sequentially" "(\<lambda>_. 0) ----> 0"
+      by (auto simp: positive_integral_positive)
+
+    show "eventually (\<lambda>i. ?f i \<le> K * (\<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M)) sequentially"
+      (is "eventually (\<lambda>i. ?f i \<le> ?g i) sequentially")
+    proof (intro always_eventually allI)
+      fix i have "?f i \<le> (\<integral>\<^sup>+ x. ereal K * norm (f x - s i x) \<partial>M)"
+        using K by (intro positive_integral_mono) (auto simp: mult_ac)
+      also have "\<dots> = ?g i"
+        using K by (intro positive_integral_cmult) auto
+      finally show "?f i \<le> ?g i" .
+    qed
+    show "?g ----> 0"
+      using ereal_lim_mult[OF f_s, of "ereal K"] by simp
+  qed
+
+  assume "(\<lambda>i. simple_bochner_integral M (s i)) ----> x"
+  with s show "(\<lambda>i. simple_bochner_integral M (\<lambda>x. T (s i x))) ----> T x"
+    by (auto intro!: T.tendsto simp: T.simple_bochner_integral_linear)
+qed
+
+lemma has_bochner_integral_zero[intro]: "has_bochner_integral M (\<lambda>x. 0) 0"
+  by (auto intro!: has_bochner_integral.intros[where s="\<lambda>_ _. 0"]
+           simp: zero_ereal_def[symmetric] simple_bochner_integrable.simps
+                 simple_bochner_integral_def image_constant_conv)
+
+lemma has_bochner_integral_scaleR_left[intro]:
+  "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x *\<^sub>R c) (x *\<^sub>R c)"
+  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_scaleR_left])
+
+lemma has_bochner_integral_scaleR_right[intro]:
+  "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c *\<^sub>R f x) (c *\<^sub>R x)"
+  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_scaleR_right])
+
+lemma has_bochner_integral_mult_left[intro]:
+  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
+  shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x * c) (x * c)"
+  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_mult_left])
+
+lemma has_bochner_integral_mult_right[intro]:
+  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
+  shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c * f x) (c * x)"
+  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_mult_right])
+
+lemmas has_bochner_integral_divide = 
+  has_bochner_integral_bounded_linear[OF bounded_linear_divide]
+
+lemma has_bochner_integral_divide_zero[intro]:
+  fixes c :: "_::{real_normed_field, field_inverse_zero, second_countable_topology}"
+  shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x / c) (x / c)"
+  using has_bochner_integral_divide by (cases "c = 0") auto
+
+lemma has_bochner_integral_inner_left[intro]:
+  "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x \<bullet> c) (x \<bullet> c)"
+  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_inner_left])
+
+lemma has_bochner_integral_inner_right[intro]:
+  "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c \<bullet> f x) (c \<bullet> x)"
+  by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_inner_right])
+
+lemmas has_bochner_integral_minus =
+  has_bochner_integral_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
+lemmas has_bochner_integral_Re =
+  has_bochner_integral_bounded_linear[OF bounded_linear_Re]
+lemmas has_bochner_integral_Im =
+  has_bochner_integral_bounded_linear[OF bounded_linear_Im]
+lemmas has_bochner_integral_cnj =
+  has_bochner_integral_bounded_linear[OF bounded_linear_cnj]
+lemmas has_bochner_integral_of_real =
+  has_bochner_integral_bounded_linear[OF bounded_linear_of_real]
+lemmas has_bochner_integral_fst =
+  has_bochner_integral_bounded_linear[OF bounded_linear_fst]
+lemmas has_bochner_integral_snd =
+  has_bochner_integral_bounded_linear[OF bounded_linear_snd]
+
+lemma has_bochner_integral_indicator:
+  "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
+    has_bochner_integral M (\<lambda>x. indicator A x *\<^sub>R c) (measure M A *\<^sub>R c)"
+  by (intro has_bochner_integral_scaleR_left has_bochner_integral_real_indicator)
+
+lemma has_bochner_integral_diff:
+  "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M g y \<Longrightarrow>
+    has_bochner_integral M (\<lambda>x. f x - g x) (x - y)"
+  unfolding diff_conv_add_uminus
+  by (intro has_bochner_integral_add has_bochner_integral_minus)
+
+lemma has_bochner_integral_setsum:
+  "(\<And>i. i \<in> I \<Longrightarrow> has_bochner_integral M (f i) (x i)) \<Longrightarrow>
+    has_bochner_integral M (\<lambda>x. \<Sum>i\<in>I. f i x) (\<Sum>i\<in>I. x i)"
+  by (induct I rule: infinite_finite_induct)
+     (auto intro: has_bochner_integral_zero has_bochner_integral_add)
+
+lemma has_bochner_integral_implies_finite_norm:
+  "has_bochner_integral M f x \<Longrightarrow> (\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
+proof (elim has_bochner_integral.cases)
+  fix s v
+  assume [measurable]: "f \<in> borel_measurable M" and s: "\<And>i. simple_bochner_integrable M (s i)" and
+    lim_0: "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
+  from order_tendstoD[OF lim_0, of "\<infinity>"]
+  obtain i where f_s_fin: "(\<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) < \<infinity>"
+    by (metis (mono_tags, lifting) eventually_False_sequentially eventually_elim1
+              less_ereal.simps(4) zero_ereal_def)
+
+  have [measurable]: "\<And>i. s i \<in> borel_measurable M"
+    using s by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
+
+  def m \<equiv> "if space M = {} then 0 else Max ((\<lambda>x. norm (s i x))`space M)"
+  have "finite (s i ` space M)"
+    using s by (auto simp: simple_function_def simple_bochner_integrable.simps)
+  then have "finite (norm ` s i ` space M)"
+    by (rule finite_imageI)
+  then have "\<And>x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> m" "0 \<le> m"
+    by (auto simp: m_def image_comp comp_def Max_ge_iff)
+  then have "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) \<le> (\<integral>\<^sup>+x. ereal m * indicator {x\<in>space M. s i x \<noteq> 0} x \<partial>M)"
+    by (auto split: split_indicator intro!: Max_ge positive_integral_mono simp:)
+  also have "\<dots> < \<infinity>"
+    using s by (subst positive_integral_cmult_indicator) (auto simp: `0 \<le> m` simple_bochner_integrable.simps)
+  finally have s_fin: "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) < \<infinity>" .
+
+  have "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x - s i x)) + ereal (norm (s i x)) \<partial>M)"
+    by (auto intro!: positive_integral_mono) (metis add_commute norm_triangle_sub)
+  also have "\<dots> = (\<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) + (\<integral>\<^sup>+x. norm (s i x) \<partial>M)"
+    by (rule positive_integral_add) auto
+  also have "\<dots> < \<infinity>"
+    using s_fin f_s_fin by auto
+  finally show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>" .
+qed
+
+lemma has_bochner_integral_norm_bound:
+  assumes i: "has_bochner_integral M f x"
+  shows "norm x \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
+using assms proof
+  fix s assume
+    x: "(\<lambda>i. simple_bochner_integral M (s i)) ----> x" (is "?s ----> x") and
+    s[simp]: "\<And>i. simple_bochner_integrable M (s i)" and
+    lim: "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0" and
+    f[measurable]: "f \<in> borel_measurable M"
+
+  have [measurable]: "\<And>i. s i \<in> borel_measurable M"
+    using s by (auto simp: simple_bochner_integrable.simps intro: borel_measurable_simple_function)
+
+  show "norm x \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
+  proof (rule LIMSEQ_le)
+    show "(\<lambda>i. ereal (norm (?s i))) ----> norm x"
+      using x by (intro tendsto_intros lim_ereal[THEN iffD2])
+    show "\<exists>N. \<forall>n\<ge>N. norm (?s n) \<le> (\<integral>\<^sup>+x. norm (f x - s n x) \<partial>M) + (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
+      (is "\<exists>N. \<forall>n\<ge>N. _ \<le> ?t n")
+    proof (intro exI allI impI)
+      fix n
+      have "ereal (norm (?s n)) \<le> simple_bochner_integral M (\<lambda>x. norm (s n x))"
+        by (auto intro!: simple_bochner_integral_norm_bound)
+      also have "\<dots> = (\<integral>\<^sup>+x. norm (s n x) \<partial>M)"
+        by (intro simple_bochner_integral_eq_positive_integral)
+           (auto intro: s simple_bochner_integrable_compose2)
+      also have "\<dots> \<le> (\<integral>\<^sup>+x. ereal (norm (f x - s n x)) + norm (f x) \<partial>M)"
+        by (auto intro!: positive_integral_mono)
+           (metis add_commute norm_minus_commute norm_triangle_sub)
+      also have "\<dots> = ?t n" 
+        by (rule positive_integral_add) auto
+      finally show "norm (?s n) \<le> ?t n" .
+    qed
+    have "?t ----> 0 + (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
+      using has_bochner_integral_implies_finite_norm[OF i]
+      by (intro tendsto_add_ereal tendsto_const lim) auto
+    then show "?t ----> \<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M"
+      by simp
+  qed
+qed
+
+lemma has_bochner_integral_eq:
+  "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M f y \<Longrightarrow> x = y"
+proof (elim has_bochner_integral.cases)
+  assume f[measurable]: "f \<in> borel_measurable M"
+
+  fix s t
+  assume "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M) ----> 0" (is "?S ----> 0")
+  assume "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - t i x) \<partial>M) ----> 0" (is "?T ----> 0")
+  assume s: "\<And>i. simple_bochner_integrable M (s i)"
+  assume t: "\<And>i. simple_bochner_integrable M (t i)"
+
+  have [measurable]: "\<And>i. s i \<in> borel_measurable M" "\<And>i. t i \<in> borel_measurable M"
+    using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
+
+  let ?s = "\<lambda>i. simple_bochner_integral M (s i)"
+  let ?t = "\<lambda>i. simple_bochner_integral M (t i)"
+  assume "?s ----> x" "?t ----> y"
+  then have "(\<lambda>i. norm (?s i - ?t i)) ----> norm (x - y)"
+    by (intro tendsto_intros)
+  moreover
+  have "(\<lambda>i. ereal (norm (?s i - ?t i))) ----> ereal 0"
+  proof (rule tendsto_sandwich)
+    show "eventually (\<lambda>i. 0 \<le> ereal (norm (?s i - ?t i))) sequentially" "(\<lambda>_. 0) ----> ereal 0"
+      by (auto simp: positive_integral_positive zero_ereal_def[symmetric])
+
+    show "eventually (\<lambda>i. norm (?s i - ?t i) \<le> ?S i + ?T i) sequentially"
+      by (intro always_eventually allI simple_bochner_integral_bounded s t f)
+    show "(\<lambda>i. ?S i + ?T i) ----> ereal 0"
+      using tendsto_add_ereal[OF _ _ `?S ----> 0` `?T ----> 0`]
+      by (simp add: zero_ereal_def[symmetric])
+  qed
+  then have "(\<lambda>i. norm (?s i - ?t i)) ----> 0"
+    by simp
+  ultimately have "norm (x - y) = 0"
+    by (rule LIMSEQ_unique)
+  then show "x = y" by simp
+qed
+
+lemma has_bochner_integralI_AE:
+  assumes f: "has_bochner_integral M f x"
+    and g: "g \<in> borel_measurable M"
+    and ae: "AE x in M. f x = g x"
+  shows "has_bochner_integral M g x"
+  using f
+proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
+  fix s assume "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
+  also have "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) = (\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (g x - s i x)) \<partial>M)"
+    using ae
+    by (intro ext positive_integral_cong_AE, eventually_elim) simp
+  finally show "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (g x - s i x)) \<partial>M) ----> 0" .
+qed (auto intro: g)
+
+lemma has_bochner_integral_eq_AE:
+  assumes f: "has_bochner_integral M f x"
+    and g: "has_bochner_integral M g y"
+    and ae: "AE x in M. f x = g x"
+  shows "x = y"
+proof -
+  from assms have "has_bochner_integral M g x"
+    by (auto intro: has_bochner_integralI_AE)
+  from this g show "x = y"
+    by (rule has_bochner_integral_eq)
+qed
+
+inductive integrable for M f where
+  "has_bochner_integral M f x \<Longrightarrow> integrable M f"
+
+definition lebesgue_integral ("integral\<^sup>L") where
+  "integral\<^sup>L M f = (THE x. has_bochner_integral M f x)"
+
+syntax
+  "_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> 'a measure \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
+
+translations
+  "\<integral> x. f \<partial>M" == "CONST lebesgue_integral M (\<lambda>x. f)"
+
+lemma has_bochner_integral_integral_eq: "has_bochner_integral M f x \<Longrightarrow> integral\<^sup>L M f = x"
+  by (metis the_equality has_bochner_integral_eq lebesgue_integral_def)
+
+lemma has_bochner_integral_integrable:
+  "integrable M f \<Longrightarrow> has_bochner_integral M f (integral\<^sup>L M f)"
+  by (auto simp: has_bochner_integral_integral_eq integrable.simps)
+
+lemma has_bochner_integral_iff:
+  "has_bochner_integral M f x \<longleftrightarrow> integrable M f \<and> integral\<^sup>L M f = x"
+  by (metis has_bochner_integral_integrable has_bochner_integral_integral_eq integrable.intros)
+
+lemma simple_bochner_integrable_eq_integral:
+  "simple_bochner_integrable M f \<Longrightarrow> simple_bochner_integral M f = integral\<^sup>L M f"
+  using has_bochner_integral_simple_bochner_integrable[of M f]
+  by (simp add: has_bochner_integral_integral_eq)
+
+lemma not_integrable_integral_eq: "\<not> integrable M f \<Longrightarrow> integral\<^sup>L M f = (THE x. False)"
+  unfolding integrable.simps lebesgue_integral_def by (auto intro!: arg_cong[where f=The])
+
+lemma integral_eq_cases:
+  "integrable M f \<longleftrightarrow> integrable N g \<Longrightarrow>
+    (integrable M f \<Longrightarrow> integrable N g \<Longrightarrow> integral\<^sup>L M f = integral\<^sup>L N g) \<Longrightarrow>
+    integral\<^sup>L M f = integral\<^sup>L N g"
+  by (metis not_integrable_integral_eq)
+
+lemma borel_measurable_integrable[measurable_dest]: "integrable M f \<Longrightarrow> f \<in> borel_measurable M"
+  by (auto elim: integrable.cases has_bochner_integral.cases)
+
+lemma integrable_cong:
+  "M = N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable N g"
+  using assms by (simp cong: has_bochner_integral_cong add: integrable.simps)
+
+lemma integrable_cong_AE:
+  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. f x = g x \<Longrightarrow>
+    integrable M f \<longleftrightarrow> integrable M g"
+  unfolding integrable.simps
+  by (intro has_bochner_integral_cong_AE arg_cong[where f=Ex] ext)
+
+lemma integral_cong:
+  "M = N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> f x = g x) \<Longrightarrow> integral\<^sup>L M f = integral\<^sup>L N g"
+  using assms by (simp cong: has_bochner_integral_cong add: lebesgue_integral_def)
+
+lemma integral_cong_AE:
+  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. f x = g x \<Longrightarrow>
+    integral\<^sup>L M f = integral\<^sup>L M g"
+  unfolding lebesgue_integral_def
+  by (intro has_bochner_integral_cong_AE arg_cong[where f=The] ext)
+
+lemma integrable_add[simp, intro]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> integrable M (\<lambda>x. f x + g x)"
+  by (auto simp: integrable.simps intro: has_bochner_integral_add)
+
+lemma integrable_zero[simp, intro]: "integrable M (\<lambda>x. 0)"
+  by (metis has_bochner_integral_zero integrable.simps) 
+
+lemma integrable_setsum[simp, intro]: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow> integrable M (\<lambda>x. \<Sum>i\<in>I. f i x)"
+  by (metis has_bochner_integral_setsum integrable.simps) 
+
+lemma integrable_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
+  integrable M (\<lambda>x. indicator A x *\<^sub>R c)"
+  by (metis has_bochner_integral_indicator integrable.simps) 
+
+lemma integrable_real_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
+  integrable M (indicator A :: 'a \<Rightarrow> real)"
+  by (metis has_bochner_integral_real_indicator integrable.simps)
+
+lemma integrable_diff[simp, intro]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> integrable M (\<lambda>x. f x - g x)"
+  by (auto simp: integrable.simps intro: has_bochner_integral_diff)
+  
+lemma integrable_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. T (f x))"
+  by (auto simp: integrable.simps intro: has_bochner_integral_bounded_linear)
+
+lemma integrable_scaleR_left[simp, intro]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x *\<^sub>R c)"
+  unfolding integrable.simps by fastforce
+
+lemma integrable_scaleR_right[simp, intro]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c *\<^sub>R f x)"
+  unfolding integrable.simps by fastforce
+
+lemma integrable_mult_left[simp, intro]:
+  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
+  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x * c)"
+  unfolding integrable.simps by fastforce
+
+lemma integrable_mult_right[simp, intro]:
+  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
+  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c * f x)"
+  unfolding integrable.simps by fastforce
+
+lemma integrable_divide_zero[simp, intro]:
+  fixes c :: "_::{real_normed_field, field_inverse_zero, second_countable_topology}"
+  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x / c)"
+  unfolding integrable.simps by fastforce
+
+lemma integrable_inner_left[simp, intro]:
+  "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x \<bullet> c)"
+  unfolding integrable.simps by fastforce
+
+lemma integrable_inner_right[simp, intro]:
+  "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c \<bullet> f x)"
+  unfolding integrable.simps by fastforce
+
+lemmas integrable_minus[simp, intro] =
+  integrable_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
+lemmas integrable_divide[simp, intro] =
+  integrable_bounded_linear[OF bounded_linear_divide]
+lemmas integrable_Re[simp, intro] =
+  integrable_bounded_linear[OF bounded_linear_Re]
+lemmas integrable_Im[simp, intro] =
+  integrable_bounded_linear[OF bounded_linear_Im]
+lemmas integrable_cnj[simp, intro] =
+  integrable_bounded_linear[OF bounded_linear_cnj]
+lemmas integrable_of_real[simp, intro] =
+  integrable_bounded_linear[OF bounded_linear_of_real]
+lemmas integrable_fst[simp, intro] =
+  integrable_bounded_linear[OF bounded_linear_fst]
+lemmas integrable_snd[simp, intro] =
+  integrable_bounded_linear[OF bounded_linear_snd]
+
+lemma integral_zero[simp]: "integral\<^sup>L M (\<lambda>x. 0) = 0"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_zero)
+
+lemma integral_add[simp]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow>
+    integral\<^sup>L M (\<lambda>x. f x + g x) = integral\<^sup>L M f + integral\<^sup>L M g"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_add has_bochner_integral_integrable)
+
+lemma integral_diff[simp]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow>
+    integral\<^sup>L M (\<lambda>x. f x - g x) = integral\<^sup>L M f - integral\<^sup>L M g"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_diff has_bochner_integral_integrable)
+
+lemma integral_setsum[simp]: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow>
+  integral\<^sup>L M (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_setsum has_bochner_integral_integrable)
+
+lemma integral_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow>
+    integral\<^sup>L M (\<lambda>x. T (f x)) = T (integral\<^sup>L M f)"
+  by (metis has_bochner_integral_bounded_linear has_bochner_integral_integrable has_bochner_integral_integral_eq)
+
+lemma integral_indicator[simp]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
+  integral\<^sup>L M (\<lambda>x. indicator A x *\<^sub>R c) = measure M A *\<^sub>R c"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_indicator has_bochner_integral_integrable)
+
+lemma integral_real_indicator[simp]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
+  integral\<^sup>L M (indicator A :: 'a \<Rightarrow> real) = measure M A"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_real_indicator has_bochner_integral_integrable)
+
+lemma integral_scaleR_left[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x *\<^sub>R c \<partial>M) = integral\<^sup>L M f *\<^sub>R c"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_scaleR_left)
+
+lemma integral_scaleR_right[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c *\<^sub>R f x \<partial>M) = c *\<^sub>R integral\<^sup>L M f"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_scaleR_right)
+
+lemma integral_mult_left[simp]:
+  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
+  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x * c \<partial>M) = integral\<^sup>L M f * c"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_mult_left)
+
+lemma integral_mult_right[simp]:
+  fixes c :: "_::{real_normed_algebra,second_countable_topology}"
+  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c * f x \<partial>M) = c * integral\<^sup>L M f"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_mult_right)
+
+lemma integral_inner_left[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x \<bullet> c \<partial>M) = integral\<^sup>L M f \<bullet> c"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_left)
+
+lemma integral_inner_right[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c \<bullet> f x \<partial>M) = c \<bullet> integral\<^sup>L M f"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_right)
+
+lemma integral_divide_zero[simp]:
+  fixes c :: "_::{real_normed_field, field_inverse_zero, second_countable_topology}"
+  shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integral\<^sup>L M (\<lambda>x. f x / c) = integral\<^sup>L M f / c"
+  by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_divide_zero)
+
+lemmas integral_minus[simp] =
+  integral_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
+lemmas integral_divide[simp] =
+  integral_bounded_linear[OF bounded_linear_divide]
+lemmas integral_Re[simp] =
+  integral_bounded_linear[OF bounded_linear_Re]
+lemmas integral_Im[simp] =
+  integral_bounded_linear[OF bounded_linear_Im]
+lemmas integral_cnj[simp] =
+  integral_bounded_linear[OF bounded_linear_cnj]
+lemmas integral_of_real[simp] =
+  integral_bounded_linear[OF bounded_linear_of_real]
+lemmas integral_fst[simp] =
+  integral_bounded_linear[OF bounded_linear_fst]
+lemmas integral_snd[simp] =
+  integral_bounded_linear[OF bounded_linear_snd]
+
+lemma integral_norm_bound_ereal:
+  "integrable M f \<Longrightarrow> norm (integral\<^sup>L M f) \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
+  by (metis has_bochner_integral_integrable has_bochner_integral_norm_bound)
+
+lemma integrableI_sequence:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  assumes f[measurable]: "f \<in> borel_measurable M"
+  assumes s: "\<And>i. simple_bochner_integrable M (s i)"
+  assumes lim: "(\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) ----> 0" (is "?S ----> 0")
+  shows "integrable M f"
+proof -
+  let ?s = "\<lambda>n. simple_bochner_integral M (s n)"
+
+  have "\<exists>x. ?s ----> x"
+    unfolding convergent_eq_cauchy
+  proof (rule metric_CauchyI)
+    fix e :: real assume "0 < e"
+    then have "0 < ereal (e / 2)" by auto
+    from order_tendstoD(2)[OF lim this]
+    obtain M where M: "\<And>n. M \<le> n \<Longrightarrow> ?S n < e / 2"
+      by (auto simp: eventually_sequentially)
+    show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (?s m) (?s n) < e"
+    proof (intro exI allI impI)
+      fix m n assume m: "M \<le> m" and n: "M \<le> n"
+      have "?S n \<noteq> \<infinity>"
+        using M[OF n] by auto
+      have "norm (?s n - ?s m) \<le> ?S n + ?S m"
+        by (intro simple_bochner_integral_bounded s f)
+      also have "\<dots> < ereal (e / 2) + e / 2"
+        using ereal_add_strict_mono[OF less_imp_le[OF M[OF n]] _ `?S n \<noteq> \<infinity>` M[OF m]]
+        by (auto simp: positive_integral_positive)
+      also have "\<dots> = e" by simp
+      finally show "dist (?s n) (?s m) < e"
+        by (simp add: dist_norm)
+    qed
+  qed
+  then obtain x where "?s ----> x" ..
+  show ?thesis
+    by (rule, rule) fact+
+qed
+
+lemma positive_integral_dominated_convergence_norm:
+  fixes u' :: "_ \<Rightarrow> _::{real_normed_vector, second_countable_topology}"
+  assumes [measurable]:
+       "\<And>i. u i \<in> borel_measurable M" "u' \<in> borel_measurable M" "w \<in> borel_measurable M"
+    and bound: "\<And>j. AE x in M. norm (u j x) \<le> w x"
+    and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
+    and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
+  shows "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) ----> 0"
+proof -
+  have "AE x in M. \<forall>j. norm (u j x) \<le> w x"
+    unfolding AE_all_countable by rule fact
+  with u' have bnd: "AE x in M. \<forall>j. norm (u' x - u j x) \<le> 2 * w x"
+  proof (eventually_elim, intro allI)
+    fix i x assume "(\<lambda>i. u i x) ----> u' x" "\<forall>j. norm (u j x) \<le> w x" "\<forall>j. norm (u j x) \<le> w x"
+    then have "norm (u' x) \<le> w x" "norm (u i x) \<le> w x"
+      by (auto intro: LIMSEQ_le_const2 tendsto_norm)
+    then have "norm (u' x) + norm (u i x) \<le> 2 * w x"
+      by simp
+    also have "norm (u' x - u i x) \<le> norm (u' x) + norm (u i x)"
+      by (rule norm_triangle_ineq4)
+    finally (xtrans) show "norm (u' x - u i x) \<le> 2 * w x" .
+  qed
+  
+  have "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) ----> (\<integral>\<^sup>+x. 0 \<partial>M)"
+  proof (rule positive_integral_dominated_convergence)  
+    show "(\<integral>\<^sup>+x. 2 * w x \<partial>M) < \<infinity>"
+      by (rule positive_integral_mult_bounded_inf[OF _ w, of 2]) auto
+    show "AE x in M. (\<lambda>i. ereal (norm (u' x - u i x))) ----> 0"
+      using u' 
+    proof eventually_elim
+      fix x assume "(\<lambda>i. u i x) ----> u' x"
+      from tendsto_diff[OF tendsto_const[of "u' x"] this]
+      show "(\<lambda>i. ereal (norm (u' x - u i x))) ----> 0"
+        by (simp add: zero_ereal_def tendsto_norm_zero_iff)
+    qed
+  qed (insert bnd, auto)
+  then show ?thesis by simp
+qed
+
+lemma integrableI_bounded:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  assumes f[measurable]: "f \<in> borel_measurable M" and fin: "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
+  shows "integrable M f"
+proof -
+  from borel_measurable_implies_sequence_metric[OF f, of 0] obtain s where
+    s: "\<And>i. simple_function M (s i)" and
+    pointwise: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x" and
+    bound: "\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)"
+    by (simp add: norm_conv_dist) metis
+  
+  show ?thesis
+  proof (rule integrableI_sequence)
+    { fix i
+      have "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) \<le> (\<integral>\<^sup>+x. 2 * ereal (norm (f x)) \<partial>M)"
+        by (intro positive_integral_mono) (simp add: bound)
+      also have "\<dots> = 2 * (\<integral>\<^sup>+x. ereal (norm (f x)) \<partial>M)"
+        by (rule positive_integral_cmult) auto
+      finally have "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) < \<infinity>"
+        using fin by auto }
+    note fin_s = this
+
+    show "\<And>i. simple_bochner_integrable M (s i)"
+      by (rule simple_bochner_integrableI_bounded) fact+
+
+    show "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
+    proof (rule positive_integral_dominated_convergence_norm)
+      show "\<And>j. AE x in M. norm (s j x) \<le> 2 * norm (f x)"
+        using bound by auto
+      show "\<And>i. s i \<in> borel_measurable M" "(\<lambda>x. 2 * norm (f x)) \<in> borel_measurable M"
+        using s by (auto intro: borel_measurable_simple_function)
+      show "(\<integral>\<^sup>+ x. ereal (2 * norm (f x)) \<partial>M) < \<infinity>"
+        using fin unfolding times_ereal.simps(1)[symmetric] by (subst positive_integral_cmult) auto
+      show "AE x in M. (\<lambda>i. s i x) ----> f x"
+        using pointwise by auto
+    qed fact
+  qed fact
+qed
+
+lemma integrableI_nonneg:
+  fixes f :: "'a \<Rightarrow> real"
+  assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>"
+  shows "integrable M f"
+proof -
+  have "(\<integral>\<^sup>+x. norm (f x) \<partial>M) = (\<integral>\<^sup>+x. f x \<partial>M)"
+    using assms by (intro positive_integral_cong_AE) auto
+  then show ?thesis
+    using assms by (intro integrableI_bounded) auto
+qed
+
+lemma integrable_iff_bounded:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  shows "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and> (\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
+  using integrableI_bounded[of f M] has_bochner_integral_implies_finite_norm[of M f]
+  unfolding integrable.simps has_bochner_integral.simps[abs_def] by auto
+
+lemma integrable_bound:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+    and g :: "'a \<Rightarrow> 'c::{banach, second_countable_topology}"
+  shows "integrable M f \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. norm (g x) \<le> norm (f x)) \<Longrightarrow>
+    integrable M g"
+  unfolding integrable_iff_bounded
+proof safe
+  assume "f \<in> borel_measurable M" "g \<in> borel_measurable M"
+  assume "AE x in M. norm (g x) \<le> norm (f x)"
+  then have "(\<integral>\<^sup>+ x. ereal (norm (g x)) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
+    by  (intro positive_integral_mono_AE) auto
+  also assume "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>"
+  finally show "(\<integral>\<^sup>+ x. ereal (norm (g x)) \<partial>M) < \<infinity>" .
+qed 
+
+lemma integrable_abs[simp, intro]:
+  fixes f :: "'a \<Rightarrow> real"
+  assumes [measurable]: "integrable M f" shows "integrable M (\<lambda>x. \<bar>f x\<bar>)"
+  using assms by (rule integrable_bound) auto
+
+lemma integrable_norm[simp, intro]:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  assumes [measurable]: "integrable M f" shows "integrable M (\<lambda>x. norm (f x))"
+  using assms by (rule integrable_bound) auto
+  
+lemma integrable_norm_cancel:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  assumes [measurable]: "integrable M (\<lambda>x. norm (f x))" "f \<in> borel_measurable M" shows "integrable M f"
+  using assms by (rule integrable_bound) auto
+
+lemma integrable_abs_cancel:
+  fixes f :: "'a \<Rightarrow> real"
+  assumes [measurable]: "integrable M (\<lambda>x. \<bar>f x\<bar>)" "f \<in> borel_measurable M" shows "integrable M f"
+  using assms by (rule integrable_bound) auto
+
+lemma integrable_max[simp, intro]:
+  fixes f :: "'a \<Rightarrow> real"
+  assumes fg[measurable]: "integrable M f" "integrable M g"
+  shows "integrable M (\<lambda>x. max (f x) (g x))"
+  using integrable_add[OF integrable_norm[OF fg(1)] integrable_norm[OF fg(2)]]
+  by (rule integrable_bound) auto
+
+lemma integrable_min[simp, intro]:
+  fixes f :: "'a \<Rightarrow> real"
+  assumes fg[measurable]: "integrable M f" "integrable M g"
+  shows "integrable M (\<lambda>x. min (f x) (g x))"
+  using integrable_add[OF integrable_norm[OF fg(1)] integrable_norm[OF fg(2)]]
+  by (rule integrable_bound) auto
+
+lemma integral_minus_iff[simp]:
+  "integrable M (\<lambda>x. - f x ::'a::{banach, second_countable_topology}) \<longleftrightarrow> integrable M f"
+  unfolding integrable_iff_bounded
+  by (auto intro: borel_measurable_uminus[of "\<lambda>x. - f x" M, simplified])
+
+lemma integrable_indicator_iff:
+  "integrable M (indicator A::_ \<Rightarrow> real) \<longleftrightarrow> A \<inter> space M \<in> sets M \<and> emeasure M (A \<inter> space M) < \<infinity>"
+  by (simp add: integrable_iff_bounded borel_measurable_indicator_iff ereal_indicator positive_integral_indicator'
+           cong: conj_cong)
+
+lemma integral_dominated_convergence:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and w :: "'a \<Rightarrow> real"
+  assumes "f \<in> borel_measurable M" "\<And>i. s i \<in> borel_measurable M" "integrable M w"
+  assumes lim: "AE x in M. (\<lambda>i. s i x) ----> f x"
+  assumes bound: "\<And>i. AE x in M. norm (s i x) \<le> w x"
+  shows "integrable M f"
+    and "\<And>i. integrable M (s i)"
+    and "(\<lambda>i. integral\<^sup>L M (s i)) ----> integral\<^sup>L M f"
+proof -
+  have "AE x in M. 0 \<le> w x"
+    using bound[of 0] by eventually_elim (auto intro: norm_ge_zero order_trans)
+  then have "(\<integral>\<^sup>+x. w x \<partial>M) = (\<integral>\<^sup>+x. norm (w x) \<partial>M)"
+    by (intro positive_integral_cong_AE) auto
+  with `integrable M w` have w: "w \<in> borel_measurable M" "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
+    unfolding integrable_iff_bounded by auto
+
+  show int_s: "\<And>i. integrable M (s i)"
+    unfolding integrable_iff_bounded
+  proof
+    fix i 
+    have "(\<integral>\<^sup>+ x. ereal (norm (s i x)) \<partial>M) \<le> (\<integral>\<^sup>+x. w x \<partial>M)"
+      using bound by (intro positive_integral_mono_AE) auto
+    with w show "(\<integral>\<^sup>+ x. ereal (norm (s i x)) \<partial>M) < \<infinity>" by auto
+  qed fact
+
+  have all_bound: "AE x in M. \<forall>i. norm (s i x) \<le> w x"
+    using bound unfolding AE_all_countable by auto
+
+  show int_f: "integrable M f"
+    unfolding integrable_iff_bounded
+  proof
+    have "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) \<le> (\<integral>\<^sup>+x. w x \<partial>M)"
+      using all_bound lim
+    proof (intro positive_integral_mono_AE, eventually_elim)
+      fix x assume "\<forall>i. norm (s i x) \<le> w x" "(\<lambda>i. s i x) ----> f x"
+      then show "ereal (norm (f x)) \<le> ereal (w x)"
+        by (intro LIMSEQ_le_const2[where X="\<lambda>i. ereal (norm (s i x))"] tendsto_intros lim_ereal[THEN iffD2]) auto
+    qed
+    with w show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>" by auto
+  qed fact
+
+  have "(\<lambda>n. ereal (norm (integral\<^sup>L M (s n) - integral\<^sup>L M f))) ----> ereal 0" (is "?d ----> ereal 0")
+  proof (rule tendsto_sandwich)
+    show "eventually (\<lambda>n. ereal 0 \<le> ?d n) sequentially" "(\<lambda>_. ereal 0) ----> ereal 0" by auto
+    show "eventually (\<lambda>n. ?d n \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)) sequentially"
+    proof (intro always_eventually allI)
+      fix n
+      have "?d n = norm (integral\<^sup>L M (\<lambda>x. s n x - f x))"
+        using int_f int_s by simp
+      also have "\<dots> \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)"
+        by (intro int_f int_s integrable_diff integral_norm_bound_ereal)
+      finally show "?d n \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)" .
+    qed
+    show "(\<lambda>n. \<integral>\<^sup>+x. norm (s n x - f x) \<partial>M) ----> ereal 0"
+      unfolding zero_ereal_def[symmetric]
+      apply (subst norm_minus_commute)
+    proof (rule positive_integral_dominated_convergence_norm[where w=w])
+      show "\<And>n. s n \<in> borel_measurable M"
+        using int_s unfolding integrable_iff_bounded by auto
+    qed fact+
+  qed
+  then have "(\<lambda>n. integral\<^sup>L M (s n) - integral\<^sup>L M f) ----> 0"
+    unfolding lim_ereal tendsto_norm_zero_iff .
+  from tendsto_add[OF this tendsto_const[of "integral\<^sup>L M f"]]
+  show "(\<lambda>i. integral\<^sup>L M (s i)) ----> integral\<^sup>L M f"  by simp
+qed
+
+lemma integrable_mult_left_iff:
+  fixes f :: "'a \<Rightarrow> real"
+  shows "integrable M (\<lambda>x. c * f x) \<longleftrightarrow> c = 0 \<or> integrable M f"
+  using integrable_mult_left[of c M f] integrable_mult_left[of "1 / c" M "\<lambda>x. c * f x"]
+  by (cases "c = 0") auto
+
+lemma positive_integral_eq_integral:
+  assumes f: "integrable M f"
+  assumes nonneg: "AE x in M. 0 \<le> f x" 
+  shows "(\<integral>\<^sup>+ x. f x \<partial>M) = integral\<^sup>L M f"
+proof -
+  { fix f :: "'a \<Rightarrow> real" assume f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "integrable M f"
+    then have "(\<integral>\<^sup>+ x. f x \<partial>M) = integral\<^sup>L M f"
+    proof (induct rule: borel_measurable_induct_real)
+      case (set A) then show ?case
+        by (simp add: integrable_indicator_iff ereal_indicator emeasure_eq_ereal_measure)
+    next
+      case (mult f c) then show ?case
+        unfolding times_ereal.simps(1)[symmetric]
+        by (subst positive_integral_cmult)
+           (auto simp add: integrable_mult_left_iff zero_ereal_def[symmetric])
+    next
+      case (add g f)
+      then have "integrable M f" "integrable M g"
+        by (auto intro!: integrable_bound[OF add(8)])
+      with add show ?case
+        unfolding plus_ereal.simps(1)[symmetric]
+        by (subst positive_integral_add) auto
+    next
+      case (seq s)
+      { fix i x assume "x \<in> space M" with seq(4) have "s i x \<le> f x"
+          by (intro LIMSEQ_le_const[OF seq(5)] exI[of _ i]) (auto simp: incseq_def le_fun_def) }
+      note s_le_f = this
+
+      show ?case
+      proof (rule LIMSEQ_unique)
+        show "(\<lambda>i. ereal (integral\<^sup>L M (s i))) ----> ereal (integral\<^sup>L M f)"
+          unfolding lim_ereal
+        proof (rule integral_dominated_convergence[where w=f])
+          show "integrable M f" by fact
+          from s_le_f seq show "\<And>i. AE x in M. norm (s i x) \<le> f x"
+            by auto
+        qed (insert seq, auto)
+        have int_s: "\<And>i. integrable M (s i)"
+          using seq f s_le_f by (intro integrable_bound[OF f(3)]) auto
+        have "(\<lambda>i. \<integral>\<^sup>+ x. s i x \<partial>M) ----> \<integral>\<^sup>+ x. f x \<partial>M"
+          using seq s_le_f f
+          by (intro positive_integral_dominated_convergence[where w=f])
+             (auto simp: integrable_iff_bounded)
+        also have "(\<lambda>i. \<integral>\<^sup>+x. s i x \<partial>M) = (\<lambda>i. \<integral>x. s i x \<partial>M)"
+          using seq int_s by simp
+        finally show "(\<lambda>i. \<integral>x. s i x \<partial>M) ----> \<integral>\<^sup>+x. f x \<partial>M"
+          by simp
+      qed
+    qed }
+  from this[of "\<lambda>x. max 0 (f x)"] assms have "(\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) = integral\<^sup>L M (\<lambda>x. max 0 (f x))"
+    by simp
+  also have "\<dots> = integral\<^sup>L M f"
+    using assms by (auto intro!: integral_cong_AE)
+  also have "(\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) = (\<integral>\<^sup>+ x. f x \<partial>M)"
+    using assms by (auto intro!: positive_integral_cong_AE simp: max_def)
+  finally show ?thesis .
+qed
+
+lemma integral_norm_bound:
+  fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
+  shows "integrable M f \<Longrightarrow> norm (integral\<^sup>L M f) \<le> (\<integral>x. norm (f x) \<partial>M)"
+  using positive_integral_eq_integral[of M "\<lambda>x. norm (f x)"]
+  using integral_norm_bound_ereal[of M f] by simp
+  
+lemma integral_eq_positive_integral:
+  "integrable M f \<Longrightarrow> (\<And>x. 0 \<le> f x) \<Longrightarrow> integral\<^sup>L M f = real (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
+  by (subst positive_integral_eq_integral) auto
+  
+lemma integrableI_simple_bochner_integrable:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  shows "simple_bochner_integrable M f \<Longrightarrow> integrable M f"
+  by (intro integrableI_sequence[where s="\<lambda>_. f"] borel_measurable_simple_function)
+     (auto simp: zero_ereal_def[symmetric] simple_bochner_integrable.simps)
+
+lemma integrable_induct[consumes 1, case_names base add lim, induct pred: integrable]:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  assumes "integrable M f"
+  assumes base: "\<And>A c. A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> P (\<lambda>x. indicator A x *\<^sub>R c)"
+  assumes add: "\<And>f g. integrable M f \<Longrightarrow> P f \<Longrightarrow> integrable M g \<Longrightarrow> P g \<Longrightarrow> P (\<lambda>x. f x + g x)"
+  assumes lim: "\<And>f s. (\<And>i. integrable M (s i)) \<Longrightarrow> (\<And>i. P (s i)) \<Longrightarrow>
+   (\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x) \<Longrightarrow>
+   (\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)) \<Longrightarrow> integrable M f \<Longrightarrow> P f"
+  shows "P f"
+proof -
+  from `integrable M f` have f: "f \<in> borel_measurable M" "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
+    unfolding integrable_iff_bounded by auto
+  from borel_measurable_implies_sequence_metric[OF f(1)]
+  obtain s where s: "\<And>i. simple_function M (s i)" "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x"
+    "\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)"
+    unfolding norm_conv_dist by metis
+
+  { fix f A 
+    have [simp]: "P (\<lambda>x. 0)"
+      using base[of "{}" undefined] by simp
+    have "(\<And>i::'b. i \<in> A \<Longrightarrow> integrable M (f i::'a \<Rightarrow> 'b)) \<Longrightarrow>
+    (\<And>i. i \<in> A \<Longrightarrow> P (f i)) \<Longrightarrow> P (\<lambda>x. \<Sum>i\<in>A. f i x)"
+    by (induct A rule: infinite_finite_induct) (auto intro!: add) }
+  note setsum = this
+
+  def s' \<equiv> "\<lambda>i z. indicator (space M) z *\<^sub>R s i z"
+  then have s'_eq_s: "\<And>i x. x \<in> space M \<Longrightarrow> s' i x = s i x"
+    by simp
+
+  have sf[measurable]: "\<And>i. simple_function M (s' i)"
+    unfolding s'_def using s(1)
+    by (intro simple_function_compose2[where h="op *\<^sub>R"] simple_function_indicator) auto
+
+  { fix i 
+    have "\<And>z. {y. s' i z = y \<and> y \<in> s' i ` space M \<and> y \<noteq> 0 \<and> z \<in> space M} =
+        (if z \<in> space M \<and> s' i z \<noteq> 0 then {s' i z} else {})"
+      by (auto simp add: s'_def split: split_indicator)
+    then have "\<And>z. s' i = (\<lambda>z. \<Sum>y\<in>s' i`space M - {0}. indicator {x\<in>space M. s' i x = y} z *\<^sub>R y)"
+      using sf by (auto simp: fun_eq_iff simple_function_def s'_def) }
+  note s'_eq = this
+
+  show "P f"
+  proof (rule lim)
+    fix i
+
+    have "(\<integral>\<^sup>+x. norm (s' i x) \<partial>M) \<le> (\<integral>\<^sup>+x. 2 * ereal (norm (f x)) \<partial>M)"
+      using s by (intro positive_integral_mono) (auto simp: s'_eq_s)
+    also have "\<dots> < \<infinity>"
+      using f by (subst positive_integral_cmult) auto
+    finally have sbi: "simple_bochner_integrable M (s' i)"
+      using sf by (intro simple_bochner_integrableI_bounded) auto
+    then show "integrable M (s' i)"
+      by (rule integrableI_simple_bochner_integrable)
+
+    { fix x assume"x \<in> space M" "s' i x \<noteq> 0"
+      then have "emeasure M {y \<in> space M. s' i y = s' i x} \<le> emeasure M {y \<in> space M. s' i y \<noteq> 0}"
+        by (intro emeasure_mono) auto
+      also have "\<dots> < \<infinity>"
+        using sbi by (auto elim: simple_bochner_integrable.cases)
+      finally have "emeasure M {y \<in> space M. s' i y = s' i x} \<noteq> \<infinity>" by simp }
+    then show "P (s' i)"
+      by (subst s'_eq) (auto intro!: setsum base)
+
+    fix x assume "x \<in> space M" with s show "(\<lambda>i. s' i x) ----> f x"
+      by (simp add: s'_eq_s)
+    show "norm (s' i x) \<le> 2 * norm (f x)"
+      using `x \<in> space M` s by (simp add: s'_eq_s)
+  qed fact
+qed
+
+lemma integral_nonneg_AE:
+  fixes f :: "'a \<Rightarrow> real"
+  assumes [measurable]: "integrable M f" "AE x in M. 0 \<le> f x"
+  shows "0 \<le> integral\<^sup>L M f"
+proof -
+  have "0 \<le> ereal (integral\<^sup>L M (\<lambda>x. max 0 (f x)))"
+    by (subst integral_eq_positive_integral)
+       (auto intro: real_of_ereal_pos positive_integral_positive integrable_max assms)
+  also have "integral\<^sup>L M (\<lambda>x. max 0 (f x)) = integral\<^sup>L M f"
+    using assms(2) by (intro integral_cong_AE assms integrable_max) auto
+  finally show ?thesis
+    by simp
+qed
+
+lemma integral_eq_zero_AE:
+  "f \<in> borel_measurable M \<Longrightarrow> (AE x in M. f x = 0) \<Longrightarrow> integral\<^sup>L M f = 0"
+  using integral_cong_AE[of f M "\<lambda>_. 0"] by simp
+
+lemma integral_nonneg_eq_0_iff_AE:
+  fixes f :: "_ \<Rightarrow> real"
+  assumes f[measurable]: "integrable M f" and nonneg: "AE x in M. 0 \<le> f x"
+  shows "integral\<^sup>L M f = 0 \<longleftrightarrow> (AE x in M. f x = 0)"
+proof
+  assume "integral\<^sup>L M f = 0"
+  then have "integral\<^sup>P M f = 0"
+    using positive_integral_eq_integral[OF f nonneg] by simp
+  then have "AE x in M. ereal (f x) \<le> 0"
+    by (simp add: positive_integral_0_iff_AE)
+  with nonneg show "AE x in M. f x = 0"
+    by auto
+qed (auto simp add: integral_eq_zero_AE)
+
+lemma integral_mono_AE:
+  fixes f :: "'a \<Rightarrow> real"
+  assumes "integrable M f" "integrable M g" "AE x in M. f x \<le> g x"
+  shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
+proof -
+  have "0 \<le> integral\<^sup>L M (\<lambda>x. g x - f x)"
+    using assms by (intro integral_nonneg_AE integrable_diff assms) auto
+  also have "\<dots> = integral\<^sup>L M g - integral\<^sup>L M f"
+    by (intro integral_diff assms)
+  finally show ?thesis by simp
+qed
+
+lemma integral_mono:
+  fixes f :: "'a \<Rightarrow> real"
+  shows "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x \<le> g x) \<Longrightarrow> 
+    integral\<^sup>L M f \<le> integral\<^sup>L M g"
+  by (intro integral_mono_AE) auto
+
+section {* Measure spaces with an associated density *}
+
+lemma integrable_density:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
+  assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
+    and nn: "AE x in M. 0 \<le> g x"
+  shows "integrable (density M g) f \<longleftrightarrow> integrable M (\<lambda>x. g x *\<^sub>R f x)"
+  unfolding integrable_iff_bounded using nn
+  apply (simp add: positive_integral_density )
+  apply (intro arg_cong2[where f="op ="] refl positive_integral_cong_AE)
+  apply auto
+  done
+
+lemma integral_density:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
+  assumes f: "f \<in> borel_measurable M"
+    and g[measurable]: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
+  shows "integral\<^sup>L (density M g) f = integral\<^sup>L M (\<lambda>x. g x *\<^sub>R f x)"
+proof (rule integral_eq_cases)
+  assume "integrable (density M g) f"
+  then show ?thesis
+  proof induct
+    case (base A c)
+    then have [measurable]: "A \<in> sets M" by auto
+  
+    have int: "integrable M (\<lambda>x. g x * indicator A x)"
+      using g base integrable_density[of "indicator A :: 'a \<Rightarrow> real" M g] by simp
+    then have "integral\<^sup>L M (\<lambda>x. g x * indicator A x) = (\<integral>\<^sup>+ x. ereal (g x * indicator A x) \<partial>M)"
+      using g by (subst positive_integral_eq_integral) auto
+    also have "\<dots> = (\<integral>\<^sup>+ x. ereal (g x) * indicator A x \<partial>M)"
+      by (intro positive_integral_cong) (auto split: split_indicator)
+    also have "\<dots> = emeasure (density M g) A"
+      by (rule emeasure_density[symmetric]) auto
+    also have "\<dots> = ereal (measure (density M g) A)"
+      using base by (auto intro: emeasure_eq_ereal_measure)
+    also have "\<dots> = integral\<^sup>L (density M g) (indicator A)"
+      using base by simp
+    finally show ?case
+      using base by (simp add: int)
+  next
+    case (add f h)
+    then have [measurable]: "f \<in> borel_measurable M" "h \<in> borel_measurable M"
+      by (auto dest!: borel_measurable_integrable)
+    from add g show ?case
+      by (simp add: scaleR_add_right integrable_density)
+  next
+    case (lim f s)
+    have [measurable]: "f \<in> borel_measurable M" "\<And>i. s i \<in> borel_measurable M"
+      using lim(1,5)[THEN borel_measurable_integrable] by auto
+  
+    show ?case
+    proof (rule LIMSEQ_unique)
+      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. g x *\<^sub>R s i x)) ----> integral\<^sup>L M (\<lambda>x. g x *\<^sub>R f x)"
+      proof (rule integral_dominated_convergence(3))
+        show "integrable M (\<lambda>x. 2 * norm (g x *\<^sub>R f x))"
+          by (intro integrable_mult_right integrable_norm integrable_density[THEN iffD1] lim g) auto
+        show "AE x in M. (\<lambda>i. g x *\<^sub>R s i x) ----> g x *\<^sub>R f x"
+          using lim(3) by (auto intro!: tendsto_scaleR AE_I2[of M])
+        show "\<And>i. AE x in M. norm (g x *\<^sub>R s i x) \<le> 2 * norm (g x *\<^sub>R f x)"
+          using lim(4) g by (auto intro!: AE_I2[of M] mult_left_mono simp: field_simps)
+      qed auto
+      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. g x *\<^sub>R s i x)) ----> integral\<^sup>L (density M g) f"
+        unfolding lim(2)[symmetric]
+        by (rule integral_dominated_convergence(3)[where w="\<lambda>x. 2 * norm (f x)"])
+           (insert lim(3-5), auto intro: integrable_norm)
+    qed
+  qed
+qed (simp add: f g integrable_density)
+
+lemma
+  fixes g :: "'a \<Rightarrow> real"
+  assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "g \<in> borel_measurable M"
+  shows integral_real_density: "integral\<^sup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)"
+    and integrable_real_density: "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)"
+  using assms integral_density[of g M f] integrable_density[of g M f] by auto
+
+lemma has_bochner_integral_density:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
+  shows "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. 0 \<le> g x) \<Longrightarrow>
+    has_bochner_integral M (\<lambda>x. g x *\<^sub>R f x) x \<Longrightarrow> has_bochner_integral (density M g) f x"
+  by (simp add: has_bochner_integral_iff integrable_density integral_density)
+
+subsection {* Distributions *}
+
+lemma integrable_distr_eq:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  assumes [measurable]: "g \<in> measurable M N" "f \<in> borel_measurable N"
+  shows "integrable (distr M N g) f \<longleftrightarrow> integrable M (\<lambda>x. f (g x))"
+  unfolding integrable_iff_bounded by (simp_all add: positive_integral_distr)
+
+lemma integrable_distr:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  shows "T \<in> measurable M M' \<Longrightarrow> integrable (distr M M' T) f \<Longrightarrow> integrable M (\<lambda>x. f (T x))"
+  by (subst integrable_distr_eq[symmetric, where g=T])
+     (auto dest: borel_measurable_integrable)
+
+lemma integral_distr:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  assumes g[measurable]: "g \<in> measurable M N" and f: "f \<in> borel_measurable N"
+  shows "integral\<^sup>L (distr M N g) f = integral\<^sup>L M (\<lambda>x. f (g x))"
+proof (rule integral_eq_cases)
+  assume "integrable (distr M N g) f"
+  then show ?thesis
+  proof induct
+    case (base A c)
+    then have [measurable]: "A \<in> sets N" by auto
+    from base have int: "integrable (distr M N g) (\<lambda>a. indicator A a *\<^sub>R c)"
+      by (intro integrable_indicator)
+  
+    have "integral\<^sup>L (distr M N g) (\<lambda>a. indicator A a *\<^sub>R c) = measure (distr M N g) A *\<^sub>R c"
+      using base by (subst integral_indicator) auto
+    also have "\<dots> = measure M (g -` A \<inter> space M) *\<^sub>R c"
+      by (subst measure_distr) auto
+    also have "\<dots> = integral\<^sup>L M (\<lambda>a. indicator (g -` A \<inter> space M) a *\<^sub>R c)"
+      using base by (subst integral_indicator) (auto simp: emeasure_distr)
+    also have "\<dots> = integral\<^sup>L M (\<lambda>a. indicator A (g a) *\<^sub>R c)"
+      using int base by (intro integral_cong_AE) (auto simp: emeasure_distr split: split_indicator)
+    finally show ?case .
+  next
+    case (add f h)
+    then have [measurable]: "f \<in> borel_measurable N" "h \<in> borel_measurable N"
+      by (auto dest!: borel_measurable_integrable)
+    from add g show ?case
+      by (simp add: scaleR_add_right integrable_distr_eq)
+  next
+    case (lim f s)
+    have [measurable]: "f \<in> borel_measurable N" "\<And>i. s i \<in> borel_measurable N"
+      using lim(1,5)[THEN borel_measurable_integrable] by auto
+  
+    show ?case
+    proof (rule LIMSEQ_unique)
+      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. s i (g x))) ----> integral\<^sup>L M (\<lambda>x. f (g x))"
+      proof (rule integral_dominated_convergence(3))
+        show "integrable M (\<lambda>x. 2 * norm (f (g x)))"
+          using lim by (auto intro!: integrable_norm simp: integrable_distr_eq) 
+        show "AE x in M. (\<lambda>i. s i (g x)) ----> f (g x)"
+          using lim(3) g[THEN measurable_space] by auto
+        show "\<And>i. AE x in M. norm (s i (g x)) \<le> 2 * norm (f (g x))"
+          using lim(4) g[THEN measurable_space] by auto
+      qed auto
+      show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. s i (g x))) ----> integral\<^sup>L (distr M N g) f"
+        unfolding lim(2)[symmetric]
+        by (rule integral_dominated_convergence(3)[where w="\<lambda>x. 2 * norm (f x)"])
+           (insert lim(3-5), auto intro: integrable_norm)
+    qed
+  qed
+qed (simp add: f g integrable_distr_eq)
+
+lemma has_bochner_integral_distr:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  shows "f \<in> borel_measurable N \<Longrightarrow> g \<in> measurable M N \<Longrightarrow>
+    has_bochner_integral M (\<lambda>x. f (g x)) x \<Longrightarrow> has_bochner_integral (distr M N g) f x"
+  by (simp add: has_bochner_integral_iff integrable_distr_eq integral_distr)
+
+section {* Lebesgue integration on @{const count_space} *}
+
+lemma integrable_count_space:
+  fixes f :: "'a \<Rightarrow> 'b::{banach,second_countable_topology}"
+  shows "finite X \<Longrightarrow> integrable (count_space X) f"
+  by (auto simp: positive_integral_count_space integrable_iff_bounded)
+
+lemma measure_count_space[simp]:
+  "B \<subseteq> A \<Longrightarrow> finite B \<Longrightarrow> measure (count_space A) B = card B"
+  unfolding measure_def by (subst emeasure_count_space ) auto
+
+lemma lebesgue_integral_count_space_finite_support:
+  assumes f: "finite {a\<in>A. f a \<noteq> 0}"
+  shows "(\<integral>x. f x \<partial>count_space A) = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. f a)"
+proof -
+  have eq: "\<And>x. x \<in> A \<Longrightarrow> (\<Sum>a | x = a \<and> a \<in> A \<and> f a \<noteq> 0. f a) = (\<Sum>x\<in>{x}. f x)"
+    by (intro setsum_mono_zero_cong_left) auto
+    
+  have "(\<integral>x. f x \<partial>count_space A) = (\<integral>x. (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. indicator {a} x *\<^sub>R f a) \<partial>count_space A)"
+    by (intro integral_cong refl) (simp add: f eq)
+  also have "\<dots> = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. measure (count_space A) {a} *\<^sub>R f a)"
+    by (subst integral_setsum) (auto intro!: setsum_cong)
+  finally show ?thesis
+    by auto
+qed
+
+lemma lebesgue_integral_count_space_finite: "finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
+  by (subst lebesgue_integral_count_space_finite_support)
+     (auto intro!: setsum_mono_zero_cong_left)
+
+section {* Point measure *}
+
+lemma lebesgue_integral_point_measure_finite:
+  fixes g :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  shows "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow>
+    integral\<^sup>L (point_measure A f) g = (\<Sum>a\<in>A. f a *\<^sub>R g a)"
+  by (simp add: lebesgue_integral_count_space_finite AE_count_space integral_density point_measure_def)
+
+lemma integrable_point_measure_finite:
+  fixes g :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and f :: "'a \<Rightarrow> real"
+  shows "finite A \<Longrightarrow> integrable (point_measure A f) g"
+  unfolding point_measure_def
+  apply (subst density_ereal_max_0)
+  apply (subst integrable_density)
+  apply (auto simp: AE_count_space integrable_count_space)
+  done
+
+subsection {* Legacy lemmas for the real-valued Lebesgue integral\<^sup>L *}
+
+lemma real_lebesgue_integral_def:
+  assumes f: "integrable M f"
+  shows "integral\<^sup>L M f = real (\<integral>\<^sup>+x. f x \<partial>M) - real (\<integral>\<^sup>+x. - f x \<partial>M)"
+proof -
+  have "integral\<^sup>L M f = integral\<^sup>L M (\<lambda>x. max 0 (f x) - max 0 (- f x))"
+    by (auto intro!: arg_cong[where f="integral\<^sup>L M"])
+  also have "\<dots> = integral\<^sup>L M (\<lambda>x. max 0 (f x)) - integral\<^sup>L M (\<lambda>x. max 0 (- f x))"
+    by (intro integral_diff integrable_max integrable_minus integrable_zero f)
+  also have "integral\<^sup>L M (\<lambda>x. max 0 (f x)) = real (\<integral>\<^sup>+x. max 0 (f x) \<partial>M)"
+    by (subst integral_eq_positive_integral[symmetric]) (auto intro!: integrable_max f)
+  also have "integral\<^sup>L M (\<lambda>x. max 0 (- f x)) = real (\<integral>\<^sup>+x. max 0 (- f x) \<partial>M)"
+    by (subst integral_eq_positive_integral[symmetric]) (auto intro!: integrable_max f)
+  also have "(\<lambda>x. ereal (max 0 (f x))) = (\<lambda>x. max 0 (ereal (f x)))"
+    by (auto simp: max_def)
+  also have "(\<lambda>x. ereal (max 0 (- f x))) = (\<lambda>x. max 0 (- ereal (f x)))"
+    by (auto simp: max_def)
+  finally show ?thesis
+    unfolding positive_integral_max_0 .
+qed
+
+lemma real_integrable_def:
+  "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
+    (\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
+  unfolding integrable_iff_bounded
+proof (safe del: notI)
+  assume *: "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>"
+  have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
+    by (intro positive_integral_mono) auto
+  also note *
+  finally show "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>"
+    by simp
+  have "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
+    by (intro positive_integral_mono) auto
+  also note *
+  finally show "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
+    by simp
+next
+  assume [measurable]: "f \<in> borel_measurable M"
+  assume fin: "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
+  have "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) = (\<integral>\<^sup>+ x. max 0 (ereal (f x)) + max 0 (ereal (- f x)) \<partial>M)"
+    by (intro positive_integral_cong) (auto simp: max_def)
+  also have"\<dots> = (\<integral>\<^sup>+ x. max 0 (ereal (f x)) \<partial>M) + (\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M)"
+    by (intro positive_integral_add) auto
+  also have "\<dots> < \<infinity>"
+    using fin by (auto simp: positive_integral_max_0)
+  finally show "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) < \<infinity>" .
+qed
+
+lemma integrableD[dest]:
+  assumes "integrable M f"
+  shows "f \<in> borel_measurable M" "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
+  using assms unfolding real_integrable_def by auto
+
+lemma integrableE:
+  assumes "integrable M f"
+  obtains r q where
+    "(\<integral>\<^sup>+x. ereal (f x)\<partial>M) = ereal r"
+    "(\<integral>\<^sup>+x. ereal (-f x)\<partial>M) = ereal q"
+    "f \<in> borel_measurable M" "integral\<^sup>L M f = r - q"
+  using assms unfolding real_integrable_def real_lebesgue_integral_def[OF assms]
+  using positive_integral_positive[of M "\<lambda>x. ereal (f x)"]
+  using positive_integral_positive[of M "\<lambda>x. ereal (-f x)"]
+  by (cases rule: ereal2_cases[of "(\<integral>\<^sup>+x. ereal (-f x)\<partial>M)" "(\<integral>\<^sup>+x. ereal (f x)\<partial>M)"]) auto
+
+lemma integral_monotone_convergence_nonneg:
+  fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
+  assumes i: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
+    and pos: "\<And>i. AE x in M. 0 \<le> f i x"
+    and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
+    and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
+    and u: "u \<in> borel_measurable M"
+  shows "integrable M u"
+  and "integral\<^sup>L M u = x"
+proof -
+  have "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = (SUP n. (\<integral>\<^sup>+ x. ereal (f n x) \<partial>M))"
+  proof (subst positive_integral_monotone_convergence_SUP_AE[symmetric])
+    fix i
+    from mono pos show "AE x in M. ereal (f i x) \<le> ereal (f (Suc i) x) \<and> 0 \<le> ereal (f i x)"
+      by eventually_elim (auto simp: mono_def)
+    show "(\<lambda>x. ereal (f i x)) \<in> borel_measurable M"
+      using i by auto
+  next
+    show "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = \<integral>\<^sup>+ x. (SUP i. ereal (f i x)) \<partial>M"
+      apply (rule positive_integral_cong_AE)
+      using lim mono
+      by eventually_elim (simp add: SUP_eq_LIMSEQ[THEN iffD2])
+  qed
+  also have "\<dots> = ereal x"
+    using mono i unfolding positive_integral_eq_integral[OF i pos]
+    by (subst SUP_eq_LIMSEQ) (auto simp: mono_def intro!: integral_mono_AE ilim)
+  finally have "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = ereal x" .
+  moreover have "(\<integral>\<^sup>+ x. ereal (- u x) \<partial>M) = 0"
+  proof (subst positive_integral_0_iff_AE)
+    show "(\<lambda>x. ereal (- u x)) \<in> borel_measurable M"
+      using u by auto
+    from mono pos[of 0] lim show "AE x in M. ereal (- u x) \<le> 0"
+    proof eventually_elim
+      fix x assume "mono (\<lambda>n. f n x)" "0 \<le> f 0 x" "(\<lambda>i. f i x) ----> u x"
+      then show "ereal (- u x) \<le> 0"
+        using incseq_le[of "\<lambda>n. f n x" "u x" 0] by (simp add: mono_def incseq_def)
+    qed
+  qed
+  ultimately show "integrable M u" "integral\<^sup>L M u = x"
+    by (auto simp: real_integrable_def real_lebesgue_integral_def u)
+qed
+
+lemma
+  fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
+  assumes f: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
+  and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
+  and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
+  and u: "u \<in> borel_measurable M"
+  shows integrable_monotone_convergence: "integrable M u"
+    and integral_monotone_convergence: "integral\<^sup>L M u = x"
+    and has_bochner_integral_monotone_convergence: "has_bochner_integral M u x"
+proof -
+  have 1: "\<And>i. integrable M (\<lambda>x. f i x - f 0 x)"
+    using f by auto
+  have 2: "AE x in M. mono (\<lambda>n. f n x - f 0 x)"
+    using mono by (auto simp: mono_def le_fun_def)
+  have 3: "\<And>n. AE x in M. 0 \<le> f n x - f 0 x"
+    using mono by (auto simp: field_simps mono_def le_fun_def)
+  have 4: "AE x in M. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
+    using lim by (auto intro!: tendsto_diff)
+  have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^sup>L M (f 0)"
+    using f ilim by (auto intro!: tendsto_diff)
+  have 6: "(\<lambda>x. u x - f 0 x) \<in> borel_measurable M"
+    using f[of 0] u by auto
+  note diff = integral_monotone_convergence_nonneg[OF 1 2 3 4 5 6]
+  have "integrable M (\<lambda>x. (u x - f 0 x) + f 0 x)"
+    using diff(1) f by (rule integrable_add)
+  with diff(2) f show "integrable M u" "integral\<^sup>L M u = x"
+    by auto
+  then show "has_bochner_integral M u x"
+    by (metis has_bochner_integral_integrable)
+qed
+
+lemma integral_norm_eq_0_iff:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  assumes f[measurable]: "integrable M f"
+  shows "(\<integral>x. norm (f x) \<partial>M) = 0 \<longleftrightarrow> emeasure M {x\<in>space M. f x \<noteq> 0} = 0"
+proof -
+  have "(\<integral>\<^sup>+x. norm (f x) \<partial>M) = (\<integral>x. norm (f x) \<partial>M)"
+    using f by (intro positive_integral_eq_integral integrable_norm) auto
+  then have "(\<integral>x. norm (f x) \<partial>M) = 0 \<longleftrightarrow> (\<integral>\<^sup>+x. norm (f x) \<partial>M) = 0"
+    by simp
+  also have "\<dots> \<longleftrightarrow> emeasure M {x\<in>space M. ereal (norm (f x)) \<noteq> 0} = 0"
+    by (intro positive_integral_0_iff) auto
+  finally show ?thesis
+    by simp
+qed
+
+lemma integral_0_iff:
+  fixes f :: "'a \<Rightarrow> real"
+  shows "integrable M f \<Longrightarrow> (\<integral>x. abs (f x) \<partial>M) = 0 \<longleftrightarrow> emeasure M {x\<in>space M. f x \<noteq> 0} = 0"
+  using integral_norm_eq_0_iff[of M f] by simp
+
+lemma (in finite_measure) lebesgue_integral_const[simp]:
+  "(\<integral>x. a \<partial>M) = measure M (space M) *\<^sub>R a"
+  using integral_indicator[of "space M" M a]
+  by (simp del: integral_indicator integral_scaleR_left cong: integral_cong)
+
+lemma (in finite_measure) integrable_const[intro!, simp]: "integrable M (\<lambda>x. a)"
+  using integrable_indicator[of "space M" M a] by (simp cong: integrable_cong)
+
+lemma (in finite_measure) integrable_const_bound:
+  fixes f :: "'a \<Rightarrow> 'b::{banach,second_countable_topology}"
+  shows "AE x in M. norm (f x) \<le> B \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> integrable M f"
+  apply (rule integrable_bound[OF integrable_const[of B], of f])
+  apply assumption
+  apply (cases "0 \<le> B")
+  apply auto
+  done
+
+lemma (in finite_measure) integral_less_AE:
+  fixes X Y :: "'a \<Rightarrow> real"
+  assumes int: "integrable M X" "integrable M Y"
+  assumes A: "(emeasure M) A \<noteq> 0" "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> X x \<noteq> Y x"
+  assumes gt: "AE x in M. X x \<le> Y x"
+  shows "integral\<^sup>L M X < integral\<^sup>L M Y"
+proof -
+  have "integral\<^sup>L M X \<le> integral\<^sup>L M Y"
+    using gt int by (intro integral_mono_AE) auto
+  moreover
+  have "integral\<^sup>L M X \<noteq> integral\<^sup>L M Y"
+  proof
+    assume eq: "integral\<^sup>L M X = integral\<^sup>L M Y"
+    have "integral\<^sup>L M (\<lambda>x. \<bar>Y x - X x\<bar>) = integral\<^sup>L M (\<lambda>x. Y x - X x)"
+      using gt int by (intro integral_cong_AE) auto
+    also have "\<dots> = 0"
+      using eq int by simp
+    finally have "(emeasure M) {x \<in> space M. Y x - X x \<noteq> 0} = 0"
+      using int by (simp add: integral_0_iff)
+    moreover
+    have "(\<integral>\<^sup>+x. indicator A x \<partial>M) \<le> (\<integral>\<^sup>+x. indicator {x \<in> space M. Y x - X x \<noteq> 0} x \<partial>M)"
+      using A by (intro positive_integral_mono_AE) auto
+    then have "(emeasure M) A \<le> (emeasure M) {x \<in> space M. Y x - X x \<noteq> 0}"
+      using int A by (simp add: integrable_def)
+    ultimately have "emeasure M A = 0"
+      using emeasure_nonneg[of M A] by simp
+    with `(emeasure M) A \<noteq> 0` show False by auto
+  qed
+  ultimately show ?thesis by auto
+qed
+
+lemma (in finite_measure) integral_less_AE_space:
+  fixes X Y :: "'a \<Rightarrow> real"
+  assumes int: "integrable M X" "integrable M Y"
+  assumes gt: "AE x in M. X x < Y x" "emeasure M (space M) \<noteq> 0"
+  shows "integral\<^sup>L M X < integral\<^sup>L M Y"
+  using gt by (intro integral_less_AE[OF int, where A="space M"]) auto
+
+(* GENERALIZE to banach, sct *)
+lemma integral_sums:
+  fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
+  assumes integrable[measurable]: "\<And>i. integrable M (f i)"
+  and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
+  and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
+  shows "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?S")
+  and "(\<lambda>i. integral\<^sup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is ?integral)
+proof -
+  have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
+    using summable unfolding summable_def by auto
+  from bchoice[OF this]
+  obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto
+  then have w_borel: "w \<in> borel_measurable M" unfolding sums_def
+    by (rule borel_measurable_LIMSEQ) auto
+
+  let ?w = "\<lambda>y. if y \<in> space M then w y else 0"
+
+  obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M)) sums x"
+    using sums unfolding summable_def ..
+
+  have 1: "\<And>n. integrable M (\<lambda>x. \<Sum>i<n. f i x)"
+    using integrable by auto
+
+  have 2: "\<And>j. AE x in M. norm (\<Sum>i<j. f i x) \<le> ?w x"
+    using AE_space
+  proof eventually_elim
+    fix j x assume [simp]: "x \<in> space M"
+    have "\<bar>\<Sum>i<j. f i x\<bar> \<le> (\<Sum>i<j. \<bar>f i x\<bar>)" by (rule setsum_abs)
+    also have "\<dots> \<le> w x" using w[of x] setsum_le_suminf[of "\<lambda>i. \<bar>f i x\<bar>"] unfolding sums_iff by auto
+    finally show "norm (\<Sum>i<j. f i x) \<le> ?w x" by simp
+  qed
+
+  have 3: "integrable M ?w"
+  proof (rule integrable_monotone_convergence(1))
+    let ?F = "\<lambda>n y. (\<Sum>i<n. \<bar>f i y\<bar>)"
+    let ?w' = "\<lambda>n y. if y \<in> space M then ?F n y else 0"
+    have "\<And>n. integrable M (?F n)"
+      using integrable by (auto intro!: integrable_abs)
+    thus "\<And>n. integrable M (?w' n)" by (simp cong: integrable_cong)
+    show "AE x in M. mono (\<lambda>n. ?w' n x)"
+      by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
+    show "AE x in M. (\<lambda>n. ?w' n x) ----> ?w x"
+        using w by (simp_all add: tendsto_const sums_def)
+    have *: "\<And>n. integral\<^sup>L M (?w' n) = (\<Sum>i< n. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
+      using integrable by (simp add: integrable_abs cong: integral_cong)
+    from abs_sum
+    show "(\<lambda>i. integral\<^sup>L M (?w' i)) ----> x" unfolding * sums_def .
+  qed (simp add: w_borel measurable_If_set)
+
+  from summable[THEN summable_rabs_cancel]
+  have 4: "AE x in M. (\<lambda>n. \<Sum>i<n. f i x) ----> (\<Sum>i. f i x)"
+    by (auto intro: summable_LIMSEQ)
+
+  note int = integral_dominated_convergence(1,3)[OF _ _ 3 4 2]
+
+  from int show "integrable M ?S" by simp
+
+  show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF integrable]
+    using int(2) by simp
+qed
+
+lemma integrable_mult_indicator:
+  fixes f :: "'a \<Rightarrow> real"
+  shows "A \<in> sets M \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. f x * indicator A x)"
+  by (rule integrable_bound[where f="\<lambda>x. \<bar>f x\<bar>"])
+     (auto intro: integrable_abs split: split_indicator)
+
+lemma tendsto_integral_at_top:
+  fixes f :: "real \<Rightarrow> real"
+  assumes M: "sets M = sets borel" and f[measurable]: "integrable M f"
+  shows "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
+proof -
+  have M_measure[simp]: "borel_measurable M = borel_measurable borel"
+    using M by (simp add: sets_eq_imp_space_eq measurable_def)
+  { fix f :: "real \<Rightarrow> real" assume f: "integrable M f" "\<And>x. 0 \<le> f x"
+    then have [measurable]: "f \<in> borel_measurable borel"
+      by (simp add: real_integrable_def)
+    have "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
+    proof (rule tendsto_at_topI_sequentially)
+      have int: "\<And>n. integrable M (\<lambda>x. f x * indicator {.. n} x)"
+        by (rule integrable_mult_indicator) (auto simp: M f)
+      show "(\<lambda>n. \<integral> x. f x * indicator {..real n} x \<partial>M) ----> integral\<^sup>L M f"
+      proof (rule integral_dominated_convergence)
+        { fix x have "eventually (\<lambda>n. f x * indicator {..real n} x = f x) sequentially"
+            by (rule eventually_sequentiallyI[of "natceiling x"])
+               (auto split: split_indicator simp: natceiling_le_eq) }
+        from filterlim_cong[OF refl refl this]
+        show "AE x in M. (\<lambda>n. f x * indicator {..real n} x) ----> f x"
+          by (simp add: tendsto_const)
+        show "\<And>j. AE x in M. norm (f x * indicator {.. j} x) \<le> f x"
+          using f(2) by (intro AE_I2) (auto split: split_indicator)
+      qed (simp | fact)+
+      show "mono (\<lambda>y. \<integral> x. f x * indicator {..y} x \<partial>M)"
+        by (intro monoI integral_mono int) (auto split: split_indicator intro: f)
+    qed }
+  note nonneg = this
+  let ?P = "\<lambda>y. \<integral> x. max 0 (f x) * indicator {..y} x \<partial>M"
+  let ?N = "\<lambda>y. \<integral> x. max 0 (- f x) * indicator {..y} x \<partial>M"
+  let ?p = "integral\<^sup>L M (\<lambda>x. max 0 (f x))"
+  let ?n = "integral\<^sup>L M (\<lambda>x. max 0 (- f x))"
+  have "(?P ---> ?p) at_top" "(?N ---> ?n) at_top"
+    by (auto intro!: nonneg integrable_max f)
+  note tendsto_diff[OF this]
+  also have "(\<lambda>y. ?P y - ?N y) = (\<lambda>y. \<integral> x. f x * indicator {..y} x \<partial>M)"
+    by (subst integral_diff[symmetric])
+       (auto intro!: integrable_mult_indicator integrable_max f integral_cong
+             simp: M split: split_max)
+  also have "?p - ?n = integral\<^sup>L M f"
+    by (subst integral_diff[symmetric])
+       (auto intro!: integrable_max f integral_cong simp: M split: split_max)
+  finally show ?thesis .
+qed
+
+lemma
+  fixes f :: "real \<Rightarrow> real"
+  assumes M: "sets M = sets borel"
+  assumes nonneg: "AE x in M. 0 \<le> f x"
+  assumes borel: "f \<in> borel_measurable borel"
+  assumes int: "\<And>y. integrable M (\<lambda>x. f x * indicator {.. y} x)"
+  assumes conv: "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> x) at_top"
+  shows has_bochner_integral_monotone_convergence_at_top: "has_bochner_integral M f x"
+    and integrable_monotone_convergence_at_top: "integrable M f"
+    and integral_monotone_convergence_at_top:"integral\<^sup>L M f = x"
+proof -
+  from nonneg have "AE x in M. mono (\<lambda>n::nat. f x * indicator {..real n} x)"
+    by (auto split: split_indicator intro!: monoI)
+  { fix x have "eventually (\<lambda>n. f x * indicator {..real n} x = f x) sequentially"
+      by (rule eventually_sequentiallyI[of "natceiling x"])
+         (auto split: split_indicator simp: natceiling_le_eq) }
+  from filterlim_cong[OF refl refl this]
+  have "AE x in M. (\<lambda>i. f x * indicator {..real i} x) ----> f x"
+    by (simp add: tendsto_const)
+  have "(\<lambda>i. \<integral> x. f x * indicator {..real i} x \<partial>M) ----> x"
+    using conv filterlim_real_sequentially by (rule filterlim_compose)
+  have M_measure[simp]: "borel_measurable M = borel_measurable borel"
+    using M by (simp add: sets_eq_imp_space_eq measurable_def)
+  have "f \<in> borel_measurable M"
+    using borel by simp
+  show "has_bochner_integral M f x"
+    by (rule has_bochner_integral_monotone_convergence) fact+
+  then show "integrable M f" "integral\<^sup>L M f = x"
+    by (auto simp: _has_bochner_integral_iff)
+qed
+
+
+section "Lebesgue integration on countable spaces"
+
+lemma integral_on_countable:
+  fixes f :: "real \<Rightarrow> real"
+  assumes f: "f \<in> borel_measurable M"
+  and bij: "bij_betw enum S (f ` space M)"
+  and enum_zero: "enum ` (-S) \<subseteq> {0}"
+  and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> (emeasure M) (f -` {x} \<inter> space M) \<noteq> \<infinity>"
+  and abs_summable: "summable (\<lambda>r. \<bar>enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))\<bar>)"
+  shows "integrable M f"
+  and "(\<lambda>r. enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))) sums integral\<^sup>L M f" (is ?sums)
+proof -
+  let ?A = "\<lambda>r. f -` {enum r} \<inter> space M"
+  let ?F = "\<lambda>r x. enum r * indicator (?A r) x"
+  have enum_eq: "\<And>r. enum r * real ((emeasure M) (?A r)) = integral\<^sup>L M (?F r)"
+    using f fin by (simp add: measure_def cong: disj_cong)
+
+  { fix x assume "x \<in> space M"
+    hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto
+    then obtain i where "i\<in>S" "enum i = f x" by auto
+    have F: "\<And>j. ?F j x = (if j = i then f x else 0)"
+    proof cases
+      fix j assume "j = i"
+      thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def)
+    next
+      fix j assume "j \<noteq> i"
+      show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero
+        by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def)
+    qed
+    hence F_abs: "\<And>j. \<bar>if j = i then f x else 0\<bar> = (if j = i then \<bar>f x\<bar> else 0)" by auto
+    have "(\<lambda>i. ?F i x) sums f x"
+         "(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>"
+      by (auto intro!: sums_single simp: F F_abs) }
+  note F_sums_f = this(1) and F_abs_sums_f = this(2)
+
+  have int_f: "integral\<^sup>L M f = (\<integral>x. (\<Sum>r. ?F r x) \<partial>M)" "integrable M f = integrable M (\<lambda>x. \<Sum>r. ?F r x)"
+    using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
+
+  { fix r
+    have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)"
+      by (auto simp: indicator_def intro!: integral_cong)
+    also have "\<dots> = \<bar>enum r\<bar> * real ((emeasure M) (?A r))"
+      using f fin by (simp add: measure_def cong: disj_cong)
+    finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real ((emeasure M) (?A r))\<bar>"
+      using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_ereal_pos measurable_sets) }
+  note int_abs_F = this
+
+  have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
+    using f fin by auto
+
+  have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
+    using F_abs_sums_f unfolding sums_iff by auto
+
+  from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
+  show ?sums unfolding enum_eq int_f by simp
+
+  from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
+  show "integrable M f" unfolding int_f by simp
+qed
+
+subsection {* Product measure *}
+
+lemma (in sigma_finite_measure) borel_measurable_lebesgue_integrable[measurable (raw)]:
+  fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes [measurable]: "split f \<in> borel_measurable (N \<Otimes>\<^sub>M M)"
+  shows "Measurable.pred N (\<lambda>x. integrable M (f x))"
+proof -
+  have [simp]: "\<And>x. x \<in> space N \<Longrightarrow> integrable M (f x) \<longleftrightarrow> (\<integral>\<^sup>+y. norm (f x y) \<partial>M) < \<infinity>"
+    unfolding integrable_iff_bounded by simp
+  show ?thesis
+    by (simp cong: measurable_cong)
+qed
+
+lemma (in sigma_finite_measure) measurable_measure[measurable (raw)]:
+  "(\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M) \<Longrightarrow>
+    {x \<in> space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M) \<Longrightarrow>
+    (\<lambda>x. measure M (A x)) \<in> borel_measurable N"
+  unfolding measure_def by (intro measurable_emeasure borel_measurable_real_of_ereal)
+
+lemma Collect_subset [simp]: "{x\<in>A. P x} \<subseteq> A" by auto
+
+lemma (in sigma_finite_measure) borel_measurable_lebesgue_integral[measurable (raw)]:
+  fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes f[measurable]: "split f \<in> borel_measurable (N \<Otimes>\<^sub>M M)"
+  shows "(\<lambda>x. \<integral>y. f x y \<partial>M) \<in> borel_measurable N"
+proof -
+  from borel_measurable_implies_sequence_metric[OF f, of 0] guess s ..
+  then have s: "\<And>i. simple_function (N \<Otimes>\<^sub>M M) (s i)"
+    "\<And>x y. x \<in> space N \<Longrightarrow> y \<in> space M \<Longrightarrow> (\<lambda>i. s i (x, y)) ----> f x y"
+    "\<And>i x y. x \<in> space N \<Longrightarrow> y \<in> space M \<Longrightarrow> norm (s i (x, y)) \<le> 2 * norm (f x y)"
+    by (auto simp: space_pair_measure norm_conv_dist)
+
+  have [measurable]: "\<And>i. s i \<in> borel_measurable (N \<Otimes>\<^sub>M M)"
+    by (rule borel_measurable_simple_function) fact
+
+  have "\<And>i. s i \<in> measurable (N \<Otimes>\<^sub>M M) (count_space UNIV)"
+    by (rule measurable_simple_function) fact
+
+  def f' \<equiv> "\<lambda>i x. if integrable M (f x)
+    then simple_bochner_integral M (\<lambda>y. s i (x, y))
+    else (THE x. False)"
+
+  { fix i x assume "x \<in> space N"
+    then have "simple_bochner_integral M (\<lambda>y. s i (x, y)) =
+      (\<Sum>z\<in>s i ` (space N \<times> space M). measure M {y \<in> space M. s i (x, y) = z} *\<^sub>R z)"
+      using s(1)[THEN simple_functionD(1)]
+      unfolding simple_bochner_integral_def
+      by (intro setsum_mono_zero_cong_left)
+         (auto simp: eq_commute space_pair_measure image_iff cong: conj_cong) }
+  note eq = this
+
+  show ?thesis
+  proof (rule borel_measurable_LIMSEQ_metric)
+    fix i show "f' i \<in> borel_measurable N"
+      unfolding f'_def by (simp_all add: eq cong: measurable_cong if_cong)
+  next
+    fix x assume x: "x \<in> space N"
+    { assume int_f: "integrable M (f x)"
+      have int_2f: "integrable M (\<lambda>y. 2 * norm (f x y))"
+        by (intro integrable_norm integrable_mult_right int_f)
+      have "(\<lambda>i. integral\<^sup>L M (\<lambda>y. s i (x, y))) ----> integral\<^sup>L M (f x)"
+      proof (rule integral_dominated_convergence)
+        from int_f show "f x \<in> borel_measurable M" by auto
+        show "\<And>i. (\<lambda>y. s i (x, y)) \<in> borel_measurable M"
+          using x by simp
+        show "AE xa in M. (\<lambda>i. s i (x, xa)) ----> f x xa"
+          using x s(2) by auto
+        show "\<And>i. AE xa in M. norm (s i (x, xa)) \<le> 2 * norm (f x xa)"
+          using x s(3) by auto
+      qed fact
+      moreover
+      { fix i
+        have "simple_bochner_integrable M (\<lambda>y. s i (x, y))"
+        proof (rule simple_bochner_integrableI_bounded)
+          have "(\<lambda>y. s i (x, y)) ` space M \<subseteq> s i ` (space N \<times> space M)"
+            using x by auto
+          then show "simple_function M (\<lambda>y. s i (x, y))"
+            using simple_functionD(1)[OF s(1), of i] x
+            by (intro simple_function_borel_measurable)
+               (auto simp: space_pair_measure dest: finite_subset)
+          have "(\<integral>\<^sup>+ y. ereal (norm (s i (x, y))) \<partial>M) \<le> (\<integral>\<^sup>+ y. 2 * norm (f x y) \<partial>M)"
+            using x s by (intro positive_integral_mono) auto
+          also have "(\<integral>\<^sup>+ y. 2 * norm (f x y) \<partial>M) < \<infinity>"
+            using int_2f by (simp add: integrable_iff_bounded)
+          finally show "(\<integral>\<^sup>+ xa. ereal (norm (s i (x, xa))) \<partial>M) < \<infinity>" .
+        qed
+        then have "integral\<^sup>L M (\<lambda>y. s i (x, y)) = simple_bochner_integral M (\<lambda>y. s i (x, y))"
+          by (rule simple_bochner_integrable_eq_integral[symmetric]) }
+      ultimately have "(\<lambda>i. simple_bochner_integral M (\<lambda>y. s i (x, y))) ----> integral\<^sup>L M (f x)"
+        by simp }
+    then 
+    show "(\<lambda>i. f' i x) ----> integral\<^sup>L M (f x)"
+      unfolding f'_def
+      by (cases "integrable M (f x)") (simp_all add: not_integrable_integral_eq tendsto_const)
+  qed
+qed
+
+lemma (in pair_sigma_finite) integrable_product_swap:
+  fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes "integrable (M1 \<Otimes>\<^sub>M M2) f"
+  shows "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x,y). f (y,x))"
+proof -
+  interpret Q: pair_sigma_finite M2 M1 by default
+  have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
+  show ?thesis unfolding *
+    by (rule integrable_distr[OF measurable_pair_swap'])
+       (simp add: distr_pair_swap[symmetric] assms)
+qed
+
+lemma (in pair_sigma_finite) integrable_product_swap_iff:
+  fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
+  shows "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable (M1 \<Otimes>\<^sub>M M2) f"
+proof -
+  interpret Q: pair_sigma_finite M2 M1 by default
+  from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
+  show ?thesis by auto
+qed
+
+lemma (in pair_sigma_finite) integral_product_swap:
+  fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
+  shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
+proof -
+  have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
+  show ?thesis unfolding *
+    by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric])
+qed
+
+lemma (in pair_sigma_finite) emeasure_pair_measure_finite:
+  assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" and finite: "emeasure (M1 \<Otimes>\<^sub>M M2) A < \<infinity>"
+  shows "AE x in M1. emeasure M2 {y\<in>space M2. (x, y) \<in> A} < \<infinity>"
+proof -
+  from M2.emeasure_pair_measure_alt[OF A] finite
+  have "(\<integral>\<^sup>+ x. emeasure M2 (Pair x -` A) \<partial>M1) \<noteq> \<infinity>"
+    by simp
+  then have "AE x in M1. emeasure M2 (Pair x -` A) \<noteq> \<infinity>"
+    by (rule positive_integral_PInf_AE[rotated]) (intro M2.measurable_emeasure_Pair A)
+  moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> Pair x -` A = {y\<in>space M2. (x, y) \<in> A}"
+    using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)
+  ultimately show ?thesis by auto
+qed
+
+lemma (in pair_sigma_finite) AE_integrable_fst':
+  fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes f[measurable]: "integrable (M1 \<Otimes>\<^sub>M M2) f"
+  shows "AE x in M1. integrable M2 (\<lambda>y. f (x, y))"
+proof -
+  have "(\<integral>\<^sup>+x. (\<integral>\<^sup>+y. norm (f (x, y)) \<partial>M2) \<partial>M1) = (\<integral>\<^sup>+x. norm (f x) \<partial>(M1 \<Otimes>\<^sub>M M2))"
+    by (rule M2.positive_integral_fst) simp
+  also have "(\<integral>\<^sup>+x. norm (f x) \<partial>(M1 \<Otimes>\<^sub>M M2)) \<noteq> \<infinity>"
+    using f unfolding integrable_iff_bounded by simp
+  finally have "AE x in M1. (\<integral>\<^sup>+y. norm (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
+    by (intro positive_integral_PInf_AE M2.borel_measurable_positive_integral )
+       (auto simp: measurable_split_conv)
+  with AE_space show ?thesis
+    by eventually_elim
+       (auto simp: integrable_iff_bounded measurable_compose[OF _ borel_measurable_integrable[OF f]])
+qed
+
+lemma (in pair_sigma_finite) integrable_fst':
+  fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes f[measurable]: "integrable (M1 \<Otimes>\<^sub>M M2) f"
+  shows "integrable M1 (\<lambda>x. \<integral>y. f (x, y) \<partial>M2)"
+  unfolding integrable_iff_bounded
+proof
+  show "(\<lambda>x. \<integral> y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
+    by (rule M2.borel_measurable_lebesgue_integral) simp
+  have "(\<integral>\<^sup>+ x. ereal (norm (\<integral> y. f (x, y) \<partial>M2)) \<partial>M1) \<le> (\<integral>\<^sup>+x. (\<integral>\<^sup>+y. norm (f (x, y)) \<partial>M2) \<partial>M1)"
+    using AE_integrable_fst'[OF f] by (auto intro!: positive_integral_mono_AE integral_norm_bound_ereal)
+  also have "(\<integral>\<^sup>+x. (\<integral>\<^sup>+y. norm (f (x, y)) \<partial>M2) \<partial>M1) = (\<integral>\<^sup>+x. norm (f x) \<partial>(M1 \<Otimes>\<^sub>M M2))"
+    by (rule M2.positive_integral_fst) simp
+  also have "(\<integral>\<^sup>+x. norm (f x) \<partial>(M1 \<Otimes>\<^sub>M M2)) < \<infinity>"
+    using f unfolding integrable_iff_bounded by simp
+  finally show "(\<integral>\<^sup>+ x. ereal (norm (\<integral> y. f (x, y) \<partial>M2)) \<partial>M1) < \<infinity>" .
+qed
+
+lemma (in pair_sigma_finite) integral_fst':
+  fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) f"
+  shows "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
+using f proof induct
+  case (base A c)
+  have A[measurable]: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" by fact
+
+  have eq: "\<And>x y. x \<in> space M1 \<Longrightarrow> indicator A (x, y) = indicator {y\<in>space M2. (x, y) \<in> A} y"
+    using sets.sets_into_space[OF A] by (auto split: split_indicator simp: space_pair_measure)
+
+  have int_A: "integrable (M1 \<Otimes>\<^sub>M M2) (indicator A :: _ \<Rightarrow> real)"
+    using base by (rule integrable_real_indicator)
+
+  have "(\<integral> x. \<integral> y. indicator A (x, y) *\<^sub>R c \<partial>M2 \<partial>M1) = (\<integral>x. measure M2 {y\<in>space M2. (x, y) \<in> A} *\<^sub>R c \<partial>M1)"
+  proof (intro integral_cong_AE, simp, simp)
+    from AE_integrable_fst'[OF int_A] AE_space
+    show "AE x in M1. (\<integral>y. indicator A (x, y) *\<^sub>R c \<partial>M2) = measure M2 {y\<in>space M2. (x, y) \<in> A} *\<^sub>R c"
+      by eventually_elim (simp add: eq integrable_indicator_iff)
+  qed
+  also have "\<dots> = measure (M1 \<Otimes>\<^sub>M M2) A *\<^sub>R c"
+  proof (subst integral_scaleR_left)
+    have "(\<integral>\<^sup>+x. ereal (measure M2 {y \<in> space M2. (x, y) \<in> A}) \<partial>M1) =
+      (\<integral>\<^sup>+x. emeasure M2 {y \<in> space M2. (x, y) \<in> A} \<partial>M1)"
+      using emeasure_pair_measure_finite[OF base]
+      by (intro positive_integral_cong_AE, eventually_elim) (simp add: emeasure_eq_ereal_measure)
+    also have "\<dots> = emeasure (M1 \<Otimes>\<^sub>M M2) A"
+      using sets.sets_into_space[OF A]
+      by (subst M2.emeasure_pair_measure_alt)
+         (auto intro!: positive_integral_cong arg_cong[where f="emeasure M2"] simp: space_pair_measure)
+    finally have *: "(\<integral>\<^sup>+x. ereal (measure M2 {y \<in> space M2. (x, y) \<in> A}) \<partial>M1) = emeasure (M1 \<Otimes>\<^sub>M M2) A" .
+
+    from base * show "integrable M1 (\<lambda>x. measure M2 {y \<in> space M2. (x, y) \<in> A})"
+      by (simp add: measure_nonneg integrable_iff_bounded)
+    then have "(\<integral>x. measure M2 {y \<in> space M2. (x, y) \<in> A} \<partial>M1) = 
+      (\<integral>\<^sup>+x. ereal (measure M2 {y \<in> space M2. (x, y) \<in> A}) \<partial>M1)"
+      by (rule positive_integral_eq_integral[symmetric]) (simp add: measure_nonneg)
+    also note *
+    finally show "(\<integral>x. measure M2 {y \<in> space M2. (x, y) \<in> A} \<partial>M1) *\<^sub>R c = measure (M1 \<Otimes>\<^sub>M M2) A *\<^sub>R c"
+      using base by (simp add: emeasure_eq_ereal_measure)
+  qed
+  also have "\<dots> = (\<integral> a. indicator A a *\<^sub>R c \<partial>(M1 \<Otimes>\<^sub>M M2))"
+    using base by simp
+  finally show ?case .
+next
+  case (add f g)
+  then have [measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" "g \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
+    by auto
+  have "(\<integral> x. \<integral> y. f (x, y) + g (x, y) \<partial>M2 \<partial>M1) = 
+    (\<integral> x. (\<integral> y. f (x, y) \<partial>M2) + (\<integral> y. g (x, y) \<partial>M2) \<partial>M1)"
+    apply (rule integral_cong_AE)
+    apply simp_all
+    using AE_integrable_fst'[OF add(1)] AE_integrable_fst'[OF add(3)]
+    apply eventually_elim
+    apply simp
+    done 
+  also have "\<dots> = (\<integral> x. f x \<partial>(M1 \<Otimes>\<^sub>M M2)) + (\<integral> x. g x \<partial>(M1 \<Otimes>\<^sub>M M2))"
+    using integrable_fst'[OF add(1)] integrable_fst'[OF add(3)] add(2,4) by simp
+  finally show ?case
+    using add by simp
+next
+  case (lim f s)
+  then have [measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" "\<And>i. s i \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
+    by auto
+  
+  show ?case
+  proof (rule LIMSEQ_unique)
+    show "(\<lambda>i. integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (s i)) ----> integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
+    proof (rule integral_dominated_convergence)
+      show "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. 2 * norm (f x))"
+        using lim(5) by (auto intro: integrable_norm)
+    qed (insert lim, auto)
+    have "(\<lambda>i. \<integral> x. \<integral> y. s i (x, y) \<partial>M2 \<partial>M1) ----> \<integral> x. \<integral> y. f (x, y) \<partial>M2 \<partial>M1"
+    proof (rule integral_dominated_convergence)
+      have "AE x in M1. \<forall>i. integrable M2 (\<lambda>y. s i (x, y))"
+        unfolding AE_all_countable using AE_integrable_fst'[OF lim(1)] ..
+      with AE_space AE_integrable_fst'[OF lim(5)]
+      show "AE x in M1. (\<lambda>i. \<integral> y. s i (x, y) \<partial>M2) ----> \<integral> y. f (x, y) \<partial>M2"
+      proof eventually_elim
+        fix x assume x: "x \<in> space M1" and
+          s: "\<forall>i. integrable M2 (\<lambda>y. s i (x, y))" and f: "integrable M2 (\<lambda>y. f (x, y))"
+        show "(\<lambda>i. \<integral> y. s i (x, y) \<partial>M2) ----> \<integral> y. f (x, y) \<partial>M2"
+        proof (rule integral_dominated_convergence)
+          show "integrable M2 (\<lambda>y. 2 * norm (f (x, y)))"
+             using f by (auto intro: integrable_norm)
+          show "AE xa in M2. (\<lambda>i. s i (x, xa)) ----> f (x, xa)"
+            using x lim(3) by (auto simp: space_pair_measure)
+          show "\<And>i. AE xa in M2. norm (s i (x, xa)) \<le> 2 * norm (f (x, xa))"
+            using x lim(4) by (auto simp: space_pair_measure)
+        qed (insert x, measurable)
+      qed
+      show "integrable M1 (\<lambda>x. (\<integral> y. 2 * norm (f (x, y)) \<partial>M2))"
+        by (intro integrable_mult_right integrable_norm integrable_fst' lim)
+      fix i show "AE x in M1. norm (\<integral> y. s i (x, y) \<partial>M2) \<le> (\<integral> y. 2 * norm (f (x, y)) \<partial>M2)"
+        using AE_space AE_integrable_fst'[OF lim(1), of i] AE_integrable_fst'[OF lim(5)]
+      proof eventually_elim 
+        fix x assume x: "x \<in> space M1"
+          and s: "integrable M2 (\<lambda>y. s i (x, y))" and f: "integrable M2 (\<lambda>y. f (x, y))"
+        from s have "norm (\<integral> y. s i (x, y) \<partial>M2) \<le> (\<integral>\<^sup>+y. norm (s i (x, y)) \<partial>M2)"
+          by (rule integral_norm_bound_ereal)
+        also have "\<dots> \<le> (\<integral>\<^sup>+y. 2 * norm (f (x, y)) \<partial>M2)"
+          using x lim by (auto intro!: positive_integral_mono simp: space_pair_measure)
+        also have "\<dots> = (\<integral>y. 2 * norm (f (x, y)) \<partial>M2)"
+          using f by (intro positive_integral_eq_integral) auto
+        finally show "norm (\<integral> y. s i (x, y) \<partial>M2) \<le> (\<integral> y. 2 * norm (f (x, y)) \<partial>M2)"
+          by simp
+      qed
+    qed simp_all
+    then show "(\<lambda>i. integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (s i)) ----> \<integral> x. \<integral> y. f (x, y) \<partial>M2 \<partial>M1"
+      using lim by simp
+  qed
+qed
+
+lemma (in pair_sigma_finite)
+  fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) (split f)"
+  shows AE_integrable_fst: "AE x in M1. integrable M2 (\<lambda>y. f x y)" (is "?AE")
+    and integrable_fst: "integrable M1 (\<lambda>x. \<integral>y. f x y \<partial>M2)" (is "?INT")
+    and integral_fst: "(\<integral>x. (\<integral>y. f x y \<partial>M2) \<partial>M1) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)" (is "?EQ")
+  using AE_integrable_fst'[OF f] integrable_fst'[OF f] integral_fst'[OF f] by auto
+
+lemma (in pair_sigma_finite)
+  fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes f[measurable]: "integrable (M1 \<Otimes>\<^sub>M M2) (split f)"
+  shows AE_integrable_snd: "AE y in M2. integrable M1 (\<lambda>x. f x y)" (is "?AE")
+    and integrable_snd: "integrable M2 (\<lambda>y. \<integral>x. f x y \<partial>M1)" (is "?INT")
+    and integral_snd: "(\<integral>y. (\<integral>x. f x y \<partial>M1) \<partial>M2) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (split f)" (is "?EQ")
+proof -
+  interpret Q: pair_sigma_finite M2 M1 by default
+  have Q_int: "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x, y). f y x)"
+    using f unfolding integrable_product_swap_iff[symmetric] by simp
+  show ?AE  using Q.AE_integrable_fst'[OF Q_int] by simp
+  show ?INT using Q.integrable_fst'[OF Q_int] by simp
+  show ?EQ using Q.integral_fst'[OF Q_int]
+    using integral_product_swap[of "split f"] by simp
+qed
+
+lemma (in pair_sigma_finite) Fubini_integral:
+  fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _ :: {banach, second_countable_topology}"
+  assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) (split f)"
+  shows "(\<integral>y. (\<integral>x. f x y \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f x y \<partial>M2) \<partial>M1)"
+  unfolding integral_snd[OF assms] integral_fst[OF assms] ..
+
+lemma (in product_sigma_finite) product_integral_singleton:
+  fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
+  shows "f \<in> borel_measurable (M i) \<Longrightarrow> (\<integral>x. f (x i) \<partial>Pi\<^sub>M {i} M) = integral\<^sup>L (M i) f"
+  apply (subst distr_singleton[symmetric])
+  apply (subst integral_distr)
+  apply simp_all
+  done
+
+lemma (in product_sigma_finite) product_integral_fold:
+  fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
+  and f: "integrable (Pi\<^sub>M (I \<union> J) M) f"
+  shows "integral\<^sup>L (Pi\<^sub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I J (x, y)) \<partial>Pi\<^sub>M J M) \<partial>Pi\<^sub>M I M)"
+proof -
+  interpret I: finite_product_sigma_finite M I by default fact
+  interpret J: finite_product_sigma_finite M J by default fact
+  have "finite (I \<union> J)" using fin by auto
+  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
+  interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
+  let ?M = "merge I J"
+  let ?f = "\<lambda>x. f (?M x)"
+  from f have f_borel: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
+    by auto
+  have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
+    using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def)
+  have f_int: "integrable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) ?f"
+    by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f)
+  show ?thesis
+    apply (subst distr_merge[symmetric, OF IJ fin])
+    apply (subst integral_distr[OF measurable_merge f_borel])
+    apply (subst P.integral_fst'[symmetric, OF f_int])
+    apply simp
+    done
+qed
+
+lemma (in product_sigma_finite) product_integral_insert:
+  fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
+  assumes I: "finite I" "i \<notin> I"
+    and f: "integrable (Pi\<^sub>M (insert i I) M) f"
+  shows "integral\<^sup>L (Pi\<^sub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^sub>M I M)"
+proof -
+  have "integral\<^sup>L (Pi\<^sub>M (insert i I) M) f = integral\<^sup>L (Pi\<^sub>M (I \<union> {i}) M) f"
+    by simp
+  also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I {i} (x,y)) \<partial>Pi\<^sub>M {i} M) \<partial>Pi\<^sub>M I M)"
+    using f I by (intro product_integral_fold) auto
+  also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^sub>M I M)"
+  proof (rule integral_cong[OF refl], subst product_integral_singleton[symmetric])
+    fix x assume x: "x \<in> space (Pi\<^sub>M I M)"
+    have f_borel: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
+      using f by auto
+    show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
+      using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
+      unfolding comp_def .
+    from x I show "(\<integral> y. f (merge I {i} (x,y)) \<partial>Pi\<^sub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^sub>M {i} M)"
+      by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def PiE_def)
+  qed
+  finally show ?thesis .
+qed
+
+lemma (in product_sigma_finite) product_integrable_setprod:
+  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> _::{real_normed_field,banach,second_countable_topology}"
+  assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
+  shows "integrable (Pi\<^sub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
+proof (unfold integrable_iff_bounded, intro conjI)
+  interpret finite_product_sigma_finite M I by default fact
+  show "?f \<in> borel_measurable (Pi\<^sub>M I M)"
+    using assms by simp
+  have "(\<integral>\<^sup>+ x. ereal (norm (\<Prod>i\<in>I. f i (x i))) \<partial>Pi\<^sub>M I M) = 
+      (\<integral>\<^sup>+ x. (\<Prod>i\<in>I. ereal (norm (f i (x i)))) \<partial>Pi\<^sub>M I M)"
+    by (simp add: setprod_norm setprod_ereal)
+  also have "\<dots> = (\<Prod>i\<in>I. \<integral>\<^sup>+ x. ereal (norm (f i x)) \<partial>M i)"
+    using assms by (intro product_positive_integral_setprod) auto
+  also have "\<dots> < \<infinity>"
+    using integrable by (simp add: setprod_PInf positive_integral_positive integrable_iff_bounded)
+  finally show "(\<integral>\<^sup>+ x. ereal (norm (\<Prod>i\<in>I. f i (x i))) \<partial>Pi\<^sub>M I M) < \<infinity>" .
+qed
+
+lemma (in product_sigma_finite) product_integral_setprod:
+  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> _::{real_normed_field,banach,second_countable_topology}"
+  assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
+  shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>L (M i) (f i))"
+using assms proof induct
+  case empty
+  interpret finite_measure "Pi\<^sub>M {} M"
+    by rule (simp add: space_PiM)
+  show ?case by (simp add: space_PiM measure_def)
+next
+  case (insert i I)
+  then have iI: "finite (insert i I)" by auto
+  then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
+    integrable (Pi\<^sub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
+    by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset)
+  interpret I: finite_product_sigma_finite M I by default fact
+  have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
+    using `i \<notin> I` by (auto intro!: setprod_cong)
+  show ?case
+    unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]]
+    by (simp add: * insert prod subset_insertI)
+qed
+
+lemma integrable_subalgebra:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  assumes borel: "f \<in> borel_measurable N"
+  and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
+  shows "integrable N f \<longleftrightarrow> integrable M f" (is ?P)
+proof -
+  have "f \<in> borel_measurable M"
+    using assms by (auto simp: measurable_def)
+  with assms show ?thesis
+    using assms by (auto simp: integrable_iff_bounded positive_integral_subalgebra)
+qed
+
+lemma integral_subalgebra:
+  fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
+  assumes borel: "f \<in> borel_measurable N"
+  and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
+  shows "integral\<^sup>L N f = integral\<^sup>L M f"
+proof cases
+  assume "integrable N f"
+  then show ?thesis
+  proof induct
+    case base with assms show ?case by (auto simp: subset_eq measure_def)
+  next
+    case (add f g)
+    then have "(\<integral> a. f a + g a \<partial>N) = integral\<^sup>L M f + integral\<^sup>L M g"
+      by simp
+    also have "\<dots> = (\<integral> a. f a + g a \<partial>M)"
+      using add integrable_subalgebra[OF _ N, of f] integrable_subalgebra[OF _ N, of g] by simp
+    finally show ?case .
+  next
+    case (lim f s)
+    then have M: "\<And>i. integrable M (s i)" "integrable M f"
+      using integrable_subalgebra[OF _ N, of f] integrable_subalgebra[OF _ N, of "s i" for i] by simp_all
+    show ?case
+    proof (intro LIMSEQ_unique)
+      show "(\<lambda>i. integral\<^sup>L N (s i)) ----> integral\<^sup>L N f"
+        apply (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
+        using lim
+        apply auto
+        done
+      show "(\<lambda>i. integral\<^sup>L N (s i)) ----> integral\<^sup>L M f"
+        unfolding lim
+        apply (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
+        using lim M N(2)
+        apply auto
+        done
+    qed
+  qed
+qed (simp add: not_integrable_integral_eq integrable_subalgebra[OF assms])
+
+hide_const simple_bochner_integral
+hide_const simple_bochner_integrable
+
+end
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Borel_Space.thy
--- a/src/HOL/Probability/Borel_Space.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Borel_Space.thy	Mon May 19 12:04:45 2014 +0200
@@ -655,6 +655,9 @@
 
 subsection "Borel measurable operators"
 
+lemma borel_measurable_norm[measurable]: "norm \<in> borel_measurable borel"
+  by (intro borel_measurable_continuous_on1 continuous_intros)
+
 lemma borel_measurable_uminus[measurable (raw)]:
   fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}"
   assumes g: "g \<in> borel_measurable M"
@@ -820,20 +823,6 @@
 lemma borel_measurable_exp[measurable]: "exp \<in> borel_measurable borel"
   by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI isCont_exp)
 
-lemma measurable_count_space_eq2_countable:
-  fixes f :: "'a => 'c::countable"
-  shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
-proof -
-  { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
-    then have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)"
-      by auto
-    moreover assume "\<And>a. a\<in>A \<Longrightarrow> f -` {a} \<inter> space M \<in> sets M"
-    ultimately have "f -` X \<inter> space M \<in> sets M"
-      using `X \<subseteq> A` by (simp add: subset_eq del: UN_simps) }
-  then show ?thesis
-    unfolding measurable_def by auto
-qed
-
 lemma measurable_real_floor[measurable]:
   "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
 proof -
@@ -1166,6 +1155,37 @@
   ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff)
 qed
 
+lemma borel_measurable_LIMSEQ_metric:
+  fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b :: metric_space"
+  assumes [measurable]: "\<And>i. f i \<in> borel_measurable M"
+  assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) ----> g x"
+  shows "g \<in> borel_measurable M"
+  unfolding borel_eq_closed
+proof (safe intro!: measurable_measure_of)
+  fix A :: "'b set" assume "closed A" 
+
+  have [measurable]: "(\<lambda>x. infdist (g x) A) \<in> borel_measurable M"
+  proof (rule borel_measurable_LIMSEQ)
+    show "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. infdist (f i x) A) ----> infdist (g x) A"
+      by (intro tendsto_infdist lim)
+    show "\<And>i. (\<lambda>x. infdist (f i x) A) \<in> borel_measurable M"
+      by (intro borel_measurable_continuous_on[where f="\<lambda>x. infdist x A"]
+        continuous_at_imp_continuous_on ballI continuous_infdist isCont_ident) auto
+  qed
+
+  show "g -` A \<inter> space M \<in> sets M"
+  proof cases
+    assume "A \<noteq> {}"
+    then have "\<And>x. infdist x A = 0 \<longleftrightarrow> x \<in> A"
+      using `closed A` by (simp add: in_closed_iff_infdist_zero)
+    then have "g -` A \<inter> space M = {x\<in>space M. infdist (g x) A = 0}"
+      by auto
+    also have "\<dots> \<in> sets M"
+      by measurable
+    finally show ?thesis .
+  qed simp
+qed auto
+
 lemma sets_Collect_Cauchy[measurable]: 
   fixes f :: "nat \<Rightarrow> 'a => real"
   assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M"
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Complete_Measure.thy
--- a/src/HOL/Probability/Complete_Measure.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Complete_Measure.thy	Mon May 19 12:04:45 2014 +0200
@@ -3,7 +3,7 @@
 *)
 
 theory Complete_Measure
-imports Lebesgue_Integration
+imports Bochner_Integration
 begin
 
 definition
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Distributions.thy
--- a/src/HOL/Probability/Distributions.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Distributions.thy	Mon May 19 12:04:45 2014 +0200
@@ -162,8 +162,8 @@
   by (auto intro!: exponential_distributedI simp: one_ereal_def emeasure_eq_measure)
 
 lemma borel_integral_x_exp:
-  "(\<integral>x. x * exp (- x) * indicator {0::real ..} x \<partial>lborel) = 1"
-proof (rule integral_monotone_convergence)
+  "has_bochner_integral lborel (\<lambda>x. x * exp (- x) * indicator {0::real ..} x) 1"
+proof (rule has_bochner_integral_monotone_convergence)
   let ?f = "\<lambda>i x. x * exp (- x) * indicator {0::real .. i} x"
   have "eventually (\<lambda>b::real. 0 \<le> b) at_top"
     by (rule eventually_ge_at_top)
@@ -172,7 +172,7 @@
    fix b :: real assume [simp]: "0 \<le> b"
     have "(\<integral>x. (exp (-x)) * indicator {0 .. b} x \<partial>lborel) - (integral\<^sup>L lborel (?f b)) = 
       (\<integral>x. (exp (-x) - x * exp (-x)) * indicator {0 .. b} x \<partial>lborel)"
-      by (subst integral_diff(2)[symmetric])
+      by (subst integral_diff[symmetric])
          (auto intro!: borel_integrable_atLeastAtMost integral_cong split: split_indicator)
     also have "\<dots> = b * exp (-b) - 0 * exp (- 0)"
     proof (rule integral_FTC_atLeastAtMost)
@@ -217,9 +217,11 @@
     using `0 < l`
     by (auto split: split_indicator simp: zero_le_mult_iff exponential_density_def)
   from borel_integral_x_exp `0 < l`
-  show "(\<integral> x. exponential_density l x * x \<partial>lborel) = 1 / l"
-    by (subst (asm) lebesgue_integral_real_affine[of "l" _ 0])
-       (simp_all add: borel_measurable_exp nonzero_eq_divide_eq ac_simps)
+  have "has_bochner_integral lborel (\<lambda>x. exponential_density l x * x) (1 / l)"
+    by (subst (asm) lborel_has_bochner_integral_real_affine_iff[of l _ _ 0])
+       (simp_all add: field_simps)
+  then show "(\<integral> x. exponential_density l x * x \<partial>lborel) = 1 / l"
+    by (metis has_bochner_integral_integral_eq)
 qed simp
 
 subsection {* Uniform distribution *}
@@ -356,7 +358,9 @@
     uniform_distributed_bounds[of X a b]
     uniform_distributed_measure[of X a b]
     distributed_measurable[of M lborel X]
-  by (auto intro!: uniform_distrI_borel_atLeastAtMost simp: one_ereal_def emeasure_eq_measure)
+  by (auto intro!: uniform_distrI_borel_atLeastAtMost 
+              simp: one_ereal_def emeasure_eq_measure
+              simp del: measure_lborel)
 
 lemma (in prob_space) uniform_distributed_expectation:
   fixes a b :: real
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Finite_Product_Measure.thy
--- a/src/HOL/Probability/Finite_Product_Measure.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Finite_Product_Measure.thy	Mon May 19 12:04:45 2014 +0200
@@ -782,7 +782,7 @@
   show ?thesis
     apply (subst distr_merge[OF IJ, symmetric])
     apply (subst positive_integral_distr[OF measurable_merge f])
-    apply (subst J.positive_integral_fst_measurable(2)[symmetric, OF P_borel])
+    apply (subst J.positive_integral_fst[symmetric, OF P_borel])
     apply simp
     done
 qed
@@ -859,17 +859,6 @@
     done
 qed (simp add: space_PiM)
 
-lemma (in product_sigma_finite) product_integral_singleton:
-  assumes f: "f \<in> borel_measurable (M i)"
-  shows "(\<integral>x. f (x i) \<partial>Pi\<^sub>M {i} M) = integral\<^sup>L (M i) f"
-proof -
-  interpret I: finite_product_sigma_finite M "{i}" by default simp
-  have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
-    "(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
-    using assms by auto
-  show ?thesis
-    unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
-qed
 lemma (in product_sigma_finite) distr_component:
   "distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P")
 proof (intro measure_eqI[symmetric])
@@ -888,32 +877,6 @@
   finally show "emeasure (Pi\<^sub>M {i} M) A = emeasure ?D A" .
 qed simp
 
-lemma (in product_sigma_finite) product_integral_fold:
-  assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
-  and f: "integrable (Pi\<^sub>M (I \<union> J) M) f"
-  shows "integral\<^sup>L (Pi\<^sub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I J (x, y)) \<partial>Pi\<^sub>M J M) \<partial>Pi\<^sub>M I M)"
-proof -
-  interpret I: finite_product_sigma_finite M I by default fact
-  interpret J: finite_product_sigma_finite M J by default fact
-  have "finite (I \<union> J)" using fin by auto
-  interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
-  interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
-  let ?M = "merge I J"
-  let ?f = "\<lambda>x. f (?M x)"
-  from f have f_borel: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
-    by auto
-  have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
-    using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def)
-  have f_int: "integrable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) ?f"
-    by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f)
-  show ?thesis
-    apply (subst distr_merge[symmetric, OF IJ fin])
-    apply (subst integral_distr[OF measurable_merge f_borel])
-    apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int])
-    apply simp
-    done
-qed
-
 lemma (in product_sigma_finite)
   assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
   shows emeasure_fold_integral:
@@ -939,82 +902,6 @@
     by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong)
 qed
 
-lemma (in product_sigma_finite) product_integral_insert:
-  assumes I: "finite I" "i \<notin> I"
-    and f: "integrable (Pi\<^sub>M (insert i I) M) f"
-  shows "integral\<^sup>L (Pi\<^sub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^sub>M I M)"
-proof -
-  have "integral\<^sup>L (Pi\<^sub>M (insert i I) M) f = integral\<^sup>L (Pi\<^sub>M (I \<union> {i}) M) f"
-    by simp
-  also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I {i} (x,y)) \<partial>Pi\<^sub>M {i} M) \<partial>Pi\<^sub>M I M)"
-    using f I by (intro product_integral_fold) auto
-  also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^sub>M I M)"
-  proof (rule integral_cong, subst product_integral_singleton[symmetric])
-    fix x assume x: "x \<in> space (Pi\<^sub>M I M)"
-    have f_borel: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
-      using f by auto
-    show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
-      using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
-      unfolding comp_def .
-    from x I show "(\<integral> y. f (merge I {i} (x,y)) \<partial>Pi\<^sub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^sub>M {i} M)"
-      by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def PiE_def)
-  qed
-  finally show ?thesis .
-qed
-
-lemma (in product_sigma_finite) product_integrable_setprod:
-  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
-  assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
-  shows "integrable (Pi\<^sub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
-proof -
-  interpret finite_product_sigma_finite M I by default fact
-  have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
-    using integrable unfolding integrable_def by auto
-  have borel: "?f \<in> borel_measurable (Pi\<^sub>M I M)"
-    using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def)
-  moreover have "integrable (Pi\<^sub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
-  proof (unfold integrable_def, intro conjI)
-    show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^sub>M I M)"
-      using borel by auto
-    have "(\<integral>\<^sup>+x. ereal (abs (?f x)) \<partial>Pi\<^sub>M I M) = (\<integral>\<^sup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>Pi\<^sub>M I M)"
-      by (simp add: setprod_ereal abs_setprod)
-    also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^sup>+x. ereal (abs (f i x)) \<partial>M i))"
-      using f by (subst product_positive_integral_setprod) auto
-    also have "\<dots> < \<infinity>"
-      using integrable[THEN integrable_abs]
-      by (simp add: setprod_PInf integrable_def positive_integral_positive)
-    finally show "(\<integral>\<^sup>+x. ereal (abs (?f x)) \<partial>(Pi\<^sub>M I M)) \<noteq> \<infinity>" by auto
-    have "(\<integral>\<^sup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^sub>M I M)) = (\<integral>\<^sup>+x. 0 \<partial>(Pi\<^sub>M I M))"
-      by (intro positive_integral_cong_pos) auto
-    then show "(\<integral>\<^sup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^sub>M I M)) \<noteq> \<infinity>" by simp
-  qed
-  ultimately show ?thesis
-    by (rule integrable_abs_iff[THEN iffD1])
-qed
-
-lemma (in product_sigma_finite) product_integral_setprod:
-  fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
-  assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
-  shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>L (M i) (f i))"
-using assms proof induct
-  case empty
-  interpret finite_measure "Pi\<^sub>M {} M"
-    by rule (simp add: space_PiM)
-  show ?case by (simp add: space_PiM measure_def)
-next
-  case (insert i I)
-  then have iI: "finite (insert i I)" by auto
-  then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
-    integrable (Pi\<^sub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
-    by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset)
-  interpret I: finite_product_sigma_finite M I by default fact
-  have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
-    using `i \<notin> I` by (auto intro!: setprod_cong)
-  show ?case
-    unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]]
-    by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI)
-qed
-
 lemma sets_Collect_single:
   "i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^sub>M I M). x i \<in> A } \<in> sets (Pi\<^sub>M I M)"
   by simp
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Information.thy
--- a/src/HOL/Probability/Information.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Information.thy	Mon May 19 12:04:45 2014 +0200
@@ -79,37 +79,26 @@
 definition
   "KL_divergence b M N = integral\<^sup>L N (entropy_density b M N)"
 
-lemma (in information_space) measurable_entropy_density:
-  assumes ac: "absolutely_continuous M N" "sets N = events"
-  shows "entropy_density b M N \<in> borel_measurable M"
-proof -
-  from borel_measurable_RN_deriv[OF ac] b_gt_1 show ?thesis
-    unfolding entropy_density_def by auto
-qed
-
-lemma borel_measurable_RN_deriv_density[measurable (raw)]:
-  "f \<in> borel_measurable M \<Longrightarrow> RN_deriv M (density M f) \<in> borel_measurable M"
-  using borel_measurable_RN_deriv_density[of "\<lambda>x. max 0 (f x )" M]
-  by (simp add: density_max_0[symmetric])
+lemma measurable_entropy_density[measurable]: "entropy_density b M N \<in> borel_measurable M"
+  unfolding entropy_density_def by auto
 
 lemma (in sigma_finite_measure) KL_density:
   fixes f :: "'a \<Rightarrow> real"
   assumes "1 < b"
-  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+  assumes f[measurable]: "f \<in> borel_measurable M" and nn: "AE x in M. 0 \<le> f x"
   shows "KL_divergence b M (density M f) = (\<integral>x. f x * log b (f x) \<partial>M)"
   unfolding KL_divergence_def
-proof (subst integral_density)
-  show "entropy_density b M (density M (\<lambda>x. ereal (f x))) \<in> borel_measurable M"
+proof (subst integral_real_density)
+  show [measurable]: "entropy_density b M (density M (\<lambda>x. ereal (f x))) \<in> borel_measurable M"
     using f
     by (auto simp: comp_def entropy_density_def)
   have "density M (RN_deriv M (density M f)) = density M f"
-    using f by (intro density_RN_deriv_density) auto
+    using f nn by (intro density_RN_deriv_density) auto
   then have eq: "AE x in M. RN_deriv M (density M f) x = f x"
-    using f
-    by (intro density_unique)
-       (auto intro!: borel_measurable_log borel_measurable_RN_deriv_density simp: RN_deriv_density_nonneg)
+    using f nn by (intro density_unique) (auto simp: RN_deriv_nonneg)
   show "(\<integral>x. f x * entropy_density b M (density M (\<lambda>x. ereal (f x))) x \<partial>M) = (\<integral>x. f x * log b (f x) \<partial>M)"
     apply (intro integral_cong_AE)
+    apply measurable
     using eq
     apply eventually_elim
     apply (auto simp: entropy_density_def)
@@ -161,8 +150,10 @@
   then have D_pos: "(\<integral>\<^sup>+ x. ereal (D x) \<partial>M) = 1"
     using N.emeasure_space_1 by simp
 
-  have "integrable M D" "integral\<^sup>L M D = 1"
-    using D D_pos D_neg unfolding integrable_def lebesgue_integral_def by simp_all
+  have "integrable M D"
+    using D D_pos D_neg unfolding real_integrable_def real_lebesgue_integral_def by simp_all
+  then have "integral\<^sup>L M D = 1"
+    using D D_pos D_neg by (simp add: real_lebesgue_integral_def)
 
   have "0 \<le> 1 - measure M ?D_set"
     using prob_le_1 by (auto simp: field_simps)
@@ -172,10 +163,9 @@
   also have "\<dots> < (\<integral> x. D x * (ln b * log b (D x)) \<partial>M)"
   proof (rule integral_less_AE)
     show "integrable M (\<lambda>x. D x - indicator ?D_set x)"
-      using `integrable M D`
-      by (intro integral_diff integral_indicator) auto
+      using `integrable M D` by auto
   next
-    from integral_cmult(1)[OF int, of "ln b"]
+    from integrable_mult_left(1)[OF int, of "ln b"]
     show "integrable M (\<lambda>x. D x * (ln b * log b (D x)))" 
       by (simp add: ac_simps)
   next
@@ -242,7 +232,7 @@
   also have "\<dots> = (\<integral> x. ln b * (D x * log b (D x)) \<partial>M)"
     by (simp add: ac_simps)
   also have "\<dots> = ln b * (\<integral> x. D x * log b (D x) \<partial>M)"
-    using int by (rule integral_cmult)
+    using int by simp
   finally show ?thesis
     using b_gt_1 D by (subst KL_density) (auto simp: zero_less_mult_iff)
 qed
@@ -260,7 +250,7 @@
   qed
   then have "AE x in M. log b (real (RN_deriv M M x)) = 0"
     by (elim AE_mp) simp
-  from integral_cong_AE[OF this]
+  from integral_cong_AE[OF _ _ this]
   have "integral\<^sup>L M (entropy_density b M M) = 0"
     by (simp add: entropy_density_def comp_def)
   then show "KL_divergence b M M = 0"
@@ -291,7 +281,7 @@
   have "N = density M (RN_deriv M N)"
     using ac by (rule density_RN_deriv[symmetric])
   also have "\<dots> = density M D"
-    using borel_measurable_RN_deriv[OF ac] D by (auto intro!: density_cong)
+    using D by (auto intro!: density_cong)
   finally have N: "N = density M D" .
 
   from absolutely_continuous_AE[OF ac(2,1) D(2)] D b_gt_1 ac measurable_entropy_density
@@ -299,7 +289,7 @@
     by (intro integrable_cong_AE[THEN iffD2, OF _ _ _ int])
        (auto simp: N entropy_density_def)
   with D b_gt_1 have "integrable M (\<lambda>x. D x * log b (D x))"
-    by (subst integral_density(2)[symmetric]) (auto simp: N[symmetric] comp_def)
+    by (subst integrable_real_density[symmetric]) (auto simp: N[symmetric] comp_def)
   with `prob_space N` D show ?thesis
     unfolding N
     by (intro KL_eq_0_iff_eq) auto
@@ -335,7 +325,7 @@
       using f g by (auto simp: AE_density)
     show "integrable (density M f) (\<lambda>x. g x / f x * log b (g x / f x))"
       using `1 < b` f g ac
-      by (subst integral_density)
+      by (subst integrable_density)
          (auto intro!: integrable_cong_AE[THEN iffD2, OF _ _ _ int] measurable_If)
   qed
   also have "\<dots> = KL_divergence b (density M f) (density M g)"
@@ -414,7 +404,7 @@
   "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow>
     integrable N (\<lambda>x. f x * g x) \<longleftrightarrow> integrable M (\<lambda>x. g (X x))"
   by (auto simp: distributed_real_AE
-                    distributed_distr_eq_density[symmetric] integral_density[symmetric] integrable_distr_eq)
+                    distributed_distr_eq_density[symmetric] integrable_real_density[symmetric] integrable_distr_eq)
   
 lemma distributed_transform_integrable:
   assumes Px: "distributed M N X Px"
@@ -431,8 +421,9 @@
   finally show ?thesis .
 qed
 
-lemma integrable_cong_AE_imp: "integrable M g \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (AE x in M. g x = f x) \<Longrightarrow> integrable M f"
-  using integrable_cong_AE by blast
+lemma integrable_cong_AE_imp:
+  "integrable M g \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (AE x in M. g x = f x) \<Longrightarrow> integrable M f"
+  using integrable_cong_AE[of f M g] by (auto simp: eq_commute)
 
 lemma (in information_space) finite_entropy_integrable:
   "finite_entropy S X Px \<Longrightarrow> integrable S (\<lambda>x. Px x * log b (Px x))"
@@ -485,7 +476,7 @@
 
     show ac: "absolutely_continuous P Q" by (simp add: absolutely_continuous_def)
     then have ed: "entropy_density b P Q \<in> borel_measurable P"
-      by (rule P.measurable_entropy_density) simp
+      by simp
 
     have "AE x in P. 1 = RN_deriv P Q x"
     proof (rule P.RN_deriv_unique)
@@ -494,13 +485,15 @@
     qed auto
     then have ae_0: "AE x in P. entropy_density b P Q x = 0"
       by eventually_elim (auto simp: entropy_density_def)
-    then have "integrable P (entropy_density b P Q) \<longleftrightarrow> integrable Q (\<lambda>x. 0)"
+    then have "integrable P (entropy_density b P Q) \<longleftrightarrow> integrable Q (\<lambda>x. 0::real)"
       using ed unfolding `Q = P` by (intro integrable_cong_AE) auto
     then show "integrable Q (entropy_density b P Q)" by simp
 
-    show "mutual_information b S T X Y = 0"
+    from ae_0 have "mutual_information b S T X Y = (\<integral>x. 0 \<partial>P)"
       unfolding mutual_information_def KL_divergence_def P_def[symmetric] Q_def[symmetric] `Q = P`
-      using ae_0 by (simp cong: integral_cong_AE) }
+      by (intro integral_cong_AE) auto
+    then show "mutual_information b S T X Y = 0"
+      by simp }
 
   { assume ac: "absolutely_continuous P Q"
     assume int: "integrable Q (entropy_density b P Q)"
@@ -722,8 +715,8 @@
 lemma (in information_space)
   fixes Pxy :: "'b \<times> 'c \<Rightarrow> real" and Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
   assumes "sigma_finite_measure S" "sigma_finite_measure T"
-  assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
-  assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+  assumes Px[measurable]: "distributed M S X Px" and Py[measurable]: "distributed M T Y Py"
+  assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
   assumes ae: "AE x in S. AE y in T. Pxy (x, y) = Px x * Py y"
   shows mutual_information_eq_0: "mutual_information b S T X Y = 0"
 proof -
@@ -743,7 +736,7 @@
   ultimately have "AE x in S \<Otimes>\<^sub>M T. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) = 0"
     by eventually_elim simp
   then have "(\<integral>x. Pxy x * log b (Pxy x / (Px (fst x) * Py (snd x))) \<partial>(S \<Otimes>\<^sub>M T)) = (\<integral>x. 0 \<partial>(S \<Otimes>\<^sub>M T))"
-    by (rule integral_cong_AE)
+    by (intro integral_cong_AE) auto
   then show ?thesis
     by (subst mutual_information_distr[OF assms(1-5)]) simp
 qed
@@ -810,7 +803,7 @@
 
 lemma (in information_space)
   fixes X :: "'a \<Rightarrow> 'b"
-  assumes X: "distributed M MX X f"
+  assumes X[measurable]: "distributed M MX X f"
   shows entropy_distr: "entropy b MX X = - (\<integral>x. f x * log b (f x) \<partial>MX)" (is ?eq)
 proof -
   note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X]
@@ -825,18 +818,15 @@
     apply auto
     done
 
-  have int_eq: "- (\<integral> x. log b (f x) \<partial>distr M MX X) = - (\<integral> x. f x * log b (f x) \<partial>MX)"
+  have int_eq: "(\<integral> x. f x * log b (f x) \<partial>MX) = (\<integral> x. log b (f x) \<partial>distr M MX X)"
     unfolding distributed_distr_eq_density[OF X]
     using D
     by (subst integral_density)
        (auto simp: borel_measurable_ereal_iff)
 
   show ?eq
-    unfolding entropy_def KL_divergence_def entropy_density_def comp_def
-    apply (subst integral_cong_AE)
-    apply (rule ae_eq)
-    apply (rule int_eq)
-    done
+    unfolding entropy_def KL_divergence_def entropy_density_def comp_def int_eq neg_equal_iff_equal
+    using ae_eq by (intro integral_cong_AE) auto
 qed
 
 lemma (in prob_space) distributed_imp_emeasure_nonzero:
@@ -861,7 +851,7 @@
 
 lemma (in information_space) entropy_le:
   fixes Px :: "'b \<Rightarrow> real" and MX :: "'b measure"
-  assumes X: "distributed M MX X Px"
+  assumes X[measurable]: "distributed M MX X Px"
   and fin: "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> \<infinity>"
   and int: "integrable MX (\<lambda>x. - Px x * log b (Px x))"
   shows "entropy b MX X \<le> log b (measure MX {x \<in> space MX. Px x \<noteq> 0})"
@@ -873,7 +863,7 @@
   have " - log b (measure MX {x \<in> space MX. Px x \<noteq> 0}) = 
     - log b (\<integral> x. indicator {x \<in> space MX. Px x \<noteq> 0} x \<partial>MX)"
     using Px fin
-    by (subst integral_indicator) (auto simp: measure_def borel_measurable_ereal_iff)
+    by (subst integral_real_indicator) (auto simp: measure_def borel_measurable_ereal_iff)
   also have "- log b (\<integral> x. indicator {x \<in> space MX. Px x \<noteq> 0} x \<partial>MX) = - log b (\<integral> x. 1 / Px x \<partial>distr M MX X)"
     unfolding distributed_distr_eq_density[OF X] using Px
     apply (intro arg_cong[where f="log b"] arg_cong[where f=uminus])
@@ -889,7 +879,7 @@
       by (intro positive_integral_cong) (auto split: split_max)
     then show "integrable (distr M MX X) (\<lambda>x. 1 / Px x)"
       unfolding distributed_distr_eq_density[OF X] using Px
-      by (auto simp: positive_integral_density integrable_def borel_measurable_ereal_iff fin positive_integral_max_0
+      by (auto simp: positive_integral_density real_integrable_def borel_measurable_ereal_iff fin positive_integral_max_0
               cong: positive_integral_cong)
     have "integrable MX (\<lambda>x. Px x * log b (1 / Px x)) =
       integrable MX (\<lambda>x. - Px x * log b (Px x))"
@@ -900,7 +890,7 @@
     then show "integrable (distr M MX X) (\<lambda>x. - log b (1 / Px x))"
       unfolding distributed_distr_eq_density[OF X]
       using Px int
-      by (subst integral_density) (auto simp: borel_measurable_ereal_iff)
+      by (subst integrable_real_density) (auto simp: borel_measurable_ereal_iff)
   qed (auto simp: minus_log_convex[OF b_gt_1])
   also have "\<dots> = (\<integral> x. log b (Px x) \<partial>distr M MX X)"
     unfolding distributed_distr_eq_density[OF X] using Px
@@ -940,11 +930,11 @@
   have eq: "(\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) =
     (\<integral> x. (- log b (measure MX A) / measure MX A) * indicator A x \<partial>MX)"
     using measure_nonneg[of MX A] uniform_distributed_params[OF X]
-    by (auto intro!: integral_cong split: split_indicator simp: log_divide_eq)
+    by (intro integral_cong) (auto split: split_indicator simp: log_divide_eq)
   show "- (\<integral> x. indicator A x / measure MX A * log b (indicator A x / measure MX A) \<partial>MX) =
     log b (measure MX A)"
     unfolding eq using uniform_distributed_params[OF X]
-    by (subst lebesgue_integral_cmult) (auto simp: measure_def)
+    by (subst integral_mult_right) (auto simp: measure_def)
 qed
 
 lemma (in information_space) entropy_simple_distributed:
@@ -1068,7 +1058,7 @@
     unfolding conditional_mutual_information_def
     apply (subst mi_eq)
     apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz])
-    apply (subst integral_diff(2)[symmetric])
+    apply (subst integral_diff[symmetric])
     apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff)
     done
 
@@ -1104,7 +1094,7 @@
     apply auto
     done
   also have "\<dots> = (\<integral>\<^sup>+(y, z). \<integral>\<^sup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^sub>M P)"
-    by (subst STP.positive_integral_snd_measurable[symmetric]) (auto simp add: split_beta')
+    by (subst STP.positive_integral_snd[symmetric]) (auto simp add: split_beta')
   also have "\<dots> = (\<integral>\<^sup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^sub>M P)"
     apply (rule positive_integral_cong_AE)
     using aeX1 aeX2 aeX3 distributed_AE[OF Pyz] AE_space
@@ -1123,7 +1113,7 @@
   also have "\<dots> < \<infinity>" by simp
   finally have fin: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>" by simp
 
-  have pos: "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> 0"
+  have pos: "(\<integral>\<^sup>+x. ?f x \<partial>?P) \<noteq> 0"
     apply (subst positive_integral_density)
     apply simp
     apply (rule distributed_AE[OF Pxyz])
@@ -1131,7 +1121,7 @@
     apply (simp add: split_beta')
   proof
     let ?g = "\<lambda>x. ereal (if Pxyz x = 0 then 0 else Pxz (fst x, snd (snd x)) * Pyz (snd x) / Pz (snd (snd x)))"
-    assume "(\<integral>\<^sup>+ x. ?g x \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)) = 0"
+    assume "(\<integral>\<^sup>+x. ?g x \<partial>(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P)) = 0"
     then have "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. ?g x \<le> 0"
       by (intro positive_integral_0_iff_AE[THEN iffD1]) auto
     then have "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxyz x = 0"
@@ -1154,19 +1144,32 @@
     done
 
   have I3: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
-    apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integral_diff(1)[OF I1 I2]])
+    apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integrable_diff[OF I1 I2]])
     using ae
     apply (auto simp: split_beta')
     done
 
   have "- log b 1 \<le> - log b (integral\<^sup>L ?P ?f)"
   proof (intro le_imp_neg_le log_le[OF b_gt_1])
-    show "0 < integral\<^sup>L ?P ?f"
-      using neg pos fin positive_integral_positive[of ?P ?f]
-      by (cases "(\<integral>\<^sup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def less_le split_beta')
-    show "integral\<^sup>L ?P ?f \<le> 1"
-      using neg le1 fin positive_integral_positive[of ?P ?f]
-      by (cases "(\<integral>\<^sup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def split_beta' one_ereal_def)
+    have If: "integrable ?P ?f"
+      unfolding real_integrable_def
+    proof (intro conjI)
+      from neg show "(\<integral>\<^sup>+ x. - ?f x \<partial>?P) \<noteq> \<infinity>"
+        by simp
+      from fin show "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>"
+        by simp
+    qed simp
+    then have "(\<integral>\<^sup>+ x. ?f x \<partial>?P) = (\<integral>x. ?f x \<partial>?P)"
+      apply (rule positive_integral_eq_integral)
+      apply (subst AE_density)
+      apply simp
+      using ae5 ae6 ae7 ae8
+      apply eventually_elim
+      apply auto
+      done
+    with positive_integral_positive[of ?P ?f] pos le1
+    show "0 < (\<integral>x. ?f x \<partial>?P)" "(\<integral>x. ?f x \<partial>?P) \<le> 1"
+      by (simp_all add: one_ereal_def)
   qed
   also have "- log b (integral\<^sup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)"
   proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"])
@@ -1175,10 +1178,10 @@
       using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
       by eventually_elim (auto)
     show "integrable ?P ?f"
-      unfolding integrable_def 
+      unfolding real_integrable_def 
       using fin neg by (auto simp: split_beta')
     show "integrable ?P (\<lambda>x. - log b (?f x))"
-      apply (subst integral_density)
+      apply (subst integrable_real_density)
       apply simp
       apply (auto intro!: distributed_real_AE[OF Pxyz]) []
       apply simp
@@ -1192,13 +1195,12 @@
   qed (auto simp: b_gt_1 minus_log_convex)
   also have "\<dots> = conditional_mutual_information b S T P X Y Z"
     unfolding `?eq`
-    apply (subst integral_density)
+    apply (subst integral_real_density)
     apply simp
     apply (auto intro!: distributed_real_AE[OF Pxyz]) []
     apply simp
     apply (intro integral_cong_AE)
     using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
-    apply eventually_elim
     apply (auto simp: log_divide_eq zero_less_mult_iff zero_less_divide_iff field_simps)
     done
   finally show ?nonneg
@@ -1316,7 +1318,7 @@
     unfolding conditional_mutual_information_def
     apply (subst mi_eq)
     apply (subst mutual_information_distr[OF S TP Px Pyz Pxyz])
-    apply (subst integral_diff(2)[symmetric])
+    apply (subst integral_diff[symmetric])
     apply (auto intro!: integral_cong_AE simp: split_beta' simp del: integral_diff)
     done
 
@@ -1349,8 +1351,7 @@
     apply auto
     done
   also have "\<dots> = (\<integral>\<^sup>+(y, z). \<integral>\<^sup>+ x. ereal (Pxz (x, z)) * ereal (Pyz (y, z) / Pz z) \<partial>S \<partial>T \<Otimes>\<^sub>M P)"
-    by (subst STP.positive_integral_snd_measurable[symmetric])
-       (auto simp add: split_beta')
+    by (subst STP.positive_integral_snd[symmetric]) (auto simp add: split_beta')
   also have "\<dots> = (\<integral>\<^sup>+x. ereal (Pyz x) * 1 \<partial>T \<Otimes>\<^sub>M P)"
     apply (rule positive_integral_cong_AE)
     using aeX1 aeX2 aeX3 distributed_AE[OF Pyz] AE_space
@@ -1399,19 +1400,32 @@
     done
 
   have I3: "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y,z) / Pz z))))"
-    apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integral_diff(1)[OF I1 I2]])
+    apply (rule integrable_cong_AE[THEN iffD1, OF _ _ _ integrable_diff[OF I1 I2]])
     using ae
     apply (auto simp: split_beta')
     done
 
   have "- log b 1 \<le> - log b (integral\<^sup>L ?P ?f)"
   proof (intro le_imp_neg_le log_le[OF b_gt_1])
-    show "0 < integral\<^sup>L ?P ?f"
-      using neg pos fin positive_integral_positive[of ?P ?f]
-      by (cases "(\<integral>\<^sup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def less_le split_beta')
-    show "integral\<^sup>L ?P ?f \<le> 1"
-      using neg le1 fin positive_integral_positive[of ?P ?f]
-      by (cases "(\<integral>\<^sup>+ x. ?f x \<partial>?P)") (auto simp add: lebesgue_integral_def split_beta' one_ereal_def)
+    have If: "integrable ?P ?f"
+      unfolding real_integrable_def
+    proof (intro conjI)
+      from neg show "(\<integral>\<^sup>+ x. - ?f x \<partial>?P) \<noteq> \<infinity>"
+        by simp
+      from fin show "(\<integral>\<^sup>+ x. ?f x \<partial>?P) \<noteq> \<infinity>"
+        by simp
+    qed simp
+    then have "(\<integral>\<^sup>+ x. ?f x \<partial>?P) = (\<integral>x. ?f x \<partial>?P)"
+      apply (rule positive_integral_eq_integral)
+      apply (subst AE_density)
+      apply simp
+      using ae5 ae6 ae7 ae8
+      apply eventually_elim
+      apply auto
+      done
+    with positive_integral_positive[of ?P ?f] pos le1
+    show "0 < (\<integral>x. ?f x \<partial>?P)" "(\<integral>x. ?f x \<partial>?P) \<le> 1"
+      by (simp_all add: one_ereal_def)
   qed
   also have "- log b (integral\<^sup>L ?P ?f) \<le> (\<integral> x. - log b (?f x) \<partial>?P)"
   proof (rule P.jensens_inequality[where a=0 and b=1 and I="{0<..}"])
@@ -1420,10 +1434,10 @@
       using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
       by eventually_elim (auto)
     show "integrable ?P ?f"
-      unfolding integrable_def
+      unfolding real_integrable_def
       using fin neg by (auto simp: split_beta')
     show "integrable ?P (\<lambda>x. - log b (?f x))"
-      apply (subst integral_density)
+      apply (subst integrable_real_density)
       apply simp
       apply (auto intro!: distributed_real_AE[OF Pxyz]) []
       apply simp
@@ -1437,13 +1451,12 @@
   qed (auto simp: b_gt_1 minus_log_convex)
   also have "\<dots> = conditional_mutual_information b S T P X Y Z"
     unfolding `?eq`
-    apply (subst integral_density)
+    apply (subst integral_real_density)
     apply simp
     apply (auto intro!: distributed_real_AE[OF Pxyz]) []
     apply simp
     apply (intro integral_cong_AE)
     using ae1 ae2 ae3 ae4 ae5 ae6 ae7 ae8 Pxyz[THEN distributed_real_AE]
-    apply eventually_elim
     apply (auto simp: log_divide_eq zero_less_mult_iff zero_less_divide_iff field_simps)
     done
   finally show ?nonneg
@@ -1532,7 +1545,7 @@
   "\<H>(X | Y) \<equiv> conditional_entropy b (count_space (X`space M)) (count_space (Y`space M)) X Y"
 
 lemma (in information_space) conditional_entropy_generic_eq:
-  fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
+  fixes Pxy :: "_ \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
   assumes Py[measurable]: "distributed M T Y Py"
   assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
@@ -1552,19 +1565,15 @@
     unfolding AE_density[OF distributed_borel_measurable, OF Pxy]
     unfolding distributed_distr_eq_density[OF Py]
     apply (rule ST.AE_pair_measure)
-    apply (auto intro!: sets.sets_Collect borel_measurable_eq measurable_compose[OF _ distributed_real_measurable[OF Py]]
-                        distributed_real_measurable[OF Pxy] distributed_real_AE[OF Py]
-                        borel_measurable_real_of_ereal measurable_compose[OF _ borel_measurable_RN_deriv_density])
+    apply auto
     using distributed_RN_deriv[OF Py]
     apply auto
     done    
   ultimately
   have "conditional_entropy b S T X Y = - (\<integral>x. Pxy x * log b (Pxy x / Py (snd x)) \<partial>(S \<Otimes>\<^sub>M T))"
     unfolding conditional_entropy_def neg_equal_iff_equal
-    apply (subst integral_density(1)[symmetric])
-    apply (auto simp: distributed_real_measurable[OF Pxy] distributed_real_AE[OF Pxy]
-                      measurable_compose[OF _ distributed_real_measurable[OF Py]]
-                      distributed_distr_eq_density[OF Pxy]
+    apply (subst integral_real_density[symmetric])
+    apply (auto simp: distributed_real_AE[OF Pxy] distributed_distr_eq_density[OF Pxy]
                 intro!: integral_cong_AE)
     done
   then show ?thesis by (simp add: split_beta')
@@ -1573,8 +1582,8 @@
 lemma (in information_space) conditional_entropy_eq_entropy:
   fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real"
   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
-  assumes Py: "distributed M T Y Py"
-  assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+  assumes Py[measurable]: "distributed M T Y Py"
+  assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
   assumes I1: "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
   assumes I2: "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
   shows "conditional_entropy b S T X Y = entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) - entropy b T Y"
@@ -1606,7 +1615,6 @@
     unfolding conditional_entropy_generic_eq[OF S T Py Pxy] neg_equal_iff_equal
     apply (intro integral_cong_AE)
     using ae
-    apply eventually_elim
     apply auto
     done
   also have "\<dots> = - (\<integral>x. Pxy x * log b (Pxy x) \<partial>(S \<Otimes>\<^sub>M T)) - - (\<integral>x.  Pxy x * log b (Py (snd x)) \<partial>(S \<Otimes>\<^sub>M T))"
@@ -1702,8 +1710,8 @@
 lemma (in information_space) mutual_information_eq_entropy_conditional_entropy_distr:
   fixes Px :: "'b \<Rightarrow> real" and Py :: "'c \<Rightarrow> real" and Pxy :: "('b \<times> 'c) \<Rightarrow> real"
   assumes S: "sigma_finite_measure S" and T: "sigma_finite_measure T"
-  assumes Px: "distributed M S X Px" and Py: "distributed M T Y Py"
-  assumes Pxy: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
+  assumes Px[measurable]: "distributed M S X Px" and Py[measurable]: "distributed M T Y Py"
+  assumes Pxy[measurable]: "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
   assumes Ix: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Px (fst x)))"
   assumes Iy: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
   assumes Ixy: "integrable(S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
@@ -1758,13 +1766,13 @@
   have "entropy b S X + entropy b T Y - entropy b (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) = integral\<^sup>L (S \<Otimes>\<^sub>M T) ?f"
     unfolding X Y XY
     apply (subst integral_diff)
-    apply (intro integral_diff Ixy Ix Iy)+
+    apply (intro integrable_diff Ixy Ix Iy)+
     apply (subst integral_diff)
-    apply (intro integral_diff Ixy Ix Iy)+
+    apply (intro Ixy Ix Iy)+
     apply (simp add: field_simps)
     done
   also have "\<dots> = integral\<^sup>L (S \<Otimes>\<^sub>M T) ?g"
-    using `AE x in _. ?f x = ?g x` by (rule integral_cong_AE)
+    using `AE x in _. ?f x = ?g x` by (intro integral_cong_AE) auto
   also have "\<dots> = mutual_information b S T X Y"
     by (rule mutual_information_distr[OF S T Px Py Pxy, symmetric])
   finally show ?thesis ..
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Lebesgue_Integration.thy
--- a/src/HOL/Probability/Lebesgue_Integration.thy	Mon May 19 11:27:02 2014 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2900 +0,0 @@
-(*  Title:      HOL/Probability/Lebesgue_Integration.thy
-    Author:     Johannes Hölzl, TU München
-    Author:     Armin Heller, TU München
-*)
-
-header {*Lebesgue Integration*}
-
-theory Lebesgue_Integration
-  imports Measure_Space Borel_Space
-begin
-
-lemma indicator_less_ereal[simp]:
-  "indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)"
-  by (simp add: indicator_def not_le)
-
-section "Simple function"
-
-text {*
-
-Our simple functions are not restricted to positive real numbers. Instead
-they are just functions with a finite range and are measurable when singleton
-sets are measurable.
-
-*}
-
-definition "simple_function M g \<longleftrightarrow>
-    finite (g ` space M) \<and>
-    (\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
-
-lemma simple_functionD:
-  assumes "simple_function M g"
-  shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
-proof -
-  show "finite (g ` space M)"
-    using assms unfolding simple_function_def by auto
-  have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
-  also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
-  finally show "g -` X \<inter> space M \<in> sets M" using assms
-    by (auto simp del: UN_simps simp: simple_function_def)
-qed
-
-lemma measurable_simple_function[measurable_dest]:
-  "simple_function M f \<Longrightarrow> f \<in> measurable M (count_space UNIV)"
-  unfolding simple_function_def measurable_def
-proof safe
-  fix A assume "finite (f ` space M)" "\<forall>x\<in>f ` space M. f -` {x} \<inter> space M \<in> sets M"
-  then have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) \<in> sets M"
-    by (intro sets.finite_UN) auto
-  also have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) = f -` A \<inter> space M"
-    by (auto split: split_if_asm)
-  finally show "f -` A \<inter> space M \<in> sets M" .
-qed simp
-
-lemma borel_measurable_simple_function:
-  "simple_function M f \<Longrightarrow> f \<in> borel_measurable M"
-  by (auto dest!: measurable_simple_function simp: measurable_def)
-
-lemma simple_function_measurable2[intro]:
-  assumes "simple_function M f" "simple_function M g"
-  shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
-proof -
-  have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
-    by auto
-  then show ?thesis using assms[THEN simple_functionD(2)] by auto
-qed
-
-lemma simple_function_indicator_representation:
-  fixes f ::"'a \<Rightarrow> ereal"
-  assumes f: "simple_function M f" and x: "x \<in> space M"
-  shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
-  (is "?l = ?r")
-proof -
-  have "?r = (\<Sum>y \<in> f ` space M.
-    (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
-    by (auto intro!: setsum_cong2)
-  also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
-    using assms by (auto dest: simple_functionD simp: setsum_delta)
-  also have "... = f x" using x by (auto simp: indicator_def)
-  finally show ?thesis by auto
-qed
-
-lemma simple_function_notspace:
-  "simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h")
-proof -
-  have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
-  hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
-  have "?h -` {0} \<inter> space M = space M" by auto
-  thus ?thesis unfolding simple_function_def by auto
-qed
-
-lemma simple_function_cong:
-  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
-  shows "simple_function M f \<longleftrightarrow> simple_function M g"
-proof -
-  have "f ` space M = g ` space M"
-    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
-    using assms by (auto intro!: image_eqI)
-  thus ?thesis unfolding simple_function_def using assms by simp
-qed
-
-lemma simple_function_cong_algebra:
-  assumes "sets N = sets M" "space N = space M"
-  shows "simple_function M f \<longleftrightarrow> simple_function N f"
-  unfolding simple_function_def assms ..
-
-lemma simple_function_borel_measurable:
-  fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
-  assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
-  shows "simple_function M f"
-  using assms unfolding simple_function_def
-  by (auto intro: borel_measurable_vimage)
-
-lemma simple_function_eq_measurable:
-  fixes f :: "'a \<Rightarrow> ereal"
-  shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> measurable M (count_space UNIV)"
-  using simple_function_borel_measurable[of f] measurable_simple_function[of M f]
-  by (fastforce simp: simple_function_def)
-
-lemma simple_function_const[intro, simp]:
-  "simple_function M (\<lambda>x. c)"
-  by (auto intro: finite_subset simp: simple_function_def)
-lemma simple_function_compose[intro, simp]:
-  assumes "simple_function M f"
-  shows "simple_function M (g \<circ> f)"
-  unfolding simple_function_def
-proof safe
-  show "finite ((g \<circ> f) ` space M)"
-    using assms unfolding simple_function_def by (auto simp: image_comp [symmetric])
-next
-  fix x assume "x \<in> space M"
-  let ?G = "g -` {g (f x)} \<inter> (f`space M)"
-  have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
-    (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
-  show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
-    using assms unfolding simple_function_def *
-    by (rule_tac sets.finite_UN) auto
-qed
-
-lemma simple_function_indicator[intro, simp]:
-  assumes "A \<in> sets M"
-  shows "simple_function M (indicator A)"
-proof -
-  have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
-    by (auto simp: indicator_def)
-  hence "finite ?S" by (rule finite_subset) simp
-  moreover have "- A \<inter> space M = space M - A" by auto
-  ultimately show ?thesis unfolding simple_function_def
-    using assms by (auto simp: indicator_def [abs_def])
-qed
-
-lemma simple_function_Pair[intro, simp]:
-  assumes "simple_function M f"
-  assumes "simple_function M g"
-  shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
-  unfolding simple_function_def
-proof safe
-  show "finite (?p ` space M)"
-    using assms unfolding simple_function_def
-    by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
-next
-  fix x assume "x \<in> space M"
-  have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
-      (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
-    by auto
-  with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
-    using assms unfolding simple_function_def by auto
-qed
-
-lemma simple_function_compose1:
-  assumes "simple_function M f"
-  shows "simple_function M (\<lambda>x. g (f x))"
-  using simple_function_compose[OF assms, of g]
-  by (simp add: comp_def)
-
-lemma simple_function_compose2:
-  assumes "simple_function M f" and "simple_function M g"
-  shows "simple_function M (\<lambda>x. h (f x) (g x))"
-proof -
-  have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
-    using assms by auto
-  thus ?thesis by (simp_all add: comp_def)
-qed
-
-lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
-  and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
-  and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
-  and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
-  and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
-  and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
-  and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
-
-lemma simple_function_setsum[intro, simp]:
-  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
-  shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
-proof cases
-  assume "finite P" from this assms show ?thesis by induct auto
-qed auto
-
-lemma simple_function_ereal[intro, simp]: 
-  fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
-  shows "simple_function M (\<lambda>x. ereal (f x))"
-  by (auto intro!: simple_function_compose1[OF sf])
-
-lemma simple_function_real_of_nat[intro, simp]: 
-  fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
-  shows "simple_function M (\<lambda>x. real (f x))"
-  by (auto intro!: simple_function_compose1[OF sf])
-
-lemma borel_measurable_implies_simple_function_sequence:
-  fixes u :: "'a \<Rightarrow> ereal"
-  assumes u: "u \<in> borel_measurable M"
-  shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
-             (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
-proof -
-  def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)"
-  { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
-    proof (split split_if, intro conjI impI)
-      assume "\<not> real j \<le> u x"
-      then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
-         by (cases "u x") (auto intro!: natfloor_mono)
-      moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
-        by (intro real_natfloor_le) auto
-      ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
-        unfolding real_of_nat_le_iff by auto
-    qed auto }
-  note f_upper = this
-
-  have real_f:
-    "\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (natfloor (real (u x) * 2 ^ i)))"
-    unfolding f_def by auto
-
-  let ?g = "\<lambda>j x. real (f x j) / 2^j :: ereal"
-  show ?thesis
-  proof (intro exI[of _ ?g] conjI allI ballI)
-    fix i
-    have "simple_function M (\<lambda>x. real (f x i))"
-    proof (intro simple_function_borel_measurable)
-      show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
-        using u by (auto simp: real_f)
-      have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
-        using f_upper[of _ i] by auto
-      then show "finite ((\<lambda>x. real (f x i))`space M)"
-        by (rule finite_subset) auto
-    qed
-    then show "simple_function M (?g i)"
-      by (auto intro: simple_function_ereal simple_function_div)
-  next
-    show "incseq ?g"
-    proof (intro incseq_ereal incseq_SucI le_funI)
-      fix x and i :: nat
-      have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
-      proof ((split split_if)+, intro conjI impI)
-        assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
-        then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)"
-          by (cases "u x") (auto intro!: le_natfloor)
-      next
-        assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x"
-        then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i"
-          by (cases "u x") auto
-      next
-        assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
-        have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
-          by simp
-        also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
-        proof cases
-          assume "0 \<le> u x" then show ?thesis
-            by (intro le_mult_natfloor) 
-        next
-          assume "\<not> 0 \<le> u x" then show ?thesis
-            by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
-        qed
-        also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
-          by (simp add: ac_simps)
-        finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" .
-      qed simp
-      then show "?g i x \<le> ?g (Suc i) x"
-        by (auto simp: field_simps)
-    qed
-  next
-    fix x show "(SUP i. ?g i x) = max 0 (u x)"
-    proof (rule SUP_eqI)
-      fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
-        by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
-                                     mult_nonpos_nonneg)
-    next
-      fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
-      have "\<And>i. 0 \<le> ?g i x" by auto
-      from order_trans[OF this *] have "0 \<le> y" by simp
-      show "max 0 (u x) \<le> y"
-      proof (cases y)
-        case (real r)
-        with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
-        from reals_Archimedean2[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
-        then have "\<exists>p. max 0 (u x) = ereal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
-        then guess p .. note ux = this
-        obtain m :: nat where m: "p < real m" using reals_Archimedean2 ..
-        have "p \<le> r"
-        proof (rule ccontr)
-          assume "\<not> p \<le> r"
-          with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
-          obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: field_simps)
-          then have "r * 2^max N m < p * 2^max N m - 1" by simp
-          moreover
-          have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m"
-            using *[of "max N m"] m unfolding real_f using ux
-            by (cases "0 \<le> u x") (simp_all add: max_def split: split_if_asm)
-          then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
-            by (metis real_natfloor_gt_diff_one less_le_trans)
-          ultimately show False by auto
-        qed
-        then show "max 0 (u x) \<le> y" using real ux by simp
-      qed (insert `0 \<le> y`, auto)
-    qed
-  qed auto
-qed
-
-lemma borel_measurable_implies_simple_function_sequence':
-  fixes u :: "'a \<Rightarrow> ereal"
-  assumes u: "u \<in> borel_measurable M"
-  obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
-    "\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
-  using borel_measurable_implies_simple_function_sequence[OF u] by auto
-
-lemma simple_function_induct[consumes 1, case_names cong set mult add, induct set: simple_function]:
-  fixes u :: "'a \<Rightarrow> ereal"
-  assumes u: "simple_function M u"
-  assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
-  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
-  assumes mult: "\<And>u c. P u \<Longrightarrow> P (\<lambda>x. c * u x)"
-  assumes add: "\<And>u v. P u \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
-  shows "P u"
-proof (rule cong)
-  from AE_space show "AE x in M. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x"
-  proof eventually_elim
-    fix x assume x: "x \<in> space M"
-    from simple_function_indicator_representation[OF u x]
-    show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
-  qed
-next
-  from u have "finite (u ` space M)"
-    unfolding simple_function_def by auto
-  then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
-  proof induct
-    case empty show ?case
-      using set[of "{}"] by (simp add: indicator_def[abs_def])
-  qed (auto intro!: add mult set simple_functionD u)
-next
-  show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
-    apply (subst simple_function_cong)
-    apply (rule simple_function_indicator_representation[symmetric])
-    apply (auto intro: u)
-    done
-qed fact
-
-lemma simple_function_induct_nn[consumes 2, case_names cong set mult add]:
-  fixes u :: "'a \<Rightarrow> ereal"
-  assumes u: "simple_function M u" and nn: "\<And>x. 0 \<le> u x"
-  assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
-  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
-  assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
-  assumes add: "\<And>u v. simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> simple_function M v \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
-  shows "P u"
-proof -
-  show ?thesis
-  proof (rule cong)
-    fix x assume x: "x \<in> space M"
-    from simple_function_indicator_representation[OF u x]
-    show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
-  next
-    show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
-      apply (subst simple_function_cong)
-      apply (rule simple_function_indicator_representation[symmetric])
-      apply (auto intro: u)
-      done
-  next
-    from u nn have "finite (u ` space M)" "\<And>x. x \<in> u ` space M \<Longrightarrow> 0 \<le> x"
-      unfolding simple_function_def by auto
-    then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
-    proof induct
-      case empty show ?case
-        using set[of "{}"] by (simp add: indicator_def[abs_def])
-    qed (auto intro!: add mult set simple_functionD u setsum_nonneg
-       simple_function_setsum)
-  qed fact
-qed
-
-lemma borel_measurable_induct[consumes 2, case_names cong set mult add seq, induct set: borel_measurable]:
-  fixes u :: "'a \<Rightarrow> ereal"
-  assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x"
-  assumes cong: "\<And>f g. f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P g \<Longrightarrow> P f"
-  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
-  assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
-  assumes add: "\<And>u v. u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
-  assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow>  (\<And>i x. 0 \<le> U i x) \<Longrightarrow>  (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> P (SUP i. U i)"
-  shows "P u"
-  using u
-proof (induct rule: borel_measurable_implies_simple_function_sequence')
-  fix U assume U: "\<And>i. simple_function M (U i)" "incseq U" "\<And>i. \<infinity> \<notin> range (U i)" and
-    sup: "\<And>x. (SUP i. U i x) = max 0 (u x)" and nn: "\<And>i x. 0 \<le> U i x"
-  have u_eq: "u = (SUP i. U i)"
-    using nn u sup by (auto simp: max_def)
-  
-  from U have "\<And>i. U i \<in> borel_measurable M"
-    by (simp add: borel_measurable_simple_function)
-
-  show "P u"
-    unfolding u_eq
-  proof (rule seq)
-    fix i show "P (U i)"
-      using `simple_function M (U i)` nn
-      by (induct rule: simple_function_induct_nn)
-         (auto intro: set mult add cong dest!: borel_measurable_simple_function)
-  qed fact+
-qed
-
-lemma simple_function_If_set:
-  assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
-  shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
-proof -
-  def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
-  show ?thesis unfolding simple_function_def
-  proof safe
-    have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
-    from finite_subset[OF this] assms
-    show "finite (?IF ` space M)" unfolding simple_function_def by auto
-  next
-    fix x assume "x \<in> space M"
-    then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
-      then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
-      else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
-      using sets.sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
-    have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
-      unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
-    show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
-  qed
-qed
-
-lemma simple_function_If:
-  assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
-  shows "simple_function M (\<lambda>x. if P x then f x else g x)"
-proof -
-  have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
-  with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
-qed
-
-lemma simple_function_subalgebra:
-  assumes "simple_function N f"
-  and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
-  shows "simple_function M f"
-  using assms unfolding simple_function_def by auto
-
-lemma simple_function_comp:
-  assumes T: "T \<in> measurable M M'"
-    and f: "simple_function M' f"
-  shows "simple_function M (\<lambda>x. f (T x))"
-proof (intro simple_function_def[THEN iffD2] conjI ballI)
-  have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
-    using T unfolding measurable_def by auto
-  then show "finite ((\<lambda>x. f (T x)) ` space M)"
-    using f unfolding simple_function_def by (auto intro: finite_subset)
-  fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
-  then have "i \<in> f ` space M'"
-    using T unfolding measurable_def by auto
-  then have "f -` {i} \<inter> space M' \<in> sets M'"
-    using f unfolding simple_function_def by auto
-  then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
-    using T unfolding measurable_def by auto
-  also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
-    using T unfolding measurable_def by auto
-  finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" .
-qed
-
-section "Simple integral"
-
-definition simple_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>S") where
-  "integral\<^sup>S M f = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M))"
-
-syntax
-  "_simple_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>S _. _ \<partial>_" [60,61] 110)
-
-translations
-  "\<integral>\<^sup>S x. f \<partial>M" == "CONST simple_integral M (%x. f)"
-
-lemma simple_integral_cong:
-  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
-  shows "integral\<^sup>S M f = integral\<^sup>S M g"
-proof -
-  have "f ` space M = g ` space M"
-    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
-    using assms by (auto intro!: image_eqI)
-  thus ?thesis unfolding simple_integral_def by simp
-qed
-
-lemma simple_integral_const[simp]:
-  "(\<integral>\<^sup>Sx. c \<partial>M) = c * (emeasure M) (space M)"
-proof (cases "space M = {}")
-  case True thus ?thesis unfolding simple_integral_def by simp
-next
-  case False hence "(\<lambda>x. c) ` space M = {c}" by auto
-  thus ?thesis unfolding simple_integral_def by simp
-qed
-
-lemma simple_function_partition:
-  assumes f: "simple_function M f" and g: "simple_function M g"
-  assumes sub: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow> g x = g y \<Longrightarrow> f x = f y"
-  assumes v: "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)"
-  shows "integral\<^sup>S M f = (\<Sum>y\<in>g ` space M. v y * emeasure M {x\<in>space M. g x = y})"
-    (is "_ = ?r")
-proof -
-  from f g have [simp]: "finite (f`space M)" "finite (g`space M)"
-    by (auto simp: simple_function_def)
-  from f g have [measurable]: "f \<in> measurable M (count_space UNIV)" "g \<in> measurable M (count_space UNIV)"
-    by (auto intro: measurable_simple_function)
-
-  { fix y assume "y \<in> space M"
-    then have "f ` space M \<inter> {i. \<exists>x\<in>space M. i = f x \<and> g y = g x} = {v (g y)}"
-      by (auto cong: sub simp: v[symmetric]) }
-  note eq = this
-
-  have "integral\<^sup>S M f =
-    (\<Sum>y\<in>f`space M. y * (\<Sum>z\<in>g`space M. 
-      if \<exists>x\<in>space M. y = f x \<and> z = g x then emeasure M {x\<in>space M. g x = z} else 0))"
-    unfolding simple_integral_def
-  proof (safe intro!: setsum_cong ereal_left_mult_cong)
-    fix y assume y: "y \<in> space M" "f y \<noteq> 0"
-    have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} = 
-        {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
-      by auto
-    have eq:"(\<Union>i\<in>{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}. {x \<in> space M. g x = i}) =
-        f -` {f y} \<inter> space M"
-      by (auto simp: eq_commute cong: sub rev_conj_cong)
-    have "finite (g`space M)" by simp
-    then have "finite {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
-      by (rule rev_finite_subset) auto
-    then show "emeasure M (f -` {f y} \<inter> space M) =
-      (\<Sum>z\<in>g ` space M. if \<exists>x\<in>space M. f y = f x \<and> z = g x then emeasure M {x \<in> space M. g x = z} else 0)"
-      apply (simp add: setsum_cases)
-      apply (subst setsum_emeasure)
-      apply (auto simp: disjoint_family_on_def eq)
-      done
-  qed
-  also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M. 
-      if \<exists>x\<in>space M. y = f x \<and> z = g x then y * emeasure M {x\<in>space M. g x = z} else 0))"
-    by (auto intro!: setsum_cong simp: setsum_ereal_right_distrib emeasure_nonneg)
-  also have "\<dots> = ?r"
-    by (subst setsum_commute)
-       (auto intro!: setsum_cong simp: setsum_cases scaleR_setsum_right[symmetric] eq)
-  finally show "integral\<^sup>S M f = ?r" .
-qed
-
-lemma simple_integral_add[simp]:
-  assumes f: "simple_function M f" and "\<And>x. 0 \<le> f x" and g: "simple_function M g" and "\<And>x. 0 \<le> g x"
-  shows "(\<integral>\<^sup>Sx. f x + g x \<partial>M) = integral\<^sup>S M f + integral\<^sup>S M g"
-proof -
-  have "(\<integral>\<^sup>Sx. f x + g x \<partial>M) =
-    (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. (fst y + snd y) * emeasure M {x\<in>space M. (f x, g x) = y})"
-    by (intro simple_function_partition) (auto intro: f g)
-  also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * emeasure M {x\<in>space M. (f x, g x) = y}) +
-    (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y})"
-    using assms(2,4) by (auto intro!: setsum_cong ereal_left_distrib simp: setsum_addf[symmetric])
-  also have "(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * emeasure M {x\<in>space M. (f x, g x) = y}) = (\<integral>\<^sup>Sx. f x \<partial>M)"
-    by (intro simple_function_partition[symmetric]) (auto intro: f g)
-  also have "(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y}) = (\<integral>\<^sup>Sx. g x \<partial>M)"
-    by (intro simple_function_partition[symmetric]) (auto intro: f g)
-  finally show ?thesis .
-qed
-
-lemma simple_integral_setsum[simp]:
-  assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
-  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
-  shows "(\<integral>\<^sup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>S M (f i))"
-proof cases
-  assume "finite P"
-  from this assms show ?thesis
-    by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
-qed auto
-
-lemma simple_integral_mult[simp]:
-  assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
-  shows "(\<integral>\<^sup>Sx. c * f x \<partial>M) = c * integral\<^sup>S M f"
-proof -
-  have "(\<integral>\<^sup>Sx. c * f x \<partial>M) = (\<Sum>y\<in>f ` space M. (c * y) * emeasure M {x\<in>space M. f x = y})"
-    using f by (intro simple_function_partition) auto
-  also have "\<dots> = c * integral\<^sup>S M f"
-    using f unfolding simple_integral_def
-    by (subst setsum_ereal_right_distrib) (auto simp: emeasure_nonneg mult_assoc Int_def conj_commute)
-  finally show ?thesis .
-qed
-
-lemma simple_integral_mono_AE:
-  assumes f[measurable]: "simple_function M f" and g[measurable]: "simple_function M g"
-  and mono: "AE x in M. f x \<le> g x"
-  shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
-proof -
-  let ?\<mu> = "\<lambda>P. emeasure M {x\<in>space M. P x}"
-  have "integral\<^sup>S M f = (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * ?\<mu> (\<lambda>x. (f x, g x) = y))"
-    using f g by (intro simple_function_partition) auto
-  also have "\<dots> \<le> (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * ?\<mu> (\<lambda>x. (f x, g x) = y))"
-  proof (clarsimp intro!: setsum_mono)
-    fix x assume "x \<in> space M"
-    let ?M = "?\<mu> (\<lambda>y. f y = f x \<and> g y = g x)"
-    show "f x * ?M \<le> g x * ?M"
-    proof cases
-      assume "?M \<noteq> 0"
-      then have "0 < ?M"
-        by (simp add: less_le emeasure_nonneg)
-      also have "\<dots> \<le> ?\<mu> (\<lambda>y. f x \<le> g x)"
-        using mono by (intro emeasure_mono_AE) auto
-      finally have "\<not> \<not> f x \<le> g x"
-        by (intro notI) auto
-      then show ?thesis
-        by (intro ereal_mult_right_mono) auto
-    qed simp
-  qed
-  also have "\<dots> = integral\<^sup>S M g"
-    using f g by (intro simple_function_partition[symmetric]) auto
-  finally show ?thesis .
-qed
-
-lemma simple_integral_mono:
-  assumes "simple_function M f" and "simple_function M g"
-  and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
-  shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
-  using assms by (intro simple_integral_mono_AE) auto
-
-lemma simple_integral_cong_AE:
-  assumes "simple_function M f" and "simple_function M g"
-  and "AE x in M. f x = g x"
-  shows "integral\<^sup>S M f = integral\<^sup>S M g"
-  using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
-
-lemma simple_integral_cong':
-  assumes sf: "simple_function M f" "simple_function M g"
-  and mea: "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0"
-  shows "integral\<^sup>S M f = integral\<^sup>S M g"
-proof (intro simple_integral_cong_AE sf AE_I)
-  show "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" by fact
-  show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
-    using sf[THEN borel_measurable_simple_function] by auto
-qed simp
-
-lemma simple_integral_indicator:
-  assumes A: "A \<in> sets M"
-  assumes f: "simple_function M f"
-  shows "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
-    (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))"
-proof -
-  have eq: "(\<lambda>x. (f x, indicator A x)) ` space M \<inter> {x. snd x = 1} = (\<lambda>x. (f x, 1::ereal))`A"
-    using A[THEN sets.sets_into_space] by (auto simp: indicator_def image_iff split: split_if_asm)
-  have eq2: "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
-    by (auto simp: image_iff)
-
-  have "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
-    (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x))`space M. (fst y * snd y) * emeasure M {x\<in>space M. (f x, indicator A x) = y})"
-    using assms by (intro simple_function_partition) auto
-  also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x::ereal))`space M.
-    if snd y = 1 then fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A) else 0)"
-    by (auto simp: indicator_def split: split_if_asm intro!: arg_cong2[where f="op *"] arg_cong2[where f=emeasure] setsum_cong)
-  also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, 1::ereal))`A. fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A))"
-    using assms by (subst setsum_cases) (auto intro!: simple_functionD(1) simp: eq)
-  also have "\<dots> = (\<Sum>y\<in>fst`(\<lambda>x. (f x, 1::ereal))`A. y * emeasure M (f -` {y} \<inter> space M \<inter> A))"
-    by (subst setsum_reindex[where f=fst]) (auto simp: inj_on_def)
-  also have "\<dots> = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))"
-    using A[THEN sets.sets_into_space]
-    by (intro setsum_mono_zero_cong_left simple_functionD f) (auto simp: image_comp comp_def eq2)
-  finally show ?thesis .
-qed
-
-lemma simple_integral_indicator_only[simp]:
-  assumes "A \<in> sets M"
-  shows "integral\<^sup>S M (indicator A) = emeasure M A"
-  using simple_integral_indicator[OF assms, of "\<lambda>x. 1"] sets.sets_into_space[OF assms]
-  by (simp_all add: image_constant_conv Int_absorb1 split: split_if_asm)
-
-lemma simple_integral_null_set:
-  assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets M"
-  shows "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = 0"
-proof -
-  have "AE x in M. indicator N x = (0 :: ereal)"
-    using `N \<in> null_sets M` by (auto simp: indicator_def intro!: AE_I[of _ _ N])
-  then have "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^sup>Sx. 0 \<partial>M)"
-    using assms apply (intro simple_integral_cong_AE) by auto
-  then show ?thesis by simp
-qed
-
-lemma simple_integral_cong_AE_mult_indicator:
-  assumes sf: "simple_function M f" and eq: "AE x in M. x \<in> S" and "S \<in> sets M"
-  shows "integral\<^sup>S M f = (\<integral>\<^sup>Sx. f x * indicator S x \<partial>M)"
-  using assms by (intro simple_integral_cong_AE) auto
-
-lemma simple_integral_cmult_indicator:
-  assumes A: "A \<in> sets M"
-  shows "(\<integral>\<^sup>Sx. c * indicator A x \<partial>M) = c * emeasure M A"
-  using simple_integral_mult[OF simple_function_indicator[OF A]]
-  unfolding simple_integral_indicator_only[OF A] by simp
-
-lemma simple_integral_positive:
-  assumes f: "simple_function M f" and ae: "AE x in M. 0 \<le> f x"
-  shows "0 \<le> integral\<^sup>S M f"
-proof -
-  have "integral\<^sup>S M (\<lambda>x. 0) \<le> integral\<^sup>S M f"
-    using simple_integral_mono_AE[OF _ f ae] by auto
-  then show ?thesis by simp
-qed
-
-section "Continuous positive integration"
-
-definition positive_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>P") where
-  "integral\<^sup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^sup>S M g)"
-
-syntax
-  "_positive_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>+ _. _ \<partial>_" [60,61] 110)
-
-translations
-  "\<integral>\<^sup>+ x. f \<partial>M" == "CONST positive_integral M (%x. f)"
-
-lemma positive_integral_positive:
-  "0 \<le> integral\<^sup>P M f"
-  by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
-
-lemma positive_integral_not_MInfty[simp]: "integral\<^sup>P M f \<noteq> -\<infinity>"
-  using positive_integral_positive[of M f] by auto
-
-lemma positive_integral_def_finite:
-  "integral\<^sup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^sup>S M g)"
-    (is "_ = SUPREMUM ?A ?f")
-  unfolding positive_integral_def
-proof (safe intro!: antisym SUP_least)
-  fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
-  let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
-  note gM = g(1)[THEN borel_measurable_simple_function]
-  have \<mu>_G_pos: "0 \<le> (emeasure M) ?G" using gM by auto
-  let ?g = "\<lambda>y x. if g x = \<infinity> then y else max 0 (g x)"
-  from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
-    apply (safe intro!: simple_function_max simple_function_If)
-    apply (force simp: max_def le_fun_def split: split_if_asm)+
-    done
-  show "integral\<^sup>S M g \<le> SUPREMUM ?A ?f"
-  proof cases
-    have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
-    assume "(emeasure M) ?G = 0"
-    with gM have "AE x in M. x \<notin> ?G"
-      by (auto simp add: AE_iff_null intro!: null_setsI)
-    with gM g show ?thesis
-      by (intro SUP_upper2[OF g0] simple_integral_mono_AE)
-         (auto simp: max_def intro!: simple_function_If)
-  next
-    assume \<mu>_G: "(emeasure M) ?G \<noteq> 0"
-    have "SUPREMUM ?A (integral\<^sup>S M) = \<infinity>"
-    proof (intro SUP_PInfty)
-      fix n :: nat
-      let ?y = "ereal (real n) / (if (emeasure M) ?G = \<infinity> then 1 else (emeasure M) ?G)"
-      have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>_G \<mu>_G_pos by (auto simp: ereal_divide_eq)
-      then have "?g ?y \<in> ?A" by (rule g_in_A)
-      have "real n \<le> ?y * (emeasure M) ?G"
-        using \<mu>_G \<mu>_G_pos by (cases "(emeasure M) ?G") (auto simp: field_simps)
-      also have "\<dots> = (\<integral>\<^sup>Sx. ?y * indicator ?G x \<partial>M)"
-        using `0 \<le> ?y` `?g ?y \<in> ?A` gM
-        by (subst simple_integral_cmult_indicator) auto
-      also have "\<dots> \<le> integral\<^sup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
-        by (intro simple_integral_mono) auto
-      finally show "\<exists>i\<in>?A. real n \<le> integral\<^sup>S M i"
-        using `?g ?y \<in> ?A` by blast
-    qed
-    then show ?thesis by simp
-  qed
-qed (auto intro: SUP_upper)
-
-lemma positive_integral_mono_AE:
-  assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^sup>P M u \<le> integral\<^sup>P M v"
-  unfolding positive_integral_def
-proof (safe intro!: SUP_mono)
-  fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
-  from ae[THEN AE_E] guess N . note N = this
-  then have ae_N: "AE x in M. x \<notin> N" by (auto intro: AE_not_in)
-  let ?n = "\<lambda>x. n x * indicator (space M - N) x"
-  have "AE x in M. n x \<le> ?n x" "simple_function M ?n"
-    using n N ae_N by auto
-  moreover
-  { fix x have "?n x \<le> max 0 (v x)"
-    proof cases
-      assume x: "x \<in> space M - N"
-      with N have "u x \<le> v x" by auto
-      with n(2)[THEN le_funD, of x] x show ?thesis
-        by (auto simp: max_def split: split_if_asm)
-    qed simp }
-  then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
-  moreover have "integral\<^sup>S M n \<le> integral\<^sup>S M ?n"
-    using ae_N N n by (auto intro!: simple_integral_mono_AE)
-  ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^sup>S M n \<le> integral\<^sup>S M m"
-    by force
-qed
-
-lemma positive_integral_mono:
-  "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^sup>P M u \<le> integral\<^sup>P M v"
-  by (auto intro: positive_integral_mono_AE)
-
-lemma positive_integral_cong_AE:
-  "AE x in M. u x = v x \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P M v"
-  by (auto simp: eq_iff intro!: positive_integral_mono_AE)
-
-lemma positive_integral_cong:
-  "(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P M v"
-  by (auto intro: positive_integral_cong_AE)
-
-lemma positive_integral_eq_simple_integral:
-  assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^sup>P M f = integral\<^sup>S M f"
-proof -
-  let ?f = "\<lambda>x. f x * indicator (space M) x"
-  have f': "simple_function M ?f" using f by auto
-  with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
-    by (auto simp: fun_eq_iff max_def split: split_indicator)
-  have "integral\<^sup>P M ?f \<le> integral\<^sup>S M ?f" using f'
-    by (force intro!: SUP_least simple_integral_mono simp: le_fun_def positive_integral_def)
-  moreover have "integral\<^sup>S M ?f \<le> integral\<^sup>P M ?f"
-    unfolding positive_integral_def
-    using f' by (auto intro!: SUP_upper)
-  ultimately show ?thesis
-    by (simp cong: positive_integral_cong simple_integral_cong)
-qed
-
-lemma positive_integral_eq_simple_integral_AE:
-  assumes f: "simple_function M f" "AE x in M. 0 \<le> f x" shows "integral\<^sup>P M f = integral\<^sup>S M f"
-proof -
-  have "AE x in M. f x = max 0 (f x)" using f by (auto split: split_max)
-  with f have "integral\<^sup>P M f = integral\<^sup>S M (\<lambda>x. max 0 (f x))"
-    by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
-             add: positive_integral_eq_simple_integral)
-  with assms show ?thesis
-    by (auto intro!: simple_integral_cong_AE split: split_max)
-qed
-
-lemma positive_integral_SUP_approx:
-  assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
-  and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
-  shows "integral\<^sup>S M u \<le> (SUP i. integral\<^sup>P M (f i))" (is "_ \<le> ?S")
-proof (rule ereal_le_mult_one_interval)
-  have "0 \<le> (SUP i. integral\<^sup>P M (f i))"
-    using f(3) by (auto intro!: SUP_upper2 positive_integral_positive)
-  then show "(SUP i. integral\<^sup>P M (f i)) \<noteq> -\<infinity>" by auto
-  have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
-    using u(3) by auto
-  fix a :: ereal assume "0 < a" "a < 1"
-  hence "a \<noteq> 0" by auto
-  let ?B = "\<lambda>i. {x \<in> space M. a * u x \<le> f i x}"
-  have B: "\<And>i. ?B i \<in> sets M"
-    using f `simple_function M u`[THEN borel_measurable_simple_function] by auto
-
-  let ?uB = "\<lambda>i x. u x * indicator (?B i) x"
-
-  { fix i have "?B i \<subseteq> ?B (Suc i)"
-    proof safe
-      fix i x assume "a * u x \<le> f i x"
-      also have "\<dots> \<le> f (Suc i) x"
-        using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
-      finally show "a * u x \<le> f (Suc i) x" .
-    qed }
-  note B_mono = this
-
-  note B_u = sets.Int[OF u(1)[THEN simple_functionD(2)] B]
-
-  let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n"
-  have measure_conv: "\<And>i. (emeasure M) (u -` {i} \<inter> space M) = (SUP n. (emeasure M) (?B' i n))"
-  proof -
-    fix i
-    have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
-    have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
-    have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
-    proof safe
-      fix x i assume x: "x \<in> space M"
-      show "x \<in> (\<Union>i. ?B' (u x) i)"
-      proof cases
-        assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
-      next
-        assume "u x \<noteq> 0"
-        with `a < 1` u_range[OF `x \<in> space M`]
-        have "a * u x < 1 * u x"
-          by (intro ereal_mult_strict_right_mono) (auto simp: image_iff)
-        also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def)
-        finally obtain i where "a * u x < f i x" unfolding SUP_def
-          by (auto simp add: less_SUP_iff)
-        hence "a * u x \<le> f i x" by auto
-        thus ?thesis using `x \<in> space M` by auto
-      qed
-    qed
-    then show "?thesis i" using SUP_emeasure_incseq[OF 1 2] by simp
-  qed
-
-  have "integral\<^sup>S M u = (SUP i. integral\<^sup>S M (?uB i))"
-    unfolding simple_integral_indicator[OF B `simple_function M u`]
-  proof (subst SUP_ereal_setsum, safe)
-    fix x n assume "x \<in> space M"
-    with u_range show "incseq (\<lambda>i. u x * (emeasure M) (?B' (u x) i))" "\<And>i. 0 \<le> u x * (emeasure M) (?B' (u x) i)"
-      using B_mono B_u by (auto intro!: emeasure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff)
-  next
-    show "integral\<^sup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * (emeasure M) (?B' i n))"
-      using measure_conv u_range B_u unfolding simple_integral_def
-      by (auto intro!: setsum_cong SUP_ereal_cmult [symmetric])
-  qed
-  moreover
-  have "a * (SUP i. integral\<^sup>S M (?uB i)) \<le> ?S"
-    apply (subst SUP_ereal_cmult [symmetric])
-  proof (safe intro!: SUP_mono bexI)
-    fix i
-    have "a * integral\<^sup>S M (?uB i) = (\<integral>\<^sup>Sx. a * ?uB i x \<partial>M)"
-      using B `simple_function M u` u_range
-      by (subst simple_integral_mult) (auto split: split_indicator)
-    also have "\<dots> \<le> integral\<^sup>P M (f i)"
-    proof -
-      have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
-      show ?thesis using f(3) * u_range `0 < a`
-        by (subst positive_integral_eq_simple_integral[symmetric])
-           (auto intro!: positive_integral_mono split: split_indicator)
-    qed
-    finally show "a * integral\<^sup>S M (?uB i) \<le> integral\<^sup>P M (f i)"
-      by auto
-  next
-    fix i show "0 \<le> \<integral>\<^sup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
-      by (intro simple_integral_positive) (auto split: split_indicator)
-  qed (insert `0 < a`, auto)
-  ultimately show "a * integral\<^sup>S M u \<le> ?S" by simp
-qed
-
-lemma incseq_positive_integral:
-  assumes "incseq f" shows "incseq (\<lambda>i. integral\<^sup>P M (f i))"
-proof -
-  have "\<And>i x. f i x \<le> f (Suc i) x"
-    using assms by (auto dest!: incseq_SucD simp: le_fun_def)
-  then show ?thesis
-    by (auto intro!: incseq_SucI positive_integral_mono)
-qed
-
-text {* Beppo-Levi monotone convergence theorem *}
-lemma positive_integral_monotone_convergence_SUP:
-  assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
-  shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
-proof (rule antisym)
-  show "(SUP j. integral\<^sup>P M (f j)) \<le> (\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M)"
-    by (auto intro!: SUP_least SUP_upper positive_integral_mono)
-next
-  show "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^sup>P M (f j))"
-    unfolding positive_integral_def_finite[of _ "\<lambda>x. SUP i. f i x"]
-  proof (safe intro!: SUP_least)
-    fix g assume g: "simple_function M g"
-      and *: "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
-    then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}"
-      using f by (auto intro!: SUP_upper2)
-    with * show "integral\<^sup>S M g \<le> (SUP j. integral\<^sup>P M (f j))"
-      by (intro  positive_integral_SUP_approx[OF f g _ g'])
-         (auto simp: le_fun_def max_def)
-  qed
-qed
-
-lemma positive_integral_monotone_convergence_SUP_AE:
-  assumes f: "\<And>i. AE x in M. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
-  shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
-proof -
-  from f have "AE x in M. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
-    by (simp add: AE_all_countable)
-  from this[THEN AE_E] guess N . note N = this
-  let ?f = "\<lambda>i x. if x \<in> space M - N then f i x else 0"
-  have f_eq: "AE x in M. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N])
-  then have "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^sup>+ x. (SUP i. ?f i x) \<partial>M)"
-    by (auto intro!: positive_integral_cong_AE)
-  also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. ?f i x \<partial>M))"
-  proof (rule positive_integral_monotone_convergence_SUP)
-    show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
-    { fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
-        using f N(3) by (intro measurable_If_set) auto
-      fix x show "0 \<le> ?f i x"
-        using N(1) by auto }
-  qed
-  also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. f i x \<partial>M))"
-    using f_eq by (force intro!: arg_cong[where f="SUPREMUM UNIV"] positive_integral_cong_AE ext)
-  finally show ?thesis .
-qed
-
-lemma positive_integral_monotone_convergence_SUP_AE_incseq:
-  assumes f: "incseq f" "\<And>i. AE x in M. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
-  shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
-  using f[unfolded incseq_Suc_iff le_fun_def]
-  by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
-     auto
-
-lemma positive_integral_monotone_convergence_simple:
-  assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
-  shows "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)"
-  using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
-    f(3)[THEN borel_measurable_simple_function] f(2)]
-  by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPREMUM UNIV"] ext)
-
-lemma positive_integral_max_0:
-  "(\<integral>\<^sup>+x. max 0 (f x) \<partial>M) = integral\<^sup>P M f"
-  by (simp add: le_fun_def positive_integral_def)
-
-lemma positive_integral_cong_pos:
-  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
-  shows "integral\<^sup>P M f = integral\<^sup>P M g"
-proof -
-  have "integral\<^sup>P M (\<lambda>x. max 0 (f x)) = integral\<^sup>P M (\<lambda>x. max 0 (g x))"
-  proof (intro positive_integral_cong)
-    fix x assume "x \<in> space M"
-    from assms[OF this] show "max 0 (f x) = max 0 (g x)"
-      by (auto split: split_max)
-  qed
-  then show ?thesis by (simp add: positive_integral_max_0)
-qed
-
-lemma SUP_simple_integral_sequences:
-  assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
-  and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
-  and eq: "AE x in M. (SUP i. f i x) = (SUP i. g i x)"
-  shows "(SUP i. integral\<^sup>S M (f i)) = (SUP i. integral\<^sup>S M (g i))"
-    (is "SUPREMUM _ ?F = SUPREMUM _ ?G")
-proof -
-  have "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)"
-    using f by (rule positive_integral_monotone_convergence_simple)
-  also have "\<dots> = (\<integral>\<^sup>+x. (SUP i. g i x) \<partial>M)"
-    unfolding eq[THEN positive_integral_cong_AE] ..
-  also have "\<dots> = (SUP i. ?G i)"
-    using g by (rule positive_integral_monotone_convergence_simple[symmetric])
-  finally show ?thesis by simp
-qed
-
-lemma positive_integral_const[simp]:
-  "0 \<le> c \<Longrightarrow> (\<integral>\<^sup>+ x. c \<partial>M) = c * (emeasure M) (space M)"
-  by (subst positive_integral_eq_simple_integral) auto
-
-lemma positive_integral_linear:
-  assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
-  and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
-  shows "(\<integral>\<^sup>+ x. a * f x + g x \<partial>M) = a * integral\<^sup>P M f + integral\<^sup>P M g"
-    (is "integral\<^sup>P M ?L = _")
-proof -
-  from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
-  note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
-  from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
-  note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
-  let ?L' = "\<lambda>i x. a * u i x + v i x"
-
-  have "?L \<in> borel_measurable M" using assms by auto
-  from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
-  note l = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
-
-  have inc: "incseq (\<lambda>i. a * integral\<^sup>S M (u i))" "incseq (\<lambda>i. integral\<^sup>S M (v i))"
-    using u v `0 \<le> a`
-    by (auto simp: incseq_Suc_iff le_fun_def
-             intro!: add_mono ereal_mult_left_mono simple_integral_mono)
-  have pos: "\<And>i. 0 \<le> integral\<^sup>S M (u i)" "\<And>i. 0 \<le> integral\<^sup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^sup>S M (u i)"
-    using u v `0 \<le> a` by (auto simp: simple_integral_positive)
-  { fix i from pos[of i] have "a * integral\<^sup>S M (u i) \<noteq> -\<infinity>" "integral\<^sup>S M (v i) \<noteq> -\<infinity>"
-      by (auto split: split_if_asm) }
-  note not_MInf = this
-
-  have l': "(SUP i. integral\<^sup>S M (l i)) = (SUP i. integral\<^sup>S M (?L' i))"
-  proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
-    show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
-      using u v  `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
-      by (auto intro!: add_mono ereal_mult_left_mono)
-    { fix x
-      { fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using `0 \<le> a` u(6)[of i x] v(6)[of i x]
-          by auto }
-      then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
-        using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
-        by (subst SUP_ereal_cmult [symmetric, OF u(6) `0 \<le> a`])
-           (auto intro!: SUP_ereal_add
-                 simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono) }
-    then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)"
-      unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
-      by (intro AE_I2) (auto split: split_max)
-  qed
-  also have "\<dots> = (SUP i. a * integral\<^sup>S M (u i) + integral\<^sup>S M (v i))"
-    using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPREMUM UNIV"] ext)
-  finally have "(\<integral>\<^sup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^sup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+x. max 0 (g x) \<partial>M)"
-    unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
-    unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
-    apply (subst SUP_ereal_cmult [symmetric, OF pos(1) `0 \<le> a`])
-    apply (subst SUP_ereal_add [symmetric, OF inc not_MInf]) .
-  then show ?thesis by (simp add: positive_integral_max_0)
-qed
-
-lemma positive_integral_cmult:
-  assumes f: "f \<in> borel_measurable M" "0 \<le> c"
-  shows "(\<integral>\<^sup>+ x. c * f x \<partial>M) = c * integral\<^sup>P M f"
-proof -
-  have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
-    by (auto split: split_max simp: ereal_zero_le_0_iff)
-  have "(\<integral>\<^sup>+ x. c * f x \<partial>M) = (\<integral>\<^sup>+ x. c * max 0 (f x) \<partial>M)"
-    by (simp add: positive_integral_max_0)
-  then show ?thesis
-    using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f
-    by (auto simp: positive_integral_max_0)
-qed
-
-lemma positive_integral_multc:
-  assumes "f \<in> borel_measurable M" "0 \<le> c"
-  shows "(\<integral>\<^sup>+ x. f x * c \<partial>M) = integral\<^sup>P M f * c"
-  unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
-
-lemma positive_integral_indicator[simp]:
-  "A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. indicator A x\<partial>M) = (emeasure M) A"
-  by (subst positive_integral_eq_simple_integral)
-     (auto simp: simple_integral_indicator)
-
-lemma positive_integral_cmult_indicator:
-  "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. c * indicator A x \<partial>M) = c * (emeasure M) A"
-  by (subst positive_integral_eq_simple_integral)
-     (auto simp: simple_function_indicator simple_integral_indicator)
-
-lemma positive_integral_indicator':
-  assumes [measurable]: "A \<inter> space M \<in> sets M"
-  shows "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = emeasure M (A \<inter> space M)"
-proof -
-  have "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = (\<integral>\<^sup>+ x. indicator (A \<inter> space M) x \<partial>M)"
-    by (intro positive_integral_cong) (simp split: split_indicator)
-  also have "\<dots> = emeasure M (A \<inter> space M)"
-    by simp
-  finally show ?thesis .
-qed
-
-lemma positive_integral_add:
-  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
-  and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
-  shows "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = integral\<^sup>P M f + integral\<^sup>P M g"
-proof -
-  have ae: "AE x in M. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
-    using assms by (auto split: split_max)
-  have "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = (\<integral>\<^sup>+ x. max 0 (f x + g x) \<partial>M)"
-    by (simp add: positive_integral_max_0)
-  also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
-    unfolding ae[THEN positive_integral_cong_AE] ..
-  also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+ x. max 0 (g x) \<partial>M)"
-    using positive_integral_linear[of "\<lambda>x. max 0 (f x)" _ 1 "\<lambda>x. max 0 (g x)"] f g
-    by auto
-  finally show ?thesis
-    by (simp add: positive_integral_max_0)
-qed
-
-lemma positive_integral_setsum:
-  assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x in M. 0 \<le> f i x"
-  shows "(\<integral>\<^sup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>P M (f i))"
-proof cases
-  assume f: "finite P"
-  from assms have "AE x in M. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
-  from f this assms(1) show ?thesis
-  proof induct
-    case (insert i P)
-    then have "f i \<in> borel_measurable M" "AE x in M. 0 \<le> f i x"
-      "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x in M. 0 \<le> (\<Sum>i\<in>P. f i x)"
-      by (auto intro!: setsum_nonneg)
-    from positive_integral_add[OF this]
-    show ?case using insert by auto
-  qed simp
-qed simp
-
-lemma positive_integral_Markov_inequality:
-  assumes u: "u \<in> borel_measurable M" "AE x in M. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c"
-  shows "(emeasure M) ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)"
-    (is "(emeasure M) ?A \<le> _ * ?PI")
-proof -
-  have "?A \<in> sets M"
-    using `A \<in> sets M` u by auto
-  hence "(emeasure M) ?A = (\<integral>\<^sup>+ x. indicator ?A x \<partial>M)"
-    using positive_integral_indicator by simp
-  also have "\<dots> \<le> (\<integral>\<^sup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
-    by (auto intro!: positive_integral_mono_AE
-      simp: indicator_def ereal_zero_le_0_iff)
-  also have "\<dots> = c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)"
-    using assms
-    by (auto intro!: positive_integral_cmult simp: ereal_zero_le_0_iff)
-  finally show ?thesis .
-qed
-
-lemma positive_integral_noteq_infinite:
-  assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
-  and "integral\<^sup>P M g \<noteq> \<infinity>"
-  shows "AE x in M. g x \<noteq> \<infinity>"
-proof (rule ccontr)
-  assume c: "\<not> (AE x in M. g x \<noteq> \<infinity>)"
-  have "(emeasure M) {x\<in>space M. g x = \<infinity>} \<noteq> 0"
-    using c g by (auto simp add: AE_iff_null)
-  moreover have "0 \<le> (emeasure M) {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
-  ultimately have "0 < (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
-  then have "\<infinity> = \<infinity> * (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
-  also have "\<dots> \<le> (\<integral>\<^sup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
-    using g by (subst positive_integral_cmult_indicator) auto
-  also have "\<dots> \<le> integral\<^sup>P M g"
-    using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
-  finally show False using `integral\<^sup>P M g \<noteq> \<infinity>` by auto
-qed
-
-lemma positive_integral_PInf:
-  assumes f: "f \<in> borel_measurable M"
-  and not_Inf: "integral\<^sup>P M f \<noteq> \<infinity>"
-  shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
-proof -
-  have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^sup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
-    using f by (subst positive_integral_cmult_indicator) (auto simp: measurable_sets)
-  also have "\<dots> \<le> integral\<^sup>P M (\<lambda>x. max 0 (f x))"
-    by (auto intro!: positive_integral_mono simp: indicator_def max_def)
-  finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^sup>P M f"
-    by (simp add: positive_integral_max_0)
-  moreover have "0 \<le> (emeasure M) (f -` {\<infinity>} \<inter> space M)"
-    by (rule emeasure_nonneg)
-  ultimately show ?thesis
-    using assms by (auto split: split_if_asm)
-qed
-
-lemma positive_integral_PInf_AE:
-  assumes "f \<in> borel_measurable M" "integral\<^sup>P M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>"
-proof (rule AE_I)
-  show "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
-    by (rule positive_integral_PInf[OF assms])
-  show "f -` {\<infinity>} \<inter> space M \<in> sets M"
-    using assms by (auto intro: borel_measurable_vimage)
-qed auto
-
-lemma simple_integral_PInf:
-  assumes "simple_function M f" "\<And>x. 0 \<le> f x"
-  and "integral\<^sup>S M f \<noteq> \<infinity>"
-  shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
-proof (rule positive_integral_PInf)
-  show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
-  show "integral\<^sup>P M f \<noteq> \<infinity>"
-    using assms by (simp add: positive_integral_eq_simple_integral)
-qed
-
-lemma positive_integral_diff:
-  assumes f: "f \<in> borel_measurable M"
-  and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
-  and fin: "integral\<^sup>P M g \<noteq> \<infinity>"
-  and mono: "AE x in M. g x \<le> f x"
-  shows "(\<integral>\<^sup>+ x. f x - g x \<partial>M) = integral\<^sup>P M f - integral\<^sup>P M g"
-proof -
-  have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x in M. 0 \<le> f x - g x"
-    using assms by (auto intro: ereal_diff_positive)
-  have pos_f: "AE x in M. 0 \<le> f x" using mono g by auto
-  { fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
-      by (cases rule: ereal2_cases[of a b]) auto }
-  note * = this
-  then have "AE x in M. f x = f x - g x + g x"
-    using mono positive_integral_noteq_infinite[OF g fin] assms by auto
-  then have **: "integral\<^sup>P M f = (\<integral>\<^sup>+x. f x - g x \<partial>M) + integral\<^sup>P M g"
-    unfolding positive_integral_add[OF diff g, symmetric]
-    by (rule positive_integral_cong_AE)
-  show ?thesis unfolding **
-    using fin positive_integral_positive[of M g]
-    by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. f x - g x \<partial>M" "integral\<^sup>P M g"]) auto
-qed
-
-lemma positive_integral_suminf:
-  assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x"
-  shows "(\<integral>\<^sup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^sup>P M (f i))"
-proof -
-  have all_pos: "AE x in M. \<forall>i. 0 \<le> f i x"
-    using assms by (auto simp: AE_all_countable)
-  have "(\<Sum>i. integral\<^sup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^sup>P M (f i))"
-    using positive_integral_positive by (rule suminf_ereal_eq_SUP)
-  also have "\<dots> = (SUP n. \<integral>\<^sup>+x. (\<Sum>i<n. f i x) \<partial>M)"
-    unfolding positive_integral_setsum[OF f] ..
-  also have "\<dots> = \<integral>\<^sup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
-    by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
-       (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
-  also have "\<dots> = \<integral>\<^sup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
-    by (intro positive_integral_cong_AE) (auto simp: suminf_ereal_eq_SUP)
-  finally show ?thesis by simp
-qed
-
-text {* Fatou's lemma: convergence theorem on limes inferior *}
-lemma positive_integral_lim_INF:
-  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
-  assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> u i x"
-  shows "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>P M (u n))"
-proof -
-  have pos: "AE x in M. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
-  have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
-    (SUP n. \<integral>\<^sup>+ x. (INF i:{n..}. u i x) \<partial>M)"
-    unfolding liminf_SUP_INF using pos u
-    by (intro positive_integral_monotone_convergence_SUP_AE)
-       (elim AE_mp, auto intro!: AE_I2 intro: INF_greatest INF_superset_mono)
-  also have "\<dots> \<le> liminf (\<lambda>n. integral\<^sup>P M (u n))"
-    unfolding liminf_SUP_INF
-    by (auto intro!: SUP_mono exI INF_greatest positive_integral_mono INF_lower)
-  finally show ?thesis .
-qed
-
-lemma positive_integral_null_set:
-  assumes "N \<in> null_sets M" shows "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = 0"
-proof -
-  have "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)"
-  proof (intro positive_integral_cong_AE AE_I)
-    show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
-      by (auto simp: indicator_def)
-    show "(emeasure M) N = 0" "N \<in> sets M"
-      using assms by auto
-  qed
-  then show ?thesis by simp
-qed
-
-lemma positive_integral_0_iff:
-  assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x"
-  shows "integral\<^sup>P M u = 0 \<longleftrightarrow> emeasure M {x\<in>space M. u x \<noteq> 0} = 0"
-    (is "_ \<longleftrightarrow> (emeasure M) ?A = 0")
-proof -
-  have u_eq: "(\<integral>\<^sup>+ x. u x * indicator ?A x \<partial>M) = integral\<^sup>P M u"
-    by (auto intro!: positive_integral_cong simp: indicator_def)
-  show ?thesis
-  proof
-    assume "(emeasure M) ?A = 0"
-    with positive_integral_null_set[of ?A M u] u
-    show "integral\<^sup>P M u = 0" by (simp add: u_eq null_sets_def)
-  next
-    { fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r"
-      then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def)
-      then have "0 \<le> r" by (auto simp add: ereal_zero_less_0_iff) }
-    note gt_1 = this
-    assume *: "integral\<^sup>P M u = 0"
-    let ?M = "\<lambda>n. {x \<in> space M. 1 \<le> real (n::nat) * u x}"
-    have "0 = (SUP n. (emeasure M) (?M n \<inter> ?A))"
-    proof -
-      { fix n :: nat
-        from positive_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"]
-        have "(emeasure M) (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
-        moreover have "0 \<le> (emeasure M) (?M n \<inter> ?A)" using u by auto
-        ultimately have "(emeasure M) (?M n \<inter> ?A) = 0" by auto }
-      thus ?thesis by simp
-    qed
-    also have "\<dots> = (emeasure M) (\<Union>n. ?M n \<inter> ?A)"
-    proof (safe intro!: SUP_emeasure_incseq)
-      fix n show "?M n \<inter> ?A \<in> sets M"
-        using u by (auto intro!: sets.Int)
-    next
-      show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
-      proof (safe intro!: incseq_SucI)
-        fix n :: nat and x
-        assume *: "1 \<le> real n * u x"
-        also from gt_1[OF *] have "real n * u x \<le> real (Suc n) * u x"
-          using `0 \<le> u x` by (auto intro!: ereal_mult_right_mono)
-        finally show "1 \<le> real (Suc n) * u x" by auto
-      qed
-    qed
-    also have "\<dots> = (emeasure M) {x\<in>space M. 0 < u x}"
-    proof (safe intro!: arg_cong[where f="(emeasure M)"] dest!: gt_1)
-      fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
-      show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
-      proof (cases "u x")
-        case (real r) with `0 < u x` have "0 < r" by auto
-        obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
-        hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
-        hence "1 \<le> real j * r" using real `0 < r` by auto
-        thus ?thesis using `0 < r` real by (auto simp: one_ereal_def)
-      qed (insert `0 < u x`, auto)
-    qed auto
-    finally have "(emeasure M) {x\<in>space M. 0 < u x} = 0" by simp
-    moreover
-    from pos have "AE x in M. \<not> (u x < 0)" by auto
-    then have "(emeasure M) {x\<in>space M. u x < 0} = 0"
-      using AE_iff_null[of M] u by auto
-    moreover have "(emeasure M) {x\<in>space M. u x \<noteq> 0} = (emeasure M) {x\<in>space M. u x < 0} + (emeasure M) {x\<in>space M. 0 < u x}"
-      using u by (subst plus_emeasure) (auto intro!: arg_cong[where f="emeasure M"])
-    ultimately show "(emeasure M) ?A = 0" by simp
-  qed
-qed
-
-lemma positive_integral_0_iff_AE:
-  assumes u: "u \<in> borel_measurable M"
-  shows "integral\<^sup>P M u = 0 \<longleftrightarrow> (AE x in M. u x \<le> 0)"
-proof -
-  have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
-    using u by auto
-  from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
-  have "integral\<^sup>P M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)"
-    unfolding positive_integral_max_0
-    using AE_iff_null[OF sets] u by auto
-  also have "\<dots> \<longleftrightarrow> (AE x in M. u x \<le> 0)" by (auto split: split_max)
-  finally show ?thesis .
-qed
-
-lemma AE_iff_positive_integral: 
-  "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> integral\<^sup>P M (indicator {x. \<not> P x}) = 0"
-  by (subst positive_integral_0_iff_AE) (auto simp: one_ereal_def zero_ereal_def
-    sets.sets_Collect_neg indicator_def[abs_def] measurable_If)
-
-lemma positive_integral_const_If:
-  "(\<integral>\<^sup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)"
-  by (auto intro!: positive_integral_0_iff_AE[THEN iffD2])
-
-lemma positive_integral_subalgebra:
-  assumes f: "f \<in> borel_measurable N" "\<And>x. 0 \<le> f x"
-  and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
-  shows "integral\<^sup>P N f = integral\<^sup>P M f"
-proof -
-  have [simp]: "\<And>f :: 'a \<Rightarrow> ereal. f \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable M"
-    using N by (auto simp: measurable_def)
-  have [simp]: "\<And>P. (AE x in N. P x) \<Longrightarrow> (AE x in M. P x)"
-    using N by (auto simp add: eventually_ae_filter null_sets_def)
-  have [simp]: "\<And>A. A \<in> sets N \<Longrightarrow> A \<in> sets M"
-    using N by auto
-  from f show ?thesis
-    apply induct
-    apply (simp_all add: positive_integral_add positive_integral_cmult positive_integral_monotone_convergence_SUP N)
-    apply (auto intro!: positive_integral_cong cong: positive_integral_cong simp: N(2)[symmetric])
-    done
-qed
-
-lemma positive_integral_nat_function:
-  fixes f :: "'a \<Rightarrow> nat"
-  assumes "f \<in> measurable M (count_space UNIV)"
-  shows "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<Sum>t. emeasure M {x\<in>space M. t < f x})"
-proof -
-  def F \<equiv> "\<lambda>i. {x\<in>space M. i < f x}"
-  with assms have [measurable]: "\<And>i. F i \<in> sets M"
-    by auto
-
-  { fix x assume "x \<in> space M"
-    have "(\<lambda>i. if i < f x then 1 else 0) sums (of_nat (f x)::real)"
-      using sums_If_finite[of "\<lambda>i. i < f x" "\<lambda>_. 1::real"] by simp
-    then have "(\<lambda>i. ereal(if i < f x then 1 else 0)) sums (ereal(of_nat(f x)))"
-      unfolding sums_ereal .
-    moreover have "\<And>i. ereal (if i < f x then 1 else 0) = indicator (F i) x"
-      using `x \<in> space M` by (simp add: one_ereal_def F_def)
-    ultimately have "ereal(of_nat(f x)) = (\<Sum>i. indicator (F i) x)"
-      by (simp add: sums_iff) }
-  then have "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. indicator (F i) x) \<partial>M)"
-    by (simp cong: positive_integral_cong)
-  also have "\<dots> = (\<Sum>i. emeasure M (F i))"
-    by (simp add: positive_integral_suminf)
-  finally show ?thesis
-    by (simp add: F_def)
-qed
-
-section "Lebesgue Integral"
-
-definition integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where
-  "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
-    (\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
-
-lemma borel_measurable_integrable[measurable_dest]:
-  "integrable M f \<Longrightarrow> f \<in> borel_measurable M"
-  by (auto simp: integrable_def)
-
-lemma integrableD[dest]:
-  assumes "integrable M f"
-  shows "f \<in> borel_measurable M" "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
-  using assms unfolding integrable_def by auto
-
-definition lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real" ("integral\<^sup>L") where
-  "integral\<^sup>L M f = real ((\<integral>\<^sup>+ x. ereal (f x) \<partial>M)) - real ((\<integral>\<^sup>+ x. ereal (- f x) \<partial>M))"
-
-syntax
-  "_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> 'a measure \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
-
-translations
-  "\<integral> x. f \<partial>M" == "CONST lebesgue_integral M (%x. f)"
-
-lemma integrableE:
-  assumes "integrable M f"
-  obtains r q where
-    "(\<integral>\<^sup>+x. ereal (f x)\<partial>M) = ereal r"
-    "(\<integral>\<^sup>+x. ereal (-f x)\<partial>M) = ereal q"
-    "f \<in> borel_measurable M" "integral\<^sup>L M f = r - q"
-  using assms unfolding integrable_def lebesgue_integral_def
-  using positive_integral_positive[of M "\<lambda>x. ereal (f x)"]
-  using positive_integral_positive[of M "\<lambda>x. ereal (-f x)"]
-  by (cases rule: ereal2_cases[of "(\<integral>\<^sup>+x. ereal (-f x)\<partial>M)" "(\<integral>\<^sup>+x. ereal (f x)\<partial>M)"]) auto
-
-lemma integral_cong:
-  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
-  shows "integral\<^sup>L M f = integral\<^sup>L M g"
-  using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def)
-
-lemma integral_cong_AE:
-  assumes cong: "AE x in M. f x = g x"
-  shows "integral\<^sup>L M f = integral\<^sup>L M g"
-proof -
-  have *: "AE x in M. ereal (f x) = ereal (g x)"
-    "AE x in M. ereal (- f x) = ereal (- g x)" using cong by auto
-  show ?thesis
-    unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
-qed
-
-lemma integrable_cong_AE:
-  assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
-  assumes "AE x in M. f x = g x"
-  shows "integrable M f = integrable M g"
-proof -
-  have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = (\<integral>\<^sup>+ x. ereal (g x) \<partial>M)"
-    "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) = (\<integral>\<^sup>+ x. ereal (- g x) \<partial>M)"
-    using assms by (auto intro!: positive_integral_cong_AE)
-  with assms show ?thesis
-    by (auto simp: integrable_def)
-qed
-
-lemma integrable_cong:
-  "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
-  by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
-
-lemma integral_mono_AE:
-  assumes fg: "integrable M f" "integrable M g"
-  and mono: "AE t in M. f t \<le> g t"
-  shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
-proof -
-  have "AE x in M. ereal (f x) \<le> ereal (g x)"
-    using mono by auto
-  moreover have "AE x in M. ereal (- g x) \<le> ereal (- f x)"
-    using mono by auto
-  ultimately show ?thesis using fg
-    by (auto intro!: add_mono positive_integral_mono_AE real_of_ereal_positive_mono
-             simp: positive_integral_positive lebesgue_integral_def algebra_simps)
-qed
-
-lemma integral_mono:
-  assumes "integrable M f" "integrable M g" "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
-  shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
-  using assms by (auto intro: integral_mono_AE)
-
-lemma positive_integral_eq_integral:
-  assumes f: "integrable M f"
-  assumes nonneg: "AE x in M. 0 \<le> f x" 
-  shows "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = integral\<^sup>L M f"
-proof -
-  have "(\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)"
-    using nonneg by (intro positive_integral_cong_AE) (auto split: split_max)
-  with f positive_integral_positive show ?thesis
-    by (cases "\<integral>\<^sup>+ x. ereal (f x) \<partial>M")
-       (auto simp add: lebesgue_integral_def positive_integral_max_0 integrable_def)
-qed
-  
-lemma integral_eq_positive_integral:
-  assumes f: "\<And>x. 0 \<le> f x"
-  shows "integral\<^sup>L M f = real (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
-proof -
-  { fix x have "max 0 (ereal (- f x)) = 0" using f[of x] by (simp split: split_max) }
-  then have "0 = (\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M)" by simp
-  also have "\<dots> = (\<integral>\<^sup>+ x. ereal (- f x) \<partial>M)" unfolding positive_integral_max_0 ..
-  finally show ?thesis
-    unfolding lebesgue_integral_def by simp
-qed
-
-lemma integral_minus[intro, simp]:
-  assumes "integrable M f"
-  shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^sup>L M f"
-  using assms by (auto simp: integrable_def lebesgue_integral_def)
-
-lemma integral_minus_iff[simp]:
-  "integrable M (\<lambda>x. - f x) \<longleftrightarrow> integrable M f"
-proof
-  assume "integrable M (\<lambda>x. - f x)"
-  then have "integrable M (\<lambda>x. - (- f x))"
-    by (rule integral_minus)
-  then show "integrable M f" by simp
-qed (rule integral_minus)
-
-lemma integral_of_positive_diff:
-  assumes integrable: "integrable M u" "integrable M v"
-  and f_def: "\<And>x. f x = u x - v x" and pos: "\<And>x. 0 \<le> u x" "\<And>x. 0 \<le> v x"
-  shows "integrable M f" and "integral\<^sup>L M f = integral\<^sup>L M u - integral\<^sup>L M v"
-proof -
-  let ?f = "\<lambda>x. max 0 (ereal (f x))"
-  let ?mf = "\<lambda>x. max 0 (ereal (- f x))"
-  let ?u = "\<lambda>x. max 0 (ereal (u x))"
-  let ?v = "\<lambda>x. max 0 (ereal (v x))"
-
-  from borel_measurable_diff[of u M v] integrable
-  have f_borel: "?f \<in> borel_measurable M" and
-    mf_borel: "?mf \<in> borel_measurable M" and
-    v_borel: "?v \<in> borel_measurable M" and
-    u_borel: "?u \<in> borel_measurable M" and
-    "f \<in> borel_measurable M"
-    by (auto simp: f_def[symmetric] integrable_def)
-
-  have "(\<integral>\<^sup>+ x. ereal (u x - v x) \<partial>M) \<le> integral\<^sup>P M ?u"
-    using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
-  moreover have "(\<integral>\<^sup>+ x. ereal (v x - u x) \<partial>M) \<le> integral\<^sup>P M ?v"
-    using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
-  ultimately show f: "integrable M f"
-    using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
-    by (auto simp: integrable_def f_def positive_integral_max_0)
-
-  have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
-    unfolding f_def using pos by (simp split: split_max)
-  then have "(\<integral>\<^sup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^sup>+ x. ?v x + ?f x \<partial>M)" by simp
-  then have "real (integral\<^sup>P M ?u + integral\<^sup>P M ?mf) =
-      real (integral\<^sup>P M ?v + integral\<^sup>P M ?f)"
-    using positive_integral_add[OF u_borel _ mf_borel]
-    using positive_integral_add[OF v_borel _ f_borel]
-    by auto
-  then show "integral\<^sup>L M f = integral\<^sup>L M u - integral\<^sup>L M v"
-    unfolding positive_integral_max_0
-    unfolding pos[THEN integral_eq_positive_integral]
-    using integrable f by (auto elim!: integrableE)
-qed
-
-lemma integral_linear:
-  assumes "integrable M f" "integrable M g" and "0 \<le> a"
-  shows "integrable M (\<lambda>t. a * f t + g t)"
-  and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^sup>L M f + integral\<^sup>L M g" (is ?EQ)
-proof -
-  let ?f = "\<lambda>x. max 0 (ereal (f x))"
-  let ?g = "\<lambda>x. max 0 (ereal (g x))"
-  let ?mf = "\<lambda>x. max 0 (ereal (- f x))"
-  let ?mg = "\<lambda>x. max 0 (ereal (- g x))"
-  let ?p = "\<lambda>t. max 0 (a * f t) + max 0 (g t)"
-  let ?n = "\<lambda>t. max 0 (- (a * f t)) + max 0 (- g t)"
-
-  from assms have linear:
-    "(\<integral>\<^sup>+ x. ereal a * ?f x + ?g x \<partial>M) = ereal a * integral\<^sup>P M ?f + integral\<^sup>P M ?g"
-    "(\<integral>\<^sup>+ x. ereal a * ?mf x + ?mg x \<partial>M) = ereal a * integral\<^sup>P M ?mf + integral\<^sup>P M ?mg"
-    by (auto intro!: positive_integral_linear simp: integrable_def)
-
-  have *: "(\<integral>\<^sup>+x. ereal (- ?p x) \<partial>M) = 0" "(\<integral>\<^sup>+x. ereal (- ?n x) \<partial>M) = 0"
-    using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def)
-  have **: "\<And>x. ereal a * ?f x + ?g x = max 0 (ereal (?p x))"
-           "\<And>x. ereal a * ?mf x + ?mg x = max 0 (ereal (?n x))"
-    using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff)
-
-  have "integrable M ?p" "integrable M ?n"
-      "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
-    using linear assms unfolding integrable_def ** *
-    by (auto simp: positive_integral_max_0)
-  note diff = integral_of_positive_diff[OF this]
-
-  show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
-  from assms linear show ?EQ
-    unfolding diff(2) ** positive_integral_max_0
-    unfolding lebesgue_integral_def *
-    by (auto elim!: integrableE simp: field_simps)
-qed
-
-lemma integral_add[simp, intro]:
-  assumes "integrable M f" "integrable M g"
-  shows "integrable M (\<lambda>t. f t + g t)"
-  and "(\<integral> t. f t + g t \<partial>M) = integral\<^sup>L M f + integral\<^sup>L M g"
-  using assms integral_linear[where a=1] by auto
-
-lemma integral_zero[simp, intro]:
-  shows "integrable M (\<lambda>x. 0)" "(\<integral> x.0 \<partial>M) = 0"
-  unfolding integrable_def lebesgue_integral_def
-  by auto
-
-lemma lebesgue_integral_uminus:
-    "(\<integral>x. - f x \<partial>M) = - integral\<^sup>L M f"
-  unfolding lebesgue_integral_def by simp
-
-lemma lebesgue_integral_cmult_nonneg:
-  assumes f: "f \<in> borel_measurable M" and "0 \<le> c"
-  shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^sup>L M f"
-proof -
-  { have "real (ereal c * integral\<^sup>P M (\<lambda>x. max 0 (ereal (f x)))) =
-      real (integral\<^sup>P M (\<lambda>x. ereal c * max 0 (ereal (f x))))"
-      using f `0 \<le> c` by (subst positive_integral_cmult) auto
-    also have "\<dots> = real (integral\<^sup>P M (\<lambda>x. max 0 (ereal (c * f x))))"
-      using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def zero_le_mult_iff)
-    finally have "real (integral\<^sup>P M (\<lambda>x. ereal (c * f x))) = c * real (integral\<^sup>P M (\<lambda>x. ereal (f x)))"
-      by (simp add: positive_integral_max_0) }
-  moreover
-  { have "real (ereal c * integral\<^sup>P M (\<lambda>x. max 0 (ereal (- f x)))) =
-      real (integral\<^sup>P M (\<lambda>x. ereal c * max 0 (ereal (- f x))))"
-      using f `0 \<le> c` by (subst positive_integral_cmult) auto
-    also have "\<dots> = real (integral\<^sup>P M (\<lambda>x. max 0 (ereal (- c * f x))))"
-      using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def mult_le_0_iff)
-    finally have "real (integral\<^sup>P M (\<lambda>x. ereal (- c * f x))) = c * real (integral\<^sup>P M (\<lambda>x. ereal (- f x)))"
-      by (simp add: positive_integral_max_0) }
-  ultimately show ?thesis
-    by (simp add: lebesgue_integral_def field_simps)
-qed
-
-lemma lebesgue_integral_cmult:
-  assumes f: "f \<in> borel_measurable M"
-  shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^sup>L M f"
-proof (cases rule: linorder_le_cases)
-  assume "0 \<le> c" with f show ?thesis by (rule lebesgue_integral_cmult_nonneg)
-next
-  assume "c \<le> 0"
-  with lebesgue_integral_cmult_nonneg[OF f, of "-c"]
-  show ?thesis
-    by (simp add: lebesgue_integral_def)
-qed
-
-lemma lebesgue_integral_multc:
-  "f \<in> borel_measurable M \<Longrightarrow> (\<integral>x. f x * c \<partial>M) = integral\<^sup>L M f * c"
-  using lebesgue_integral_cmult[of f M c] by (simp add: ac_simps)
-
-lemma integral_multc:
-  "integrable M f \<Longrightarrow> (\<integral> x. f x * c \<partial>M) = integral\<^sup>L M f * c"
-  by (simp add: lebesgue_integral_multc)
-
-lemma integral_cmult[simp, intro]:
-  assumes "integrable M f"
-  shows "integrable M (\<lambda>t. a * f t)" (is ?P)
-  and "(\<integral> t. a * f t \<partial>M) = a * integral\<^sup>L M f" (is ?I)
-proof -
-  have "integrable M (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t \<partial>M) = a * integral\<^sup>L M f"
-  proof (cases rule: le_cases)
-    assume "0 \<le> a" show ?thesis
-      using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
-      by simp
-  next
-    assume "a \<le> 0" hence "0 \<le> - a" by auto
-    have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
-    show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
-        integral_minus(1)[of M "\<lambda>t. - a * f t"]
-      unfolding * integral_zero by simp
-  qed
-  thus ?P ?I by auto
-qed
-
-lemma integral_diff[simp, intro]:
-  assumes f: "integrable M f" and g: "integrable M g"
-  shows "integrable M (\<lambda>t. f t - g t)"
-  and "(\<integral> t. f t - g t \<partial>M) = integral\<^sup>L M f - integral\<^sup>L M g"
-  using integral_add[OF f integral_minus(1)[OF g]]
-  unfolding integral_minus(2)[OF g]
-  by auto
-
-lemma integral_indicator[simp, intro]:
-  assumes "A \<in> sets M" and "(emeasure M) A \<noteq> \<infinity>"
-  shows "integral\<^sup>L M (indicator A) = real (emeasure M A)" (is ?int)
-  and "integrable M (indicator A)" (is ?able)
-proof -
-  from `A \<in> sets M` have *:
-    "\<And>x. ereal (indicator A x) = indicator A x"
-    "(\<integral>\<^sup>+x. ereal (- indicator A x) \<partial>M) = 0"
-    by (auto split: split_indicator simp: positive_integral_0_iff_AE one_ereal_def)
-  show ?int ?able
-    using assms unfolding lebesgue_integral_def integrable_def
-    by (auto simp: *)
-qed
-
-lemma integral_cmul_indicator:
-  assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> (emeasure M) A \<noteq> \<infinity>"
-  shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P)
-  and "(\<integral>x. c * indicator A x \<partial>M) = c * real ((emeasure M) A)" (is ?I)
-proof -
-  show ?P
-  proof (cases "c = 0")
-    case False with assms show ?thesis by simp
-  qed simp
-
-  show ?I
-  proof (cases "c = 0")
-    case False with assms show ?thesis by simp
-  qed simp
-qed
-
-lemma integral_setsum[simp, intro]:
-  assumes "\<And>n. n \<in> S \<Longrightarrow> integrable M (f n)"
-  shows "(\<integral>x. (\<Sum> i \<in> S. f i x) \<partial>M) = (\<Sum> i \<in> S. integral\<^sup>L M (f i))" (is "?int S")
-    and "integrable M (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
-proof -
-  have "?int S \<and> ?I S"
-  proof (cases "finite S")
-    assume "finite S"
-    from this assms show ?thesis by (induct S) simp_all
-  qed simp
-  thus "?int S" and "?I S" by auto
-qed
-
-lemma integrable_bound:
-  assumes "integrable M f" and f: "AE x in M. \<bar>g x\<bar> \<le> f x"
-  assumes borel: "g \<in> borel_measurable M"
-  shows "integrable M g"
-proof -
-  have "(\<integral>\<^sup>+ x. ereal (g x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal \<bar>g x\<bar> \<partial>M)"
-    by (auto intro!: positive_integral_mono)
-  also have "\<dots> \<le> (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
-    using f by (auto intro!: positive_integral_mono_AE)
-  also have "\<dots> < \<infinity>"
-    using `integrable M f` unfolding integrable_def by auto
-  finally have pos: "(\<integral>\<^sup>+ x. ereal (g x) \<partial>M) < \<infinity>" .
-
-  have "(\<integral>\<^sup>+ x. ereal (- g x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (\<bar>g x\<bar>) \<partial>M)"
-    by (auto intro!: positive_integral_mono)
-  also have "\<dots> \<le> (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
-    using f by (auto intro!: positive_integral_mono_AE)
-  also have "\<dots> < \<infinity>"
-    using `integrable M f` unfolding integrable_def by auto
-  finally have neg: "(\<integral>\<^sup>+ x. ereal (- g x) \<partial>M) < \<infinity>" .
-
-  from neg pos borel show ?thesis
-    unfolding integrable_def by auto
-qed
-
-lemma integrable_abs:
-  assumes f[measurable]: "integrable M f"
-  shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
-proof -
-  from assms have *: "(\<integral>\<^sup>+x. ereal (- \<bar>f x\<bar>)\<partial>M) = 0"
-    "\<And>x. ereal \<bar>f x\<bar> = max 0 (ereal (f x)) + max 0 (ereal (- f x))"
-    by (auto simp: integrable_def positive_integral_0_iff_AE split: split_max)
-  with assms show ?thesis
-    by (simp add: positive_integral_add positive_integral_max_0 integrable_def)
-qed
-
-lemma integral_subalgebra:
-  assumes borel: "f \<in> borel_measurable N"
-  and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
-  shows "integrable N f \<longleftrightarrow> integrable M f" (is ?P)
-    and "integral\<^sup>L N f = integral\<^sup>L M f" (is ?I)
-proof -
-  have "(\<integral>\<^sup>+ x. max 0 (ereal (f x)) \<partial>N) = (\<integral>\<^sup>+ x. max 0 (ereal (f x)) \<partial>M)"
-       "(\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>N) = (\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M)"
-    using borel by (auto intro!: positive_integral_subalgebra N)
-  moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
-    using assms unfolding measurable_def by auto
-  ultimately show ?P ?I
-    by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0)
-qed
-
-lemma lebesgue_integral_nonneg:
-  assumes ae: "(AE x in M. 0 \<le> f x)" shows "0 \<le> integral\<^sup>L M f"
-proof -
-  have "(\<integral>\<^sup>+x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^sup>+x. 0 \<partial>M)"
-    using ae by (intro positive_integral_cong_AE) (auto simp: max_def)
-  then show ?thesis
-    by (auto simp: lebesgue_integral_def positive_integral_max_0
-             intro!: real_of_ereal_pos positive_integral_positive)
-qed
-
-lemma integrable_abs_iff:
-  "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
-  by (auto intro!: integrable_bound[where g=f] integrable_abs)
-
-lemma integrable_max:
-  assumes int: "integrable M f" "integrable M g"
-  shows "integrable M (\<lambda> x. max (f x) (g x))"
-proof (rule integrable_bound)
-  show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
-    using int by (simp add: integrable_abs)
-  show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
-    using int unfolding integrable_def by auto
-qed auto
-
-lemma integrable_min:
-  assumes int: "integrable M f" "integrable M g"
-  shows "integrable M (\<lambda> x. min (f x) (g x))"
-proof (rule integrable_bound)
-  show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
-    using int by (simp add: integrable_abs)
-  show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
-    using int unfolding integrable_def by auto
-qed auto
-
-lemma integral_triangle_inequality:
-  assumes "integrable M f"
-  shows "\<bar>integral\<^sup>L M f\<bar> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
-proof -
-  have "\<bar>integral\<^sup>L M f\<bar> = max (integral\<^sup>L M f) (- integral\<^sup>L M f)" by auto
-  also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
-      using assms integral_minus(2)[of M f, symmetric]
-      by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
-  finally show ?thesis .
-qed
-
-lemma integrable_nonneg:
-  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "(\<integral>\<^sup>+ x. f x \<partial>M) \<noteq> \<infinity>"
-  shows "integrable M f"
-  unfolding integrable_def
-proof (intro conjI f)
-  have "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) = 0"
-    using f by (subst positive_integral_0_iff_AE) auto
-  then show "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>" by simp
-qed
-
-lemma integral_positive:
-  assumes "integrable M f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
-  shows "0 \<le> integral\<^sup>L M f"
-proof -
-  have "0 = (\<integral>x. 0 \<partial>M)" by auto
-  also have "\<dots> \<le> integral\<^sup>L M f"
-    using assms by (rule integral_mono[OF integral_zero(1)])
-  finally show ?thesis .
-qed
-
-lemma integral_monotone_convergence_pos:
-  assumes i: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
-    and pos: "\<And>i. AE x in M. 0 \<le> f i x"
-    and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
-    and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
-    and u: "u \<in> borel_measurable M"
-  shows "integrable M u"
-  and "integral\<^sup>L M u = x"
-proof -
-  have "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = (SUP n. (\<integral>\<^sup>+ x. ereal (f n x) \<partial>M))"
-  proof (subst positive_integral_monotone_convergence_SUP_AE[symmetric])
-    fix i
-    from mono pos show "AE x in M. ereal (f i x) \<le> ereal (f (Suc i) x) \<and> 0 \<le> ereal (f i x)"
-      by eventually_elim (auto simp: mono_def)
-    show "(\<lambda>x. ereal (f i x)) \<in> borel_measurable M"
-      using i by auto
-  next
-    show "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = \<integral>\<^sup>+ x. (SUP i. ereal (f i x)) \<partial>M"
-      apply (rule positive_integral_cong_AE)
-      using lim mono
-      by eventually_elim (simp add: SUP_eq_LIMSEQ[THEN iffD2])
-  qed
-  also have "\<dots> = ereal x"
-    using mono i unfolding positive_integral_eq_integral[OF i pos]
-    by (subst SUP_eq_LIMSEQ) (auto simp: mono_def intro!: integral_mono_AE ilim)
-  finally have "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = ereal x" .
-  moreover have "(\<integral>\<^sup>+ x. ereal (- u x) \<partial>M) = 0"
-  proof (subst positive_integral_0_iff_AE)
-    show "(\<lambda>x. ereal (- u x)) \<in> borel_measurable M"
-      using u by auto
-    from mono pos[of 0] lim show "AE x in M. ereal (- u x) \<le> 0"
-    proof eventually_elim
-      fix x assume "mono (\<lambda>n. f n x)" "0 \<le> f 0 x" "(\<lambda>i. f i x) ----> u x"
-      then show "ereal (- u x) \<le> 0"
-        using incseq_le[of "\<lambda>n. f n x" "u x" 0] by (simp add: mono_def incseq_def)
-    qed
-  qed
-  ultimately show "integrable M u" "integral\<^sup>L M u = x"
-    by (auto simp: integrable_def lebesgue_integral_def u)
-qed
-
-lemma integral_monotone_convergence:
-  assumes f: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
-  and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
-  and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
-  and u: "u \<in> borel_measurable M"
-  shows "integrable M u"
-  and "integral\<^sup>L M u = x"
-proof -
-  have 1: "\<And>i. integrable M (\<lambda>x. f i x - f 0 x)"
-    using f by auto
-  have 2: "AE x in M. mono (\<lambda>n. f n x - f 0 x)"
-    using mono by (auto simp: mono_def le_fun_def)
-  have 3: "\<And>n. AE x in M. 0 \<le> f n x - f 0 x"
-    using mono by (auto simp: field_simps mono_def le_fun_def)
-  have 4: "AE x in M. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
-    using lim by (auto intro!: tendsto_diff)
-  have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^sup>L M (f 0)"
-    using f ilim by (auto intro!: tendsto_diff)
-  have 6: "(\<lambda>x. u x - f 0 x) \<in> borel_measurable M"
-    using f[of 0] u by auto
-  note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5 6]
-  have "integrable M (\<lambda>x. (u x - f 0 x) + f 0 x)"
-    using diff(1) f by (rule integral_add(1))
-  with diff(2) f show "integrable M u" "integral\<^sup>L M u = x"
-    by auto
-qed
-
-lemma integral_0_iff:
-  assumes "integrable M f"
-  shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> (emeasure M) {x\<in>space M. f x \<noteq> 0} = 0"
-proof -
-  have *: "(\<integral>\<^sup>+x. ereal (- \<bar>f x\<bar>) \<partial>M) = 0"
-    using assms by (auto simp: positive_integral_0_iff_AE integrable_def)
-  have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
-  hence "(\<lambda>x. ereal (\<bar>f x\<bar>)) \<in> borel_measurable M"
-    "(\<integral>\<^sup>+ x. ereal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto
-  from positive_integral_0_iff[OF this(1)] this(2)
-  show ?thesis unfolding lebesgue_integral_def *
-    using positive_integral_positive[of M "\<lambda>x. ereal \<bar>f x\<bar>"]
-    by (auto simp add: real_of_ereal_eq_0)
-qed
-
-lemma integral_real:
-  "AE x in M. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^sup>P M f) - real (integral\<^sup>P M (\<lambda>x. - f x))"
-  using assms unfolding lebesgue_integral_def
-  by (subst (1 2) positive_integral_cong_AE) (auto simp add: ereal_real)
-
-lemma (in finite_measure) lebesgue_integral_const[simp]:
-  shows "integrable M (\<lambda>x. a)"
-  and  "(\<integral>x. a \<partial>M) = a * measure M (space M)"
-proof -
-  { fix a :: real assume "0 \<le> a"
-    then have "(\<integral>\<^sup>+ x. ereal a \<partial>M) = ereal a * (emeasure M) (space M)"
-      by (subst positive_integral_const) auto
-    moreover
-    from `0 \<le> a` have "(\<integral>\<^sup>+ x. ereal (-a) \<partial>M) = 0"
-      by (subst positive_integral_0_iff_AE) auto
-    ultimately have "integrable M (\<lambda>x. a)" by (auto simp: integrable_def) }
-  note * = this
-  show "integrable M (\<lambda>x. a)"
-  proof cases
-    assume "0 \<le> a" with * show ?thesis .
-  next
-    assume "\<not> 0 \<le> a"
-    then have "0 \<le> -a" by auto
-    from *[OF this] show ?thesis by simp
-  qed
-  show "(\<integral>x. a \<partial>M) = a * measure M (space M)"
-    by (simp add: lebesgue_integral_def positive_integral_const_If emeasure_eq_measure)
-qed
-
-lemma (in finite_measure) integrable_const_bound:
-  assumes "AE x in M. \<bar>f x\<bar> \<le> B" and "f \<in> borel_measurable M" shows "integrable M f"
-  by (auto intro: integrable_bound[where f="\<lambda>x. B"] lebesgue_integral_const assms)
-
-lemma (in finite_measure) integral_less_AE:
-  assumes int: "integrable M X" "integrable M Y"
-  assumes A: "(emeasure M) A \<noteq> 0" "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> X x \<noteq> Y x"
-  assumes gt: "AE x in M. X x \<le> Y x"
-  shows "integral\<^sup>L M X < integral\<^sup>L M Y"
-proof -
-  have "integral\<^sup>L M X \<le> integral\<^sup>L M Y"
-    using gt int by (intro integral_mono_AE) auto
-  moreover
-  have "integral\<^sup>L M X \<noteq> integral\<^sup>L M Y"
-  proof
-    assume eq: "integral\<^sup>L M X = integral\<^sup>L M Y"
-    have "integral\<^sup>L M (\<lambda>x. \<bar>Y x - X x\<bar>) = integral\<^sup>L M (\<lambda>x. Y x - X x)"
-      using gt by (intro integral_cong_AE) auto
-    also have "\<dots> = 0"
-      using eq int by simp
-    finally have "(emeasure M) {x \<in> space M. Y x - X x \<noteq> 0} = 0"
-      using int by (simp add: integral_0_iff)
-    moreover
-    have "(\<integral>\<^sup>+x. indicator A x \<partial>M) \<le> (\<integral>\<^sup>+x. indicator {x \<in> space M. Y x - X x \<noteq> 0} x \<partial>M)"
-      using A by (intro positive_integral_mono_AE) auto
-    then have "(emeasure M) A \<le> (emeasure M) {x \<in> space M. Y x - X x \<noteq> 0}"
-      using int A by (simp add: integrable_def)
-    ultimately have "emeasure M A = 0"
-      using emeasure_nonneg[of M A] by simp
-    with `(emeasure M) A \<noteq> 0` show False by auto
-  qed
-  ultimately show ?thesis by auto
-qed
-
-lemma (in finite_measure) integral_less_AE_space:
-  assumes int: "integrable M X" "integrable M Y"
-  assumes gt: "AE x in M. X x < Y x" "(emeasure M) (space M) \<noteq> 0"
-  shows "integral\<^sup>L M X < integral\<^sup>L M Y"
-  using gt by (intro integral_less_AE[OF int, where A="space M"]) auto
-
-lemma integral_dominated_convergence:
-  assumes u[measurable]: "\<And>i. integrable M (u i)" and bound: "\<And>j. AE x in M. \<bar>u j x\<bar> \<le> w x"
-  and w[measurable]: "integrable M w"
-  and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
-  and [measurable]: "u' \<in> borel_measurable M"
-  shows "integrable M u'"
-  and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar> \<partial>M)) ----> 0" (is "?lim_diff")
-  and "(\<lambda>i. integral\<^sup>L M (u i)) ----> integral\<^sup>L M u'" (is ?lim)
-proof -
-  have all_bound: "AE x in M. \<forall>j. \<bar>u j x\<bar> \<le> w x"
-    using bound by (auto simp: AE_all_countable)
-  with u' have u'_bound: "AE x in M. \<bar>u' x\<bar> \<le> w x"
-    by eventually_elim (auto intro: LIMSEQ_le_const2 tendsto_rabs)
-
-  from bound[of 0] have w_pos: "AE x in M. 0 \<le> w x"
-    by eventually_elim auto
-
-  show "integrable M u'"
-    by (rule integrable_bound) fact+
-
-  let ?diff = "\<lambda>n x. 2 * w x - \<bar>u n x - u' x\<bar>"
-  have diff: "\<And>n. integrable M (\<lambda>x. \<bar>u n x - u' x\<bar>)"
-    using w u `integrable M u'` by (auto intro!: integrable_abs)
-
-  from u'_bound all_bound
-  have diff_less_2w: "AE x in M. \<forall>j. \<bar>u j x - u' x\<bar> \<le> 2 * w x"
-  proof (eventually_elim, intro allI)
-    fix x j assume *: "\<bar>u' x\<bar> \<le> w x" "\<forall>j. \<bar>u j x\<bar> \<le> w x"
-    then have "\<bar>u j x - u' x\<bar> \<le> \<bar>u j x\<bar> + \<bar>u' x\<bar>" by auto
-    also have "\<dots> \<le> w x + w x"
-      using * by (intro add_mono) auto
-    finally show "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp
-  qed
-
-  have PI_diff: "\<And>n. (\<integral>\<^sup>+ x. ereal (?diff n x) \<partial>M) =
-    (\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) - (\<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
-    using diff w diff_less_2w w_pos
-    by (subst positive_integral_diff[symmetric])
-       (auto simp: integrable_def intro!: positive_integral_cong_AE)
-
-  have "integrable M (\<lambda>x. 2 * w x)"
-    using w by auto
-  hence I2w_fin: "(\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) \<noteq> \<infinity>" and
-    borel_2w: "(\<lambda>x. ereal (2 * w x)) \<in> borel_measurable M"
-    unfolding integrable_def by auto
-
-  have "limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = 0" (is "limsup ?f = 0")
-  proof cases
-    assume eq_0: "(\<integral>\<^sup>+ x. max 0 (ereal (2 * w x)) \<partial>M) = 0" (is "?wx = 0")
-    { fix n
-      have "?f n \<le> ?wx" (is "integral\<^sup>P M ?f' \<le> _")
-        using diff_less_2w unfolding positive_integral_max_0
-        by (intro positive_integral_mono_AE) auto
-      then have "?f n = 0"
-        using positive_integral_positive[of M ?f'] eq_0 by auto }
-    then show ?thesis by (simp add: Limsup_const)
-  next
-    assume neq_0: "(\<integral>\<^sup>+ x. max 0 (ereal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0")
-    have "0 = limsup (\<lambda>n. 0 :: ereal)" by (simp add: Limsup_const)
-    also have "\<dots> \<le> limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
-      by (simp add: Limsup_mono  positive_integral_positive)
-    finally have pos: "0 \<le> limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)" .
-    have "?wx = (\<integral>\<^sup>+ x. liminf (\<lambda>n. max 0 (ereal (?diff n x))) \<partial>M)"
-      using u'
-    proof (intro positive_integral_cong_AE, eventually_elim)
-      fix x assume u': "(\<lambda>i. u i x) ----> u' x"
-      show "max 0 (ereal (2 * w x)) = liminf (\<lambda>n. max 0 (ereal (?diff n x)))"
-        unfolding ereal_max_0
-      proof (rule lim_imp_Liminf[symmetric], unfold lim_ereal)
-        have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
-          using u' by (safe intro!: tendsto_intros)
-        then show "(\<lambda>i. max 0 (?diff i x)) ----> max 0 (2 * w x)"
-          by (auto intro!: tendsto_max)
-      qed (rule trivial_limit_sequentially)
-    qed
-    also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^sup>+ x. max 0 (ereal (?diff n x)) \<partial>M)"
-      using w u unfolding integrable_def
-      by (intro positive_integral_lim_INF) (auto intro!: positive_integral_lim_INF)
-    also have "\<dots> = (\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) -
-        limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
-      unfolding PI_diff positive_integral_max_0
-      using positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"]
-      by (subst liminf_ereal_cminus) auto
-    finally show ?thesis
-      using neq_0 I2w_fin positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"] pos
-      unfolding positive_integral_max_0
-      by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"])
-         auto
-  qed
-
-  have "liminf ?f \<le> limsup ?f"
-    by (intro Liminf_le_Limsup trivial_limit_sequentially)
-  moreover
-  { have "0 = liminf (\<lambda>n. 0 :: ereal)" by (simp add: Liminf_const)
-    also have "\<dots> \<le> liminf ?f"
-      by (simp add: Liminf_mono positive_integral_positive)
-    finally have "0 \<le> liminf ?f" . }
-  ultimately have liminf_limsup_eq: "liminf ?f = ereal 0" "limsup ?f = ereal 0"
-    using `limsup ?f = 0` by auto
-  have "\<And>n. (\<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = ereal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)"
-    using diff positive_integral_positive[of M]
-    by (subst integral_eq_positive_integral[of _ M]) (auto simp: ereal_real integrable_def)
-  then show ?lim_diff
-    using Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq]
-    by simp
-
-  show ?lim
-  proof (rule LIMSEQ_I)
-    fix r :: real assume "0 < r"
-    from LIMSEQ_D[OF `?lim_diff` this]
-    obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M) < r"
-      using diff by (auto simp: integral_positive)
-
-    show "\<exists>N. \<forall>n\<ge>N. norm (integral\<^sup>L M (u n) - integral\<^sup>L M u') < r"
-    proof (safe intro!: exI[of _ N])
-      fix n assume "N \<le> n"
-      have "\<bar>integral\<^sup>L M (u n) - integral\<^sup>L M u'\<bar> = \<bar>(\<integral>x. u n x - u' x \<partial>M)\<bar>"
-        using u `integrable M u'` by auto
-      also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)" using u `integrable M u'`
-        by (rule_tac integral_triangle_inequality) auto
-      also note N[OF `N \<le> n`]
-      finally show "norm (integral\<^sup>L M (u n) - integral\<^sup>L M u') < r" by simp
-    qed
-  qed
-qed
-
-lemma integral_sums:
-  assumes integrable[measurable]: "\<And>i. integrable M (f i)"
-  and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
-  and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
-  shows "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?S")
-  and "(\<lambda>i. integral\<^sup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is ?integral)
-proof -
-  have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
-    using summable unfolding summable_def by auto
-  from bchoice[OF this]
-  obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto
-  then have w_borel: "w \<in> borel_measurable M" unfolding sums_def
-    by (rule borel_measurable_LIMSEQ) auto
-
-  let ?w = "\<lambda>y. if y \<in> space M then w y else 0"
-
-  obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M)) sums x"
-    using sums unfolding summable_def ..
-
-  have 1: "\<And>n. integrable M (\<lambda>x. \<Sum>i<n. f i x)"
-    using integrable by auto
-
-  have 2: "\<And>j. AE x in M. \<bar>\<Sum>i<j. f i x\<bar> \<le> ?w x"
-    using AE_space
-  proof eventually_elim
-    fix j x assume [simp]: "x \<in> space M"
-    have "\<bar>\<Sum>i<j. f i x\<bar> \<le> (\<Sum>i<j. \<bar>f i x\<bar>)" by (rule setsum_abs)
-    also have "\<dots> \<le> w x" using w[of x] setsum_le_suminf[of "\<lambda>i. \<bar>f i x\<bar>"] unfolding sums_iff by auto
-    finally show "\<bar>\<Sum>i<j. f i x\<bar> \<le> ?w x" by simp
-  qed
-
-  have 3: "integrable M ?w"
-  proof (rule integral_monotone_convergence(1))
-    let ?F = "\<lambda>n y. (\<Sum>i<n. \<bar>f i y\<bar>)"
-    let ?w' = "\<lambda>n y. if y \<in> space M then ?F n y else 0"
-    have "\<And>n. integrable M (?F n)"
-      using integrable by (auto intro!: integrable_abs)
-    thus "\<And>n. integrable M (?w' n)" by (simp cong: integrable_cong)
-    show "AE x in M. mono (\<lambda>n. ?w' n x)"
-      by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
-    show "AE x in M. (\<lambda>n. ?w' n x) ----> ?w x"
-        using w by (simp_all add: tendsto_const sums_def)
-    have *: "\<And>n. integral\<^sup>L M (?w' n) = (\<Sum>i< n. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
-      using integrable by (simp add: integrable_abs cong: integral_cong)
-    from abs_sum
-    show "(\<lambda>i. integral\<^sup>L M (?w' i)) ----> x" unfolding * sums_def .
-  qed (simp add: w_borel measurable_If_set)
-
-  from summable[THEN summable_rabs_cancel]
-  have 4: "AE x in M. (\<lambda>n. \<Sum>i<n. f i x) ----> (\<Sum>i. f i x)"
-    by (auto intro: summable_LIMSEQ)
-
-  note int = integral_dominated_convergence(1,3)[OF 1 2 3 4
-    borel_measurable_suminf[OF integrableD(1)[OF integrable]]]
-
-  from int show "integrable M ?S" by simp
-
-  show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF integrable]
-    using int(2) by simp
-qed
-
-lemma integrable_mult_indicator:
-  "A \<in> sets M \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. f x * indicator A x)"
-  by (rule integrable_bound[where f="\<lambda>x. \<bar>f x\<bar>"])
-     (auto intro: integrable_abs split: split_indicator)
-
-lemma tendsto_integral_at_top:
-  fixes M :: "real measure"
-  assumes M: "sets M = sets borel" and f[measurable]: "integrable M f"
-  shows "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
-proof -
-  have M_measure[simp]: "borel_measurable M = borel_measurable borel"
-    using M by (simp add: sets_eq_imp_space_eq measurable_def)
-  { fix f assume f: "integrable M f" "\<And>x. 0 \<le> f x"
-    then have [measurable]: "f \<in> borel_measurable borel"
-      by (simp add: integrable_def)
-    have "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
-    proof (rule tendsto_at_topI_sequentially)
-      have "\<And>j. AE x in M. \<bar>f x * indicator {.. j} x\<bar> \<le> f x"
-        using f(2) by (intro AE_I2) (auto split: split_indicator)
-      have int: "\<And>n. integrable M (\<lambda>x. f x * indicator {.. n} x)"
-        by (rule integrable_mult_indicator) (auto simp: M f)
-      show "(\<lambda>n. \<integral> x. f x * indicator {..real n} x \<partial>M) ----> integral\<^sup>L M f"
-      proof (rule integral_dominated_convergence)
-        { fix x have "eventually (\<lambda>n. f x * indicator {..real n} x = f x) sequentially"
-            by (rule eventually_sequentiallyI[of "natceiling x"])
-               (auto split: split_indicator simp: natceiling_le_eq) }
-        from filterlim_cong[OF refl refl this]
-        show "AE x in M. (\<lambda>n. f x * indicator {..real n} x) ----> f x"
-          by (simp add: tendsto_const)
-      qed (fact+, simp)
-      show "mono (\<lambda>y. \<integral> x. f x * indicator {..y} x \<partial>M)"
-        by (intro monoI integral_mono int) (auto split: split_indicator intro: f)
-    qed }
-  note nonneg = this
-  let ?P = "\<lambda>y. \<integral> x. max 0 (f x) * indicator {..y} x \<partial>M"
-  let ?N = "\<lambda>y. \<integral> x. max 0 (- f x) * indicator {..y} x \<partial>M"
-  let ?p = "integral\<^sup>L M (\<lambda>x. max 0 (f x))"
-  let ?n = "integral\<^sup>L M (\<lambda>x. max 0 (- f x))"
-  have "(?P ---> ?p) at_top" "(?N ---> ?n) at_top"
-    by (auto intro!: nonneg integrable_max f)
-  note tendsto_diff[OF this]
-  also have "(\<lambda>y. ?P y - ?N y) = (\<lambda>y. \<integral> x. f x * indicator {..y} x \<partial>M)"
-    by (subst integral_diff(2)[symmetric])
-       (auto intro!: integrable_mult_indicator integrable_max f integral_cong ext
-             simp: M split: split_max)
-  also have "?p - ?n = integral\<^sup>L M f"
-    by (subst integral_diff(2)[symmetric])
-       (auto intro!: integrable_max f integral_cong ext simp: M split: split_max)
-  finally show ?thesis .
-qed
-
-lemma integral_monotone_convergence_at_top:
-  fixes M :: "real measure"
-  assumes M: "sets M = sets borel"
-  assumes nonneg: "AE x in M. 0 \<le> f x"
-  assumes borel: "f \<in> borel_measurable borel"
-  assumes int: "\<And>y. integrable M (\<lambda>x. f x * indicator {.. y} x)"
-  assumes conv: "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> x) at_top"
-  shows "integrable M f" "integral\<^sup>L M f = x"
-proof -
-  from nonneg have "AE x in M. mono (\<lambda>n::nat. f x * indicator {..real n} x)"
-    by (auto split: split_indicator intro!: monoI)
-  { fix x have "eventually (\<lambda>n. f x * indicator {..real n} x = f x) sequentially"
-      by (rule eventually_sequentiallyI[of "natceiling x"])
-         (auto split: split_indicator simp: natceiling_le_eq) }
-  from filterlim_cong[OF refl refl this]
-  have "AE x in M. (\<lambda>i. f x * indicator {..real i} x) ----> f x"
-    by (simp add: tendsto_const)
-  have "(\<lambda>i. \<integral> x. f x * indicator {..real i} x \<partial>M) ----> x"
-    using conv filterlim_real_sequentially by (rule filterlim_compose)
-  have M_measure[simp]: "borel_measurable M = borel_measurable borel"
-    using M by (simp add: sets_eq_imp_space_eq measurable_def)
-  have "f \<in> borel_measurable M"
-    using borel by simp
-  show "integrable M f"
-    by (rule integral_monotone_convergence) fact+
-  show "integral\<^sup>L M f = x"
-    by (rule integral_monotone_convergence) fact+
-qed
-
-
-section "Lebesgue integration on countable spaces"
-
-lemma integral_on_countable:
-  assumes f: "f \<in> borel_measurable M"
-  and bij: "bij_betw enum S (f ` space M)"
-  and enum_zero: "enum ` (-S) \<subseteq> {0}"
-  and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> (emeasure M) (f -` {x} \<inter> space M) \<noteq> \<infinity>"
-  and abs_summable: "summable (\<lambda>r. \<bar>enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))\<bar>)"
-  shows "integrable M f"
-  and "(\<lambda>r. enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))) sums integral\<^sup>L M f" (is ?sums)
-proof -
-  let ?A = "\<lambda>r. f -` {enum r} \<inter> space M"
-  let ?F = "\<lambda>r x. enum r * indicator (?A r) x"
-  have enum_eq: "\<And>r. enum r * real ((emeasure M) (?A r)) = integral\<^sup>L M (?F r)"
-    using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
-
-  { fix x assume "x \<in> space M"
-    hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto
-    then obtain i where "i\<in>S" "enum i = f x" by auto
-    have F: "\<And>j. ?F j x = (if j = i then f x else 0)"
-    proof cases
-      fix j assume "j = i"
-      thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def)
-    next
-      fix j assume "j \<noteq> i"
-      show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero
-        by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def)
-    qed
-    hence F_abs: "\<And>j. \<bar>if j = i then f x else 0\<bar> = (if j = i then \<bar>f x\<bar> else 0)" by auto
-    have "(\<lambda>i. ?F i x) sums f x"
-         "(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>"
-      by (auto intro!: sums_single simp: F F_abs) }
-  note F_sums_f = this(1) and F_abs_sums_f = this(2)
-
-  have int_f: "integral\<^sup>L M f = (\<integral>x. (\<Sum>r. ?F r x) \<partial>M)" "integrable M f = integrable M (\<lambda>x. \<Sum>r. ?F r x)"
-    using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
-
-  { fix r
-    have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)"
-      by (auto simp: indicator_def intro!: integral_cong)
-    also have "\<dots> = \<bar>enum r\<bar> * real ((emeasure M) (?A r))"
-      using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
-    finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real ((emeasure M) (?A r))\<bar>"
-      using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_ereal_pos measurable_sets) }
-  note int_abs_F = this
-
-  have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
-    using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
-
-  have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
-    using F_abs_sums_f unfolding sums_iff by auto
-
-  from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
-  show ?sums unfolding enum_eq int_f by simp
-
-  from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
-  show "integrable M f" unfolding int_f by simp
-qed
-
-section {* Distributions *}
-
-lemma positive_integral_distr':
-  assumes T: "T \<in> measurable M M'"
-  and f: "f \<in> borel_measurable (distr M M' T)" "\<And>x. 0 \<le> f x"
-  shows "integral\<^sup>P (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
-  using f 
-proof induct
-  case (cong f g)
-  with T show ?case
-    apply (subst positive_integral_cong[of _ f g])
-    apply simp
-    apply (subst positive_integral_cong[of _ "\<lambda>x. f (T x)" "\<lambda>x. g (T x)"])
-    apply (simp add: measurable_def Pi_iff)
-    apply simp
-    done
-next
-  case (set A)
-  then have eq: "\<And>x. x \<in> space M \<Longrightarrow> indicator A (T x) = indicator (T -` A \<inter> space M) x"
-    by (auto simp: indicator_def)
-  from set T show ?case
-    by (subst positive_integral_cong[OF eq])
-       (auto simp add: emeasure_distr intro!: positive_integral_indicator[symmetric] measurable_sets)
-qed (simp_all add: measurable_compose[OF T] T positive_integral_cmult positive_integral_add
-                   positive_integral_monotone_convergence_SUP le_fun_def incseq_def)
-
-lemma positive_integral_distr:
-  "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^sup>P (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
-  by (subst (1 2) positive_integral_max_0[symmetric])
-     (simp add: positive_integral_distr')
-
-lemma integral_distr:
-  "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^sup>L (distr M M' T) f = (\<integral> x. f (T x) \<partial>M)"
-  unfolding lebesgue_integral_def
-  by (subst (1 2) positive_integral_distr) auto
-
-lemma integrable_distr_eq:
-  "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integrable (distr M M' T) f \<longleftrightarrow> integrable M (\<lambda>x. f (T x))"
-  unfolding integrable_def 
-  by (subst (1 2) positive_integral_distr) (auto simp: comp_def)
-
-lemma integrable_distr:
-  "T \<in> measurable M M' \<Longrightarrow> integrable (distr M M' T) f \<Longrightarrow> integrable M (\<lambda>x. f (T x))"
-  by (subst integrable_distr_eq[symmetric]) auto
-
-section {* Lebesgue integration on @{const count_space} *}
-
-lemma simple_function_count_space[simp]:
-  "simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)"
-  unfolding simple_function_def by simp
-
-lemma positive_integral_count_space:
-  assumes A: "finite {a\<in>A. 0 < f a}"
-  shows "integral\<^sup>P (count_space A) f = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
-proof -
-  have *: "(\<integral>\<^sup>+x. max 0 (f x) \<partial>count_space A) =
-    (\<integral>\<^sup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)"
-    by (auto intro!: positive_integral_cong
-             simp add: indicator_def if_distrib setsum_cases[OF A] max_def le_less)
-  also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. \<integral>\<^sup>+ x. f a * indicator {a} x \<partial>count_space A)"
-    by (subst positive_integral_setsum)
-       (simp_all add: AE_count_space ereal_zero_le_0_iff less_imp_le)
-  also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
-    by (auto intro!: setsum_cong simp: positive_integral_cmult_indicator one_ereal_def[symmetric])
-  finally show ?thesis by (simp add: positive_integral_max_0)
-qed
-
-lemma integrable_count_space:
-  "finite X \<Longrightarrow> integrable (count_space X) f"
-  by (auto simp: positive_integral_count_space integrable_def)
-
-lemma positive_integral_count_space_finite:
-    "finite A \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. max 0 (f a))"
-  by (subst positive_integral_max_0[symmetric])
-     (auto intro!: setsum_mono_zero_left simp: positive_integral_count_space less_le)
-
-lemma lebesgue_integral_count_space_finite_support:
-  assumes f: "finite {a\<in>A. f a \<noteq> 0}" shows "(\<integral>x. f x \<partial>count_space A) = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. f a)"
-proof -
-  have *: "\<And>r::real. 0 < max 0 r \<longleftrightarrow> 0 < r" "\<And>x. max 0 (ereal x) = ereal (max 0 x)"
-    "\<And>a. a \<in> A \<and> 0 < f a \<Longrightarrow> max 0 (f a) = f a"
-    "\<And>a. a \<in> A \<and> f a < 0 \<Longrightarrow> max 0 (- f a) = - f a"
-    "{a \<in> A. f a \<noteq> 0} = {a \<in> A. 0 < f a} \<union> {a \<in> A. f a < 0}"
-    "({a \<in> A. 0 < f a} \<inter> {a \<in> A. f a < 0}) = {}"
-    by (auto split: split_max)
-  have "finite {a \<in> A. 0 < f a}" "finite {a \<in> A. f a < 0}"
-    by (auto intro: finite_subset[OF _ f])
-  then show ?thesis
-    unfolding lebesgue_integral_def
-    apply (subst (1 2) positive_integral_max_0[symmetric])
-    apply (subst (1 2) positive_integral_count_space)
-    apply (auto simp add: * setsum_negf setsum_Un
-                simp del: ereal_max)
-    done
-qed
-
-lemma lebesgue_integral_count_space_finite:
-    "finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
-  apply (auto intro!: setsum_mono_zero_left
-              simp: positive_integral_count_space_finite lebesgue_integral_def)
-  apply (subst (1 2)  setsum_real_of_ereal[symmetric])
-  apply (auto simp: max_def setsum_subtractf[symmetric] intro!: setsum_cong)
-  done
-
-lemma emeasure_UN_countable:
-  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M" and I: "countable I" 
-  assumes disj: "disjoint_family_on X I"
-  shows "emeasure M (UNION I X) = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)"
-proof cases
-  assume "finite I" with sets disj show ?thesis
-    by (subst setsum_emeasure[symmetric])
-       (auto intro!: setsum_cong simp add: max_def subset_eq positive_integral_count_space_finite emeasure_nonneg)
-next
-  assume f: "\<not> finite I"
-  then have [intro]: "I \<noteq> {}" by auto
-  from from_nat_into_inj_infinite[OF I f] from_nat_into[OF this] disj
-  have disj2: "disjoint_family (\<lambda>i. X (from_nat_into I i))"
-    unfolding disjoint_family_on_def by metis
-
-  from f have "bij_betw (from_nat_into I) UNIV I"
-    using bij_betw_from_nat_into[OF I] by simp
-  then have "(\<Union>i\<in>I. X i) = (\<Union>i. (X \<circ> from_nat_into I) i)"
-    unfolding SUP_def image_comp [symmetric] by (simp add: bij_betw_def)
-  then have "emeasure M (UNION I X) = emeasure M (\<Union>i. X (from_nat_into I i))"
-    by simp
-  also have "\<dots> = (\<Sum>i. emeasure M (X (from_nat_into I i)))"
-    by (intro suminf_emeasure[symmetric] disj disj2) (auto intro!: sets from_nat_into[OF `I \<noteq> {}`])
-  also have "\<dots> = (\<Sum>n. \<integral>\<^sup>+i. emeasure M (X i) * indicator {from_nat_into I n} i \<partial>count_space I)"
-  proof (intro arg_cong[where f=suminf] ext)
-    fix i
-    have eq: "{a \<in> I. 0 < emeasure M (X a) * indicator {from_nat_into I i} a}
-     = (if 0 < emeasure M (X (from_nat_into I i)) then {from_nat_into I i} else {})"
-     using ereal_0_less_1
-     by (auto simp: ereal_zero_less_0_iff indicator_def from_nat_into `I \<noteq> {}` simp del: ereal_0_less_1)
-    have "(\<integral>\<^sup>+ ia. emeasure M (X ia) * indicator {from_nat_into I i} ia \<partial>count_space I) =
-      (if 0 < emeasure M (X (from_nat_into I i)) then emeasure M (X (from_nat_into I i)) else 0)"
-      by (subst positive_integral_count_space) (simp_all add: eq)
-    also have "\<dots> = emeasure M (X (from_nat_into I i))"
-      by (simp add: less_le emeasure_nonneg)
-    finally show "emeasure M (X (from_nat_into I i)) =
-         \<integral>\<^sup>+ ia. emeasure M (X ia) * indicator {from_nat_into I i} ia \<partial>count_space I" ..
-  qed
-  also have "\<dots> = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)"
-    apply (subst positive_integral_suminf[symmetric])
-    apply (auto simp: emeasure_nonneg intro!: positive_integral_cong)
-  proof -
-    fix x assume "x \<in> I"
-    then have "(\<Sum>i. emeasure M (X x) * indicator {from_nat_into I i} x) = (\<Sum>i\<in>{to_nat_on I x}. emeasure M (X x) * indicator {from_nat_into I i} x)"
-      by (intro suminf_finite) (auto simp: indicator_def I f)
-    also have "\<dots> = emeasure M (X x)"
-      by (simp add: I f `x\<in>I`)
-    finally show "(\<Sum>i. emeasure M (X x) * indicator {from_nat_into I i} x) = emeasure M (X x)" .
-  qed
-  finally show ?thesis .
-qed
-
-section {* Measures with Restricted Space *}
-
-lemma positive_integral_restrict_space:
-  assumes \<Omega>: "\<Omega> \<in> sets M" and f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "\<And>x. x \<in> space M - \<Omega> \<Longrightarrow> f x = 0"
-  shows "positive_integral (restrict_space M \<Omega>) f = positive_integral M f"
-using f proof (induct rule: borel_measurable_induct)
-  case (cong f g) then show ?case
-    using positive_integral_cong[of M f g] positive_integral_cong[of "restrict_space M \<Omega>" f g]
-      sets.sets_into_space[OF `\<Omega> \<in> sets M`]
-    by (simp add: subset_eq space_restrict_space)
-next
-  case (set A)
-  then have "A \<subseteq> \<Omega>"
-    unfolding indicator_eq_0_iff by (auto dest: sets.sets_into_space)
-  with set `\<Omega> \<in> sets M` sets.sets_into_space[OF `\<Omega> \<in> sets M`] show ?case
-    by (subst positive_integral_indicator')
-       (auto simp add: sets_restrict_space_iff space_restrict_space
-                  emeasure_restrict_space Int_absorb2
-                dest: sets.sets_into_space)
-next
-  case (mult f c) then show ?case
-    by (cases "c = 0") (simp_all add: measurable_restrict_space1 \<Omega> positive_integral_cmult)
-next
-  case (add f g) then show ?case
-    by (simp add: measurable_restrict_space1 \<Omega> positive_integral_add ereal_add_nonneg_eq_0_iff)
-next
-  case (seq F) then show ?case
-    by (auto simp add: SUP_eq_iff measurable_restrict_space1 \<Omega> positive_integral_monotone_convergence_SUP)
-qed
-
-section {* Measure spaces with an associated density *}
-
-definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
-  "density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
-
-lemma 
-  shows sets_density[simp]: "sets (density M f) = sets M"
-    and space_density[simp]: "space (density M f) = space M"
-  by (auto simp: density_def)
-
-(* FIXME: add conversion to simplify space, sets and measurable *)
-lemma space_density_imp[measurable_dest]:
-  "\<And>x M f. x \<in> space (density M f) \<Longrightarrow> x \<in> space M" by auto
-
-lemma 
-  shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'"
-    and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'"
-    and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u"
-  unfolding measurable_def simple_function_def by simp_all
-
-lemma density_cong: "f \<in> borel_measurable M \<Longrightarrow> f' \<in> borel_measurable M \<Longrightarrow>
-  (AE x in M. f x = f' x) \<Longrightarrow> density M f = density M f'"
-  unfolding density_def by (auto intro!: measure_of_eq positive_integral_cong_AE sets.space_closed)
-
-lemma density_max_0: "density M f = density M (\<lambda>x. max 0 (f x))"
-proof -
-  have "\<And>x A. max 0 (f x) * indicator A x = max 0 (f x * indicator A x)"
-    by (auto simp: indicator_def)
-  then show ?thesis
-    unfolding density_def by (simp add: positive_integral_max_0)
-qed
-
-lemma density_ereal_max_0: "density M (\<lambda>x. ereal (f x)) = density M (\<lambda>x. ereal (max 0 (f x)))"
-  by (subst density_max_0) (auto intro!: arg_cong[where f="density M"] split: split_max)
-
-lemma emeasure_density:
-  assumes f[measurable]: "f \<in> borel_measurable M" and A[measurable]: "A \<in> sets M"
-  shows "emeasure (density M f) A = (\<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
-    (is "_ = ?\<mu> A")
-  unfolding density_def
-proof (rule emeasure_measure_of_sigma)
-  show "sigma_algebra (space M) (sets M)" ..
-  show "positive (sets M) ?\<mu>"
-    using f by (auto simp: positive_def intro!: positive_integral_positive)
-  have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^sup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'")
-    apply (subst positive_integral_max_0[symmetric])
-    apply (intro ext positive_integral_cong_AE AE_I2)
-    apply (auto simp: indicator_def)
-    done
-  show "countably_additive (sets M) ?\<mu>"
-    unfolding \<mu>_eq
-  proof (intro countably_additiveI)
-    fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M"
-    then have "\<And>i. A i \<in> sets M" by auto
-    then have *: "\<And>i. (\<lambda>x. max 0 (f x) * indicator (A i) x) \<in> borel_measurable M"
-      by (auto simp: set_eq_iff)
-    assume disj: "disjoint_family A"
-    have "(\<Sum>n. ?\<mu>' (A n)) = (\<integral>\<^sup>+ x. (\<Sum>n. max 0 (f x) * indicator (A n) x) \<partial>M)"
-      using f * by (simp add: positive_integral_suminf)
-    also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * (\<Sum>n. indicator (A n) x) \<partial>M)" using f
-      by (auto intro!: suminf_cmult_ereal positive_integral_cong_AE)
-    also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * indicator (\<Union>n. A n) x \<partial>M)"
-      unfolding suminf_indicator[OF disj] ..
-    finally show "(\<Sum>n. ?\<mu>' (A n)) = ?\<mu>' (\<Union>x. A x)" by simp
-  qed
-qed fact
-
-lemma null_sets_density_iff:
-  assumes f: "f \<in> borel_measurable M"
-  shows "A \<in> null_sets (density M f) \<longleftrightarrow> A \<in> sets M \<and> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
-proof -
-  { assume "A \<in> sets M"
-    have eq: "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. max 0 (f x) * indicator A x \<partial>M)"
-      apply (subst positive_integral_max_0[symmetric])
-      apply (intro positive_integral_cong)
-      apply (auto simp: indicator_def)
-      done
-    have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> 
-      emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0"
-      unfolding eq
-      using f `A \<in> sets M`
-      by (intro positive_integral_0_iff) auto
-    also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)"
-      using f `A \<in> sets M`
-      by (intro AE_iff_measurable[OF _ refl, symmetric]) auto
-    also have "(AE x in M. max 0 (f x) * indicator A x = 0) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
-      by (auto simp add: indicator_def max_def split: split_if_asm)
-    finally have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)" . }
-  with f show ?thesis
-    by (simp add: null_sets_def emeasure_density cong: conj_cong)
-qed
-
-lemma AE_density:
-  assumes f: "f \<in> borel_measurable M"
-  shows "(AE x in density M f. P x) \<longleftrightarrow> (AE x in M. 0 < f x \<longrightarrow> P x)"
-proof
-  assume "AE x in density M f. P x"
-  with f obtain N where "{x \<in> space M. \<not> P x} \<subseteq> N" "N \<in> sets M" and ae: "AE x in M. x \<in> N \<longrightarrow> f x \<le> 0"
-    by (auto simp: eventually_ae_filter null_sets_density_iff)
-  then have "AE x in M. x \<notin> N \<longrightarrow> P x" by auto
-  with ae show "AE x in M. 0 < f x \<longrightarrow> P x"
-    by (rule eventually_elim2) auto
-next
-  fix N assume ae: "AE x in M. 0 < f x \<longrightarrow> P x"
-  then obtain N where "{x \<in> space M. \<not> (0 < f x \<longrightarrow> P x)} \<subseteq> N" "N \<in> null_sets M"
-    by (auto simp: eventually_ae_filter)
-  then have *: "{x \<in> space (density M f). \<not> P x} \<subseteq> N \<union> {x\<in>space M. \<not> 0 < f x}"
-    "N \<union> {x\<in>space M. \<not> 0 < f x} \<in> sets M" and ae2: "AE x in M. x \<notin> N"
-    using f by (auto simp: subset_eq intro!: sets.sets_Collect_neg AE_not_in)
-  show "AE x in density M f. P x"
-    using ae2
-    unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f]
-    by (intro exI[of _ "N \<union> {x\<in>space M. \<not> 0 < f x}"] conjI *)
-       (auto elim: eventually_elim2)
-qed
-
-lemma positive_integral_density':
-  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
-  assumes g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
-  shows "integral\<^sup>P (density M f) g = (\<integral>\<^sup>+ x. f x * g x \<partial>M)"
-using g proof induct
-  case (cong u v)
-  then show ?case
-    apply (subst positive_integral_cong[OF cong(3)])
-    apply (simp_all cong: positive_integral_cong)
-    done
-next
-  case (set A) then show ?case
-    by (simp add: emeasure_density f)
-next
-  case (mult u c)
-  moreover have "\<And>x. f x * (c * u x) = c * (f x * u x)" by (simp add: field_simps)
-  ultimately show ?case
-    using f by (simp add: positive_integral_cmult)
-next
-  case (add u v)
-  then have "\<And>x. f x * (v x + u x) = f x * v x + f x * u x"
-    by (simp add: ereal_right_distrib)
-  with add f show ?case
-    by (auto simp add: positive_integral_add ereal_zero_le_0_iff intro!: positive_integral_add[symmetric])
-next
-  case (seq U)
-  from f(2) have eq: "AE x in M. f x * (SUP i. U i x) = (SUP i. f x * U i x)"
-    by eventually_elim (simp add: SUP_ereal_cmult seq)
-  from seq f show ?case
-    apply (simp add: positive_integral_monotone_convergence_SUP)
-    apply (subst positive_integral_cong_AE[OF eq])
-    apply (subst positive_integral_monotone_convergence_SUP_AE)
-    apply (auto simp: incseq_def le_fun_def intro!: ereal_mult_left_mono)
-    done
-qed
-
-lemma positive_integral_density:
-  "f \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> g' \<in> borel_measurable M \<Longrightarrow> 
-    integral\<^sup>P (density M f) g' = (\<integral>\<^sup>+ x. f x * g' x \<partial>M)"
-  by (subst (1 2) positive_integral_max_0[symmetric])
-     (auto intro!: positive_integral_cong_AE
-           simp: measurable_If max_def ereal_zero_le_0_iff positive_integral_density')
-
-lemma integral_density:
-  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
-    and g: "g \<in> borel_measurable M"
-  shows "integral\<^sup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)"
-    and "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)"
-  unfolding lebesgue_integral_def integrable_def using f g
-  by (auto simp: positive_integral_density)
-
-lemma emeasure_restricted:
-  assumes S: "S \<in> sets M" and X: "X \<in> sets M"
-  shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)"
-proof -
-  have "emeasure (density M (indicator S)) X = (\<integral>\<^sup>+x. indicator S x * indicator X x \<partial>M)"
-    using S X by (simp add: emeasure_density)
-  also have "\<dots> = (\<integral>\<^sup>+x. indicator (S \<inter> X) x \<partial>M)"
-    by (auto intro!: positive_integral_cong simp: indicator_def)
-  also have "\<dots> = emeasure M (S \<inter> X)"
-    using S X by (simp add: sets.Int)
-  finally show ?thesis .
-qed
-
-lemma measure_restricted:
-  "S \<in> sets M \<Longrightarrow> X \<in> sets M \<Longrightarrow> measure (density M (indicator S)) X = measure M (S \<inter> X)"
-  by (simp add: emeasure_restricted measure_def)
-
-lemma (in finite_measure) finite_measure_restricted:
-  "S \<in> sets M \<Longrightarrow> finite_measure (density M (indicator S))"
-  by default (simp add: emeasure_restricted)
-
-lemma emeasure_density_const:
-  "A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> emeasure (density M (\<lambda>_. c)) A = c * emeasure M A"
-  by (auto simp: positive_integral_cmult_indicator emeasure_density)
-
-lemma measure_density_const:
-  "A \<in> sets M \<Longrightarrow> 0 < c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real c * measure M A"
-  by (auto simp: emeasure_density_const measure_def)
-
-lemma density_density_eq:
-   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow>
-   density (density M f) g = density M (\<lambda>x. f x * g x)"
-  by (auto intro!: measure_eqI simp: emeasure_density positive_integral_density ac_simps)
-
-lemma distr_density_distr:
-  assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
-    and inv: "\<forall>x\<in>space M. T' (T x) = x"
-  assumes f: "f \<in> borel_measurable M'"
-  shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)" (is "?R = ?L")
-proof (rule measure_eqI)
-  fix A assume A: "A \<in> sets ?R"
-  { fix x assume "x \<in> space M"
-    with sets.sets_into_space[OF A]
-    have "indicator (T' -` A \<inter> space M') (T x) = (indicator A x :: ereal)"
-      using T inv by (auto simp: indicator_def measurable_space) }
-  with A T T' f show "emeasure ?R A = emeasure ?L A"
-    by (simp add: measurable_comp emeasure_density emeasure_distr
-                  positive_integral_distr measurable_sets cong: positive_integral_cong)
-qed simp
-
-lemma density_density_divide:
-  fixes f g :: "'a \<Rightarrow> real"
-  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
-  assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
-  assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
-  shows "density (density M f) (\<lambda>x. g x / f x) = density M g"
-proof -
-  have "density M g = density M (\<lambda>x. f x * (g x / f x))"
-    using f g ac by (auto intro!: density_cong measurable_If)
-  then show ?thesis
-    using f g by (subst density_density_eq) auto
-qed
-
-section {* Point measure *}
-
-definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
-  "point_measure A f = density (count_space A) f"
-
-lemma
-  shows space_point_measure: "space (point_measure A f) = A"
-    and sets_point_measure: "sets (point_measure A f) = Pow A"
-  by (auto simp: point_measure_def)
-
-lemma measurable_point_measure_eq1[simp]:
-  "g \<in> measurable (point_measure A f) M \<longleftrightarrow> g \<in> A \<rightarrow> space M"
-  unfolding point_measure_def by simp
-
-lemma measurable_point_measure_eq2_finite[simp]:
-  "finite A \<Longrightarrow>
-   g \<in> measurable M (point_measure A f) \<longleftrightarrow>
-    (g \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. g -` {a} \<inter> space M \<in> sets M))"
-  unfolding point_measure_def by (simp add: measurable_count_space_eq2)
-
-lemma simple_function_point_measure[simp]:
-  "simple_function (point_measure A f) g \<longleftrightarrow> finite (g ` A)"
-  by (simp add: point_measure_def)
-
-lemma emeasure_point_measure:
-  assumes A: "finite {a\<in>X. 0 < f a}" "X \<subseteq> A"
-  shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)"
-proof -
-  have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}"
-    using `X \<subseteq> A` by auto
-  with A show ?thesis
-    by (simp add: emeasure_density positive_integral_count_space ereal_zero_le_0_iff
-                  point_measure_def indicator_def)
-qed
-
-lemma emeasure_point_measure_finite:
-  "finite A \<Longrightarrow> (\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
-  by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
-
-lemma emeasure_point_measure_finite2:
-  "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> (\<And>i. i \<in> X \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
-  by (subst emeasure_point_measure)
-     (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
-
-lemma null_sets_point_measure_iff:
-  "X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x \<le> 0)"
- by (auto simp: AE_count_space null_sets_density_iff point_measure_def)
-
-lemma AE_point_measure:
-  "(AE x in point_measure A f. P x) \<longleftrightarrow> (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)"
-  unfolding point_measure_def
-  by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def)
-
-lemma positive_integral_point_measure:
-  "finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow>
-    integral\<^sup>P (point_measure A f) g = (\<Sum>a|a\<in>A \<and> 0 < f a \<and> 0 < g a. f a * g a)"
-  unfolding point_measure_def
-  apply (subst density_max_0)
-  apply (subst positive_integral_density)
-  apply (simp_all add: AE_count_space positive_integral_density)
-  apply (subst positive_integral_count_space )
-  apply (auto intro!: setsum_cong simp: max_def ereal_zero_less_0_iff)
-  apply (rule finite_subset)
-  prefer 2
-  apply assumption
-  apply auto
-  done
-
-lemma positive_integral_point_measure_finite:
-  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> g a) \<Longrightarrow>
-    integral\<^sup>P (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
-  by (subst positive_integral_point_measure) (auto intro!: setsum_mono_zero_left simp: less_le)
-
-lemma lebesgue_integral_point_measure_finite:
-  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> integral\<^sup>L (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
-  by (simp add: lebesgue_integral_count_space_finite AE_count_space integral_density point_measure_def)
-
-lemma integrable_point_measure_finite:
-  "finite A \<Longrightarrow> integrable (point_measure A (\<lambda>x. ereal (f x))) g"
-  unfolding point_measure_def
-  apply (subst density_ereal_max_0)
-  apply (subst integral_density)
-  apply (auto simp: AE_count_space integrable_count_space)
-  done
-
-section {* Uniform measure *}
-
-definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
-
-lemma
-  shows sets_uniform_measure[simp]: "sets (uniform_measure M A) = sets M"
-    and space_uniform_measure[simp]: "space (uniform_measure M A) = space M"
-  by (auto simp: uniform_measure_def)
-
-lemma emeasure_uniform_measure[simp]:
-  assumes A: "A \<in> sets M" and B: "B \<in> sets M"
-  shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A"
-proof -
-  from A B have "emeasure (uniform_measure M A) B = (\<integral>\<^sup>+x. (1 / emeasure M A) * indicator (A \<inter> B) x \<partial>M)"
-    by (auto simp add: uniform_measure_def emeasure_density split: split_indicator
-             intro!: positive_integral_cong)
-  also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A"
-    using A B
-    by (subst positive_integral_cmult_indicator) (simp_all add: sets.Int emeasure_nonneg)
-  finally show ?thesis .
-qed
-
-lemma measure_uniform_measure[simp]:
-  assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>" and B: "B \<in> sets M"
-  shows "measure (uniform_measure M A) B = measure M (A \<inter> B) / measure M A"
-  using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A
-  by (cases "emeasure M A" "emeasure M (A \<inter> B)" rule: ereal2_cases) (simp_all add: measure_def)
-
-section {* Uniform count measure *}
-
-definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)"
- 
-lemma 
-  shows space_uniform_count_measure: "space (uniform_count_measure A) = A"
-    and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A"
-    unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure)
- 
-lemma emeasure_uniform_count_measure:
-  "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A"
-  by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def)
- 
-lemma measure_uniform_count_measure:
-  "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A"
-  by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def)
-
-end
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Lebesgue_Measure.thy
--- a/src/HOL/Probability/Lebesgue_Measure.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Lebesgue_Measure.thy	Mon May 19 12:04:45 2014 +0200
@@ -6,7 +6,7 @@
 header {* Lebsegue measure *}
 
 theory Lebesgue_Measure
-  imports Finite_Product_Measure
+  imports Finite_Product_Measure Bochner_Integration
 begin
 
 lemma absolutely_integrable_on_indicator[simp]:
@@ -483,10 +483,14 @@
   using sets.sigma_sets_eq[of borel]
   by (auto simp add: lborel_def measurable_def[abs_def])
 
+(* TODO: switch these rules! *)
 lemma emeasure_lborel[simp]: "A \<in> sets borel \<Longrightarrow> emeasure lborel A = emeasure lebesgue A"
   by (rule emeasure_measure_of[OF lborel_def])
      (auto simp: positive_def emeasure_nonneg countably_additive_def intro!: suminf_emeasure)
 
+lemma measure_lborel[simp]: "A \<in> sets borel \<Longrightarrow> measure lborel A = measure lebesgue A"
+  unfolding measure_def by simp
+
 interpretation lborel: sigma_finite_measure lborel
 proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
   show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed)
@@ -527,7 +531,9 @@
       by (auto simp: emeasure_eq) }
 qed
 
-lemma lebesgue_real_affine:
+
+(* GENEREALIZE to euclidean_spaces *)
+lemma lborel_real_affine:
   fixes c :: real assumes "c \<noteq> 0"
   shows "lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. \<bar>c\<bar>)" (is "_ = ?D")
 proof (rule lborel_eqI)
@@ -551,200 +557,193 @@
   qed
 qed simp
 
-lemma lebesgue_integral_real_affine:
-  fixes c :: real assumes c: "c \<noteq> 0" and f: "f \<in> borel_measurable borel"
-  shows "(\<integral> x. f x \<partial> lborel) = \<bar>c\<bar> * (\<integral> x. f (t + c * x) \<partial>lborel)"
-  by (subst lebesgue_real_affine[OF c, of t])
-     (simp add: f integral_density integral_distr lebesgue_integral_cmult)
+lemma positive_integral_real_affine:
+  fixes c :: real assumes [measurable]: "f \<in> borel_measurable borel" and c: "c \<noteq> 0"
+  shows "(\<integral>\<^sup>+x. f x \<partial>lborel) = \<bar>c\<bar> * (\<integral>\<^sup>+x. f (t + c * x) \<partial>lborel)"
+  by (subst lborel_real_affine[OF c, of t])
+     (simp add: positive_integral_density positive_integral_distr positive_integral_cmult)
+
+lemma lborel_integrable_real_affine:
+  fixes f :: "real \<Rightarrow> _ :: {banach, second_countable_topology}"
+  assumes f: "integrable lborel f"
+  shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x))"
+  using f f[THEN borel_measurable_integrable] unfolding integrable_iff_bounded
+  by (subst (asm) positive_integral_real_affine[where c=c and t=t]) auto
+
+lemma lborel_integrable_real_affine_iff:
+  fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
+  shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x)) \<longleftrightarrow> integrable lborel f"
+  using
+    lborel_integrable_real_affine[of f c t]
+    lborel_integrable_real_affine[of "\<lambda>x. f (t + c * x)" "1/c" "-t/c"]
+  by (auto simp add: field_simps)
+
+lemma lborel_integral_real_affine:
+  fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}" and c :: real
+  assumes c: "c \<noteq> 0" and f[measurable]: "integrable lborel f"
+  shows "(\<integral>x. f x \<partial> lborel) = \<bar>c\<bar> *\<^sub>R (\<integral>x. f (t + c * x) \<partial>lborel)"
+  using c f f[THEN borel_measurable_integrable] f[THEN lborel_integrable_real_affine, of c t]
+  by (subst lborel_real_affine[OF c, of t]) (simp add: integral_density integral_distr)
+
+lemma divideR_right: 
+  fixes x y :: "'a::real_normed_vector"
+  shows "r \<noteq> 0 \<Longrightarrow> y = x /\<^sub>R r \<longleftrightarrow> r *\<^sub>R y = x"
+  using scaleR_cancel_left[of r y "x /\<^sub>R r"] by simp
+
+lemma integrable_on_cmult_iff2:
+  fixes c :: real
+  shows "(\<lambda>x. c * f x) integrable_on s \<longleftrightarrow> c = 0 \<or> f integrable_on s"
+  using integrable_cmul[of "\<lambda>x. c * f x" s "1 / c"] integrable_cmul[of f s c]
+  by (cases "c = 0") auto
+
+lemma lborel_has_bochner_integral_real_affine_iff:
+  fixes x :: "'a :: {banach, second_countable_topology}"
+  shows "c \<noteq> 0 \<Longrightarrow>
+    has_bochner_integral lborel f x \<longleftrightarrow>
+    has_bochner_integral lborel (\<lambda>x. f (t + c * x)) (x /\<^sub>R \<bar>c\<bar>)"
+  unfolding has_bochner_integral_iff lborel_integrable_real_affine_iff
+  by (simp_all add: lborel_integral_real_affine[symmetric] divideR_right cong: conj_cong)
 
 subsection {* Lebesgue integrable implies Gauge integrable *}
 
-lemma simple_function_has_integral:
-  fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal"
-  assumes f:"simple_function lebesgue f"
-  and f':"range f \<subseteq> {0..<\<infinity>}"
-  and om:"\<And>x. x \<in> range f \<Longrightarrow> emeasure lebesgue (f -` {x} \<inter> UNIV) = \<infinity> \<Longrightarrow> x = 0"
-  shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^sup>S lebesgue f))) UNIV"
-  unfolding simple_integral_def space_lebesgue
-proof (subst lebesgue_simple_function_indicator)
-  let ?M = "\<lambda>x. emeasure lebesgue (f -` {x} \<inter> UNIV)"
-  let ?F = "\<lambda>x. indicator (f -` {x})"
-  { fix x y assume "y \<in> range f"
-    from subsetD[OF f' this] have "y * ?F y x = ereal (real y * ?F y x)"
-      by (cases rule: ereal2_cases[of y "?F y x"])
-         (auto simp: indicator_def one_ereal_def split: split_if_asm) }
-  moreover
-  { fix x assume x: "x\<in>range f"
-    have "x * ?M x = real x * real (?M x)"
-    proof cases
-      assume "x \<noteq> 0" with om[OF x] have "?M x \<noteq> \<infinity>" by auto
-      with subsetD[OF f' x] f[THEN simple_functionD(2)] show ?thesis
-        by (cases rule: ereal2_cases[of x "?M x"]) auto
-    qed simp }
-  ultimately
-  have "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV \<longleftrightarrow>
-    ((\<lambda>x. \<Sum>y\<in>range f. real y * ?F y x) has_integral (\<Sum>x\<in>range f. real x * real (?M x))) UNIV"
-    by simp
-  also have \<dots>
-  proof (intro has_integral_setsum has_integral_cmult_real lmeasure_finite_has_integral
-               real_of_ereal_pos emeasure_nonneg ballI)
-    show *: "finite (range f)" "\<And>y. f -` {y} \<in> sets lebesgue"
-      using simple_functionD[OF f] by auto
-    fix y assume "real y \<noteq> 0" "y \<in> range f"
-    with * om[OF this(2)] show "emeasure lebesgue (f -` {y}) = ereal (real (?M y))"
-      by (auto simp: ereal_real)
+lemma has_integral_scaleR_left: 
+  "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) s"
+  using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)
+
+lemma has_integral_mult_left:
+  fixes c :: "_ :: {real_normed_algebra}"
+  shows "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) s"
+  using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)
+
+(* GENERALIZE Integration.dominated_convergence, then generalize the following theorems *)
+lemma has_integral_dominated_convergence:
+  fixes f :: "nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
+  assumes "\<And>k. (f k has_integral y k) s" "h integrable_on s"
+    "\<And>k. \<forall>x\<in>s. norm (f k x) \<le> h x" "\<forall>x\<in>s. (\<lambda>k. f k x) ----> g x"
+    and x: "y ----> x"
+  shows "(g has_integral x) s"
+proof -
+  have int_f: "\<And>k. (f k) integrable_on s"
+    using assms by (auto simp: integrable_on_def)
+  have "(g has_integral (integral s g)) s"
+    by (intro integrable_integral dominated_convergence[OF int_f assms(2)]) fact+
+  moreover have "integral s g = x"
+  proof (rule LIMSEQ_unique)
+    show "(\<lambda>i. integral s (f i)) ----> x"
+      using integral_unique[OF assms(1)] x by simp
+    show "(\<lambda>i. integral s (f i)) ----> integral s g"
+      by (intro dominated_convergence[OF int_f assms(2)]) fact+
   qed
-  finally show "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV" .
-qed fact
-
-lemma simple_function_has_integral':
-  fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal"
-  assumes f: "simple_function lebesgue f" "\<And>x. 0 \<le> f x"
-  and i: "integral\<^sup>S lebesgue f \<noteq> \<infinity>"
-  shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^sup>S lebesgue f))) UNIV"
-proof -
-  let ?f = "\<lambda>x. if x \<in> f -` {\<infinity>} then 0 else f x"
-  note f(1)[THEN simple_functionD(2)]
-  then have [simp, intro]: "\<And>X. f -` X \<in> sets lebesgue" by auto
-  have f': "simple_function lebesgue ?f"
-    using f by (intro simple_function_If_set) auto
-  have rng: "range ?f \<subseteq> {0..<\<infinity>}" using f by auto
-  have "AE x in lebesgue. f x = ?f x"
-    using simple_integral_PInf[OF f i]
-    by (intro AE_I[where N="f -` {\<infinity>} \<inter> space lebesgue"]) auto
-  from f(1) f' this have eq: "integral\<^sup>S lebesgue f = integral\<^sup>S lebesgue ?f"
-    by (rule simple_integral_cong_AE)
-  have real_eq: "\<And>x. real (f x) = real (?f x)" by auto
-
-  show ?thesis
-    unfolding eq real_eq
-  proof (rule simple_function_has_integral[OF f' rng])
-    fix x assume x: "x \<in> range ?f" and inf: "emeasure lebesgue (?f -` {x} \<inter> UNIV) = \<infinity>"
-    have "x * emeasure lebesgue (?f -` {x} \<inter> UNIV) = (\<integral>\<^sup>S y. x * indicator (?f -` {x}) y \<partial>lebesgue)"
-      using f'[THEN simple_functionD(2)]
-      by (simp add: simple_integral_cmult_indicator)
-    also have "\<dots> \<le> integral\<^sup>S lebesgue f"
-      using f'[THEN simple_functionD(2)] f
-      by (intro simple_integral_mono simple_function_mult simple_function_indicator)
-         (auto split: split_indicator)
-    finally show "x = 0" unfolding inf using i subsetD[OF rng x] by (auto split: split_if_asm)
-  qed
+  ultimately show ?thesis
+    by simp
 qed
 
 lemma positive_integral_has_integral:
-  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal"
-  assumes f: "f \<in> borel_measurable lebesgue" "range f \<subseteq> {0..<\<infinity>}" "integral\<^sup>P lebesgue f \<noteq> \<infinity>"
-  shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^sup>P lebesgue f))) UNIV"
-proof -
-  from borel_measurable_implies_simple_function_sequence'[OF f(1)]
-  guess u . note u = this
-  have SUP_eq: "\<And>x. (SUP i. u i x) = f x"
-    using u(4) f(2)[THEN subsetD] by (auto split: split_max)
-  let ?u = "\<lambda>i x. real (u i x)"
-  note u_eq = positive_integral_eq_simple_integral[OF u(1,5), symmetric]
-  { fix i
-    note u_eq
-    also have "integral\<^sup>P lebesgue (u i) \<le> (\<integral>\<^sup>+x. max 0 (f x) \<partial>lebesgue)"
-      by (intro positive_integral_mono) (auto intro: SUP_upper simp: u(4)[symmetric])
-    finally have "integral\<^sup>S lebesgue (u i) \<noteq> \<infinity>"
-      unfolding positive_integral_max_0 using f by auto }
-  note u_fin = this
-  then have u_int: "\<And>i. (?u i has_integral real (integral\<^sup>S lebesgue (u i))) UNIV"
-    by (rule simple_function_has_integral'[OF u(1,5)])
-  have "\<forall>x. \<exists>r\<ge>0. f x = ereal r"
-  proof
-    fix x from f(2) have "0 \<le> f x" "f x \<noteq> \<infinity>" by (auto simp: subset_eq)
-    then show "\<exists>r\<ge>0. f x = ereal r" by (cases "f x") auto
-  qed
-  from choice[OF this] obtain f' where f': "f = (\<lambda>x. ereal (f' x))" "\<And>x. 0 \<le> f' x" by auto
-
-  have "\<forall>i. \<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = ereal (r x)"
-  proof
-    fix i show "\<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = ereal (r x)"
-    proof (intro choice allI)
-      fix i x have "u i x \<noteq> \<infinity>" using u(3)[of i] by (auto simp: image_iff) metis
-      then show "\<exists>r\<ge>0. u i x = ereal r" using u(5)[of i x] by (cases "u i x") auto
-    qed
-  qed
-  from choice[OF this] obtain u' where
-      u': "u = (\<lambda>i x. ereal (u' i x))" "\<And>i x. 0 \<le> u' i x" by (auto simp: fun_eq_iff)
+  fixes f::"'a::ordered_euclidean_space \<Rightarrow> real"
+  assumes f: "f \<in> borel_measurable lebesgue" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lebesgue) = ereal r"
+  shows "(f has_integral r) UNIV"
+using f proof (induct arbitrary: r rule: borel_measurable_induct_real)
+  case (set A) then show ?case
+    by (auto simp add: ereal_indicator has_integral_iff_lmeasure)
+next
+  case (mult g c)
+  then have "ereal c * (\<integral>\<^sup>+ x. g x \<partial>lebesgue) = ereal r"
+    by (subst positive_integral_cmult[symmetric]) auto
+  then obtain r' where "(c = 0 \<and> r = 0) \<or> ((\<integral>\<^sup>+ x. ereal (g x) \<partial>lebesgue) = ereal r' \<and> r = c * r')"
+    by (cases "\<integral>\<^sup>+ x. ereal (g x) \<partial>lebesgue") (auto split: split_if_asm)
+  with mult show ?case
+    by (auto intro!: has_integral_cmult_real)
+next
+  case (add g h)
+  moreover
+  then have "(\<integral>\<^sup>+ x. h x + g x \<partial>lebesgue) = (\<integral>\<^sup>+ x. h x \<partial>lebesgue) + (\<integral>\<^sup>+ x. g x \<partial>lebesgue)"
+    unfolding plus_ereal.simps[symmetric] by (subst positive_integral_add) auto
+  with add obtain a b where "(\<integral>\<^sup>+ x. h x \<partial>lebesgue) = ereal a" "(\<integral>\<^sup>+ x. g x \<partial>lebesgue) = ereal b" "r = a + b"
+    by (cases "\<integral>\<^sup>+ x. h x \<partial>lebesgue" "\<integral>\<^sup>+ x. g x \<partial>lebesgue" rule: ereal2_cases) auto
+  ultimately show ?case
+    by (auto intro!: has_integral_add)
+next
+  case (seq U)
+  note seq(1)[measurable] and f[measurable]
 
-  have convergent: "f' integrable_on UNIV \<and>
-    (\<lambda>k. integral UNIV (u' k)) ----> integral UNIV f'"
-  proof (intro monotone_convergence_increasing allI ballI)
-    show int: "\<And>k. (u' k) integrable_on UNIV"
-      using u_int unfolding integrable_on_def u' by auto
-    show "\<And>k x. u' k x \<le> u' (Suc k) x" using u(2,3,5)
-      by (auto simp: incseq_Suc_iff le_fun_def image_iff u' intro!: real_of_ereal_positive_mono)
-    show "\<And>x. (\<lambda>k. u' k x) ----> f' x"
-      using SUP_eq u(2)
-      by (intro SUP_eq_LIMSEQ[THEN iffD1]) (auto simp: u' f' incseq_Suc_iff le_fun_def)
-    show "bounded {integral UNIV (u' k)|k. True}"
-    proof (safe intro!: bounded_realI)
-      fix k
-      have "\<bar>integral UNIV (u' k)\<bar> = integral UNIV (u' k)"
-        by (intro abs_of_nonneg integral_nonneg int ballI u')
-      also have "\<dots> = real (integral\<^sup>S lebesgue (u k))"
-        using u_int[THEN integral_unique] by (simp add: u')
-      also have "\<dots> = real (integral\<^sup>P lebesgue (u k))"
-        using positive_integral_eq_simple_integral[OF u(1,5)] by simp
-      also have "\<dots> \<le> real (integral\<^sup>P lebesgue f)" using f
-        by (auto intro!: real_of_ereal_positive_mono positive_integral_positive
-             positive_integral_mono SUP_upper simp: SUP_eq[symmetric])
-      finally show "\<bar>integral UNIV (u' k)\<bar> \<le> real (integral\<^sup>P lebesgue f)" .
-    qed
+  { fix i x 
+    have "U i x \<le> f x"
+      using seq(5)
+      apply (rule LIMSEQ_le_const)
+      using seq(4)
+      apply (auto intro!: exI[of _ i] simp: incseq_def le_fun_def)
+      done }
+  note U_le_f = this
+  
+  { fix i
+    have "(\<integral>\<^sup>+x. ereal (U i x) \<partial>lebesgue) \<le> (\<integral>\<^sup>+x. ereal (f x) \<partial>lebesgue)"
+      using U_le_f by (intro positive_integral_mono) simp
+    then obtain p where "(\<integral>\<^sup>+x. U i x \<partial>lebesgue) = ereal p" "p \<le> r"
+      using seq(6) by (cases "\<integral>\<^sup>+x. U i x \<partial>lebesgue") auto
+    moreover then have "0 \<le> p"
+      by (metis ereal_less_eq(5) positive_integral_positive)
+    moreover note seq
+    ultimately have "\<exists>p. (\<integral>\<^sup>+x. U i x \<partial>lebesgue) = ereal p \<and> 0 \<le> p \<and> p \<le> r \<and> (U i has_integral p) UNIV"
+      by auto }
+  then obtain p where p: "\<And>i. (\<integral>\<^sup>+x. ereal (U i x) \<partial>lebesgue) = ereal (p i)"
+    and bnd: "\<And>i. p i \<le> r" "\<And>i. 0 \<le> p i"
+    and U_int: "\<And>i.(U i has_integral (p i)) UNIV" by metis
+
+  have int_eq: "\<And>i. integral UNIV (U i) = p i" using U_int by (rule integral_unique)
+
+  have *: "f integrable_on UNIV \<and> (\<lambda>k. integral UNIV (U k)) ----> integral UNIV f"
+  proof (rule monotone_convergence_increasing)
+    show "\<forall>k. U k integrable_on UNIV" using U_int by auto
+    show "\<forall>k. \<forall>x\<in>UNIV. U k x \<le> U (Suc k) x" using `incseq U` by (auto simp: incseq_def le_fun_def)
+    then show "bounded {integral UNIV (U k) |k. True}"
+      using bnd int_eq by (auto simp: bounded_real intro!: exI[of _ r])
+    show "\<forall>x\<in>UNIV. (\<lambda>k. U k x) ----> f x"
+      using seq by auto
   qed
-
-  have "integral\<^sup>P lebesgue f = ereal (integral UNIV f')"
-  proof (rule tendsto_unique[OF trivial_limit_sequentially])
-    have "(\<lambda>i. integral\<^sup>S lebesgue (u i)) ----> (SUP i. integral\<^sup>P lebesgue (u i))"
-      unfolding u_eq by (intro LIMSEQ_SUP incseq_positive_integral u)
-    also note positive_integral_monotone_convergence_SUP
-      [OF u(2)  borel_measurable_simple_function[OF u(1)] u(5), symmetric]
-    finally show "(\<lambda>k. integral\<^sup>S lebesgue (u k)) ----> integral\<^sup>P lebesgue f"
-      unfolding SUP_eq .
-
-    { fix k
-      have "0 \<le> integral\<^sup>S lebesgue (u k)"
-        using u by (auto intro!: simple_integral_positive)
-      then have "integral\<^sup>S lebesgue (u k) = ereal (real (integral\<^sup>S lebesgue (u k)))"
-        using u_fin by (auto simp: ereal_real) }
-    note * = this
-    show "(\<lambda>k. integral\<^sup>S lebesgue (u k)) ----> ereal (integral UNIV f')"
-      using convergent using u_int[THEN integral_unique, symmetric]
-      by (subst *) (simp add: u')
-  qed
-  then show ?thesis using convergent by (simp add: f' integrable_integral)
+  moreover have "(\<lambda>i. (\<integral>\<^sup>+x. U i x \<partial>lebesgue)) ----> (\<integral>\<^sup>+x. f x \<partial>lebesgue)"
+    using seq U_le_f by (intro positive_integral_dominated_convergence[where w=f]) auto
+  ultimately have "integral UNIV f = r"
+    by (auto simp add: int_eq p seq intro: LIMSEQ_unique)
+  with * show ?case
+    by (simp add: has_integral_integral)
 qed
 
-lemma lebesgue_integral_has_integral:
-  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
+lemma has_integral_integrable_lebesgue_nonneg:
+  fixes f::"'a::ordered_euclidean_space \<Rightarrow> real"
+  assumes f: "integrable lebesgue f" "\<And>x. 0 \<le> f x"
+  shows "(f has_integral integral\<^sup>L lebesgue f) UNIV"
+proof (rule positive_integral_has_integral)
+  show "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lebesgue) = ereal (integral\<^sup>L lebesgue f)"
+    using f by (intro positive_integral_eq_integral) auto
+qed (insert f, auto)
+
+lemma has_integral_lebesgue_integral_lebesgue:
+  fixes f::"'a::ordered_euclidean_space \<Rightarrow> real"
   assumes f: "integrable lebesgue f"
   shows "(f has_integral (integral\<^sup>L lebesgue f)) UNIV"
-proof -
-  let ?n = "\<lambda>x. real (ereal (max 0 (- f x)))" and ?p = "\<lambda>x. real (ereal (max 0 (f x)))"
-  have *: "f = (\<lambda>x. ?p x - ?n x)" by (auto simp del: ereal_max)
-  { fix f :: "'a \<Rightarrow> real" have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lebesgue) = (\<integral>\<^sup>+ x. ereal (max 0 (f x)) \<partial>lebesgue)"
-      by (intro positive_integral_cong_pos) (auto split: split_max) }
-  note eq = this
-  show ?thesis
-    unfolding lebesgue_integral_def
-    apply (subst *)
-    apply (rule has_integral_sub)
-    unfolding eq[of f] eq[of "\<lambda>x. - f x"]
-    apply (safe intro!: positive_integral_has_integral)
-    using integrableD[OF f]
-    by (auto simp: zero_ereal_def[symmetric] positive_integral_max_0  split: split_max
-             intro!: measurable_If)
+using f proof induct
+  case (base A c) then show ?case
+    by (auto intro!: has_integral_mult_left simp: has_integral_iff_lmeasure)
+       (simp add: emeasure_eq_ereal_measure)
+next
+  case (add f g) then show ?case
+    by (auto intro!: has_integral_add)
+next
+  case (lim f s)
+  show ?case
+  proof (rule has_integral_dominated_convergence)
+    show "\<And>i. (s i has_integral integral\<^sup>L lebesgue (s i)) UNIV" by fact
+    show "(\<lambda>x. norm (2 * f x)) integrable_on UNIV"
+      using lim by (intro has_integral_integrable[OF has_integral_integrable_lebesgue_nonneg]) auto
+    show "\<And>k. \<forall>x\<in>UNIV. norm (s k x) \<le> norm (2 * f x)"
+      using lim by (auto simp add: abs_mult)
+    show "\<forall>x\<in>UNIV. (\<lambda>k. s k x) ----> f x"
+      using lim by auto
+    show "(\<lambda>k. integral\<^sup>L lebesgue (s k)) ----> integral\<^sup>L lebesgue f"
+      using lim by (intro integral_dominated_convergence(3)[where w="\<lambda>x. 2 * norm (f x)"]) auto
+  qed
 qed
 
-lemma lebesgue_simple_integral_eq_borel:
-  assumes f: "f \<in> borel_measurable borel"
-  shows "integral\<^sup>S lebesgue f = integral\<^sup>S lborel f"
-  using f[THEN measurable_sets]
-  by (auto intro!: setsum_cong arg_cong2[where f="op *"] emeasure_lborel[symmetric]
-           simp: simple_integral_def)
-
 lemma lebesgue_positive_integral_eq_borel:
   assumes f: "f \<in> borel_measurable borel"
   shows "integral\<^sup>P lebesgue f = integral\<^sup>P lborel f"
@@ -755,40 +754,70 @@
 qed
 
 lemma lebesgue_integral_eq_borel:
+  fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
   assumes "f \<in> borel_measurable borel"
   shows "integrable lebesgue f \<longleftrightarrow> integrable lborel f" (is ?P)
     and "integral\<^sup>L lebesgue f = integral\<^sup>L lborel f" (is ?I)
 proof -
   have "sets lborel \<subseteq> sets lebesgue" by auto
-  from integral_subalgebra[of f lborel, OF _ this _ _] assms
+  from integral_subalgebra[of f lborel, OF _ this _ _]
+       integrable_subalgebra[of f lborel, OF _ this _ _] assms
   show ?P ?I by auto
 qed
 
-lemma borel_integral_has_integral:
+lemma has_integral_lebesgue_integral:
   fixes f::"'a::ordered_euclidean_space => real"
   assumes f:"integrable lborel f"
   shows "(f has_integral (integral\<^sup>L lborel f)) UNIV"
 proof -
   have borel: "f \<in> borel_measurable borel"
-    using f unfolding integrable_def by auto
+    using f unfolding integrable_iff_bounded by auto
   from f show ?thesis
-    using lebesgue_integral_has_integral[of f]
+    using has_integral_lebesgue_integral_lebesgue[of f]
     unfolding lebesgue_integral_eq_borel[OF borel] by simp
 qed
 
-lemma positive_integral_lebesgue_has_integral:
+lemma positive_integral_bound_simple_function:
+  assumes bnd: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" "\<And>x. x \<in> space M \<Longrightarrow> f x < \<infinity>"
+  assumes f[measurable]: "simple_function M f"
+  assumes supp: "emeasure M {x\<in>space M. f x \<noteq> 0} < \<infinity>"
+  shows "positive_integral M f < \<infinity>"
+proof cases
+  assume "space M = {}"
+  then have "positive_integral M f = (\<integral>\<^sup>+x. 0 \<partial>M)"
+    by (intro positive_integral_cong) auto
+  then show ?thesis by simp
+next
+  assume "space M \<noteq> {}"
+  with simple_functionD(1)[OF f] bnd have bnd: "0 \<le> Max (f`space M) \<and> Max (f`space M) < \<infinity>"
+    by (subst Max_less_iff) (auto simp: Max_ge_iff)
+  
+  have "positive_integral M f \<le> (\<integral>\<^sup>+x. Max (f`space M) * indicator {x\<in>space M. f x \<noteq> 0} x \<partial>M)"
+  proof (rule positive_integral_mono)
+    fix x assume "x \<in> space M"
+    with f show "f x \<le> Max (f ` space M) * indicator {x \<in> space M. f x \<noteq> 0} x"
+      by (auto split: split_indicator intro!: Max_ge simple_functionD)
+  qed
+  also have "\<dots> < \<infinity>"
+    using bnd supp by (subst positive_integral_cmult) auto
+  finally show ?thesis .
+qed
+
+
+lemma
   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
   assumes f_borel: "f \<in> borel_measurable lebesgue" and nonneg: "\<And>x. 0 \<le> f x"
   assumes I: "(f has_integral I) UNIV"
-  shows "(\<integral>\<^sup>+x. f x \<partial>lebesgue) = I"
+  shows integrable_has_integral_lebesgue_nonneg: "integrable lebesgue f"
+    and integral_has_integral_lebesgue_nonneg: "integral\<^sup>L lebesgue f = I"
 proof -
   from f_borel have "(\<lambda>x. ereal (f x)) \<in> borel_measurable lebesgue" by auto
   from borel_measurable_implies_simple_function_sequence'[OF this] guess F . note F = this
 
-  have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lebesgue) = (SUP i. integral\<^sup>S lebesgue (F i))"
+  have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lebesgue) = (SUP i. integral\<^sup>P lebesgue (F i))"
     using F
-    by (subst positive_integral_monotone_convergence_simple)
-       (simp_all add: positive_integral_max_0 simple_function_def)
+    by (subst positive_integral_monotone_convergence_SUP[symmetric])
+       (simp_all add: positive_integral_max_0 borel_measurable_simple_function)
   also have "\<dots> \<le> ereal I"
   proof (rule SUP_least)
     fix i :: nat
@@ -835,58 +864,58 @@
         by (simp add: integrable_on_cmult_iff) }
     note F_finite = lmeasure_finite[OF this]
 
-    have "((\<lambda>x. real (F i x)) has_integral real (integral\<^sup>S lebesgue (F i))) UNIV"
-    proof (rule simple_function_has_integral[of "F i"])
-      show "simple_function lebesgue (F i)"
-        using F(1) by (simp add: simple_function_def)
-      show "range (F i) \<subseteq> {0..<\<infinity>}"
-        using F(3,5)[of i] by (auto simp: image_iff) metis
-    next
-      fix x assume "x \<in> range (F i)" "emeasure lebesgue (F i -` {x} \<inter> UNIV) = \<infinity>"
-      with F_finite[of x] show "x = 0" by auto
-    qed
-    from this I have "real (integral\<^sup>S lebesgue (F i)) \<le> I"
+    have F_eq: "\<And>x. F i x = ereal (norm (real (F i x)))"
+      using F(3,5) by (auto simp: fun_eq_iff ereal_real image_iff eq_commute)
+    have F_eq2: "\<And>x. F i x = ereal (real (F i x))"
+      using F(3,5) by (auto simp: fun_eq_iff ereal_real image_iff eq_commute)
+
+    have int: "integrable lebesgue (\<lambda>x. real (F i x))"
+      unfolding integrable_iff_bounded
+    proof
+      have "(\<integral>\<^sup>+x. F i x \<partial>lebesgue) < \<infinity>"
+      proof (rule positive_integral_bound_simple_function)
+        fix x::'a assume "x \<in> space lebesgue" then show "0 \<le> F i x" "F i x < \<infinity>"
+          using F by (auto simp: image_iff eq_commute)
+      next
+        have eq: "{x \<in> space lebesgue. F i x \<noteq> 0} = (\<Union>x\<in>F i ` space lebesgue - {0}. F i -` {x} \<inter> space lebesgue)"
+          by auto
+        show "emeasure lebesgue {x \<in> space lebesgue. F i x \<noteq> 0} < \<infinity>"
+          unfolding eq using simple_functionD[OF F(1)]
+          by (subst setsum_emeasure[symmetric])
+             (auto simp: disjoint_family_on_def setsum_Pinfty F_finite)
+      qed fact
+      with F_eq show "(\<integral>\<^sup>+x. norm (real (F i x)) \<partial>lebesgue) < \<infinity>" by simp
+    qed (insert F(1), auto intro!: borel_measurable_real_of_ereal dest: borel_measurable_simple_function)
+    then have "((\<lambda>x. real (F i x)) has_integral integral\<^sup>L lebesgue (\<lambda>x. real (F i x))) UNIV"
+      by (rule has_integral_lebesgue_integral_lebesgue)
+    from this I have "integral\<^sup>L lebesgue (\<lambda>x. real (F i x)) \<le> I"
       by (rule has_integral_le) (intro ballI F_bound)
-    moreover
-    { fix x assume x: "x \<in> range (F i)"
-      with F(3,5)[of i] have "x = 0 \<or> (0 < x \<and> x \<noteq> \<infinity>)"
-        by (auto  simp: image_iff le_less) metis
-      with F_finite[OF _ x] x have "x * emeasure lebesgue (F i -` {x} \<inter> UNIV) \<noteq> \<infinity>"
-        by auto }
-    then have "integral\<^sup>S lebesgue (F i) \<noteq> \<infinity>"
-      unfolding simple_integral_def setsum_Pinfty space_lebesgue by blast
-    moreover have "0 \<le> integral\<^sup>S lebesgue (F i)"
-      using F(1,5) by (intro simple_integral_positive) (auto simp: simple_function_def)
-    ultimately show "integral\<^sup>S lebesgue (F i) \<le> ereal I"
-      by (cases "integral\<^sup>S lebesgue (F i)") auto
+    moreover have "integral\<^sup>P lebesgue (F i) = integral\<^sup>L lebesgue (\<lambda>x. real (F i x))"
+      using int F by (subst positive_integral_eq_integral[symmetric])  (auto simp: F_eq2[symmetric] real_of_ereal_pos)
+    ultimately show "integral\<^sup>P lebesgue (F i) \<le> ereal I"
+      by simp
   qed
-  also have "\<dots> < \<infinity>" by simp
-  finally have finite: "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lebesgue) \<noteq> \<infinity>" by simp
-  have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable lebesgue"
-    using f_borel by (auto intro: borel_measurable_lebesgueI)
-  from positive_integral_has_integral[OF borel _ finite]
-  have "(f has_integral real (\<integral>\<^sup>+ x. ereal (f x) \<partial>lebesgue)) UNIV"
-    using nonneg by (simp add: subset_eq)
-  with I have "I = real (\<integral>\<^sup>+ x. ereal (f x) \<partial>lebesgue)"
-    by (rule has_integral_unique)
-  with finite positive_integral_positive[of _ "\<lambda>x. ereal (f x)"] show ?thesis
-    by (cases "\<integral>\<^sup>+ x. ereal (f x) \<partial>lebesgue") auto
+  finally show "integrable lebesgue f"
+    using f_borel by (auto simp: integrable_iff_bounded nonneg)
+  from has_integral_lebesgue_integral_lebesgue[OF this] I
+  show "integral\<^sup>L lebesgue f = I"
+    by (metis has_integral_unique)
 qed
 
-lemma has_integral_iff_positive_integral_lebesgue:
+lemma has_integral_iff_has_bochner_integral_lebesgue_nonneg:
   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
-  assumes f: "f \<in> borel_measurable lebesgue" "\<And>x. 0 \<le> f x"
-  shows "(f has_integral I) UNIV \<longleftrightarrow> integral\<^sup>P lebesgue f = I"
-  using f positive_integral_lebesgue_has_integral[of f I] positive_integral_has_integral[of f]
-  by (auto simp: subset_eq)
+  shows "f \<in> borel_measurable lebesgue \<Longrightarrow> (\<And>x. 0 \<le> f x) \<Longrightarrow>
+    (f has_integral I) UNIV \<longleftrightarrow> has_bochner_integral lebesgue f I"
+  by (metis has_bochner_integral_iff has_integral_unique has_integral_lebesgue_integral_lebesgue
+            integrable_has_integral_lebesgue_nonneg)
 
-lemma has_integral_iff_positive_integral_lborel:
+lemma
   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
-  assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x"
-  shows "(f has_integral I) UNIV \<longleftrightarrow> integral\<^sup>P lborel f = I"
-  using assms
-  by (subst has_integral_iff_positive_integral_lebesgue)
-     (auto simp: borel_measurable_lebesgueI lebesgue_positive_integral_eq_borel)
+  assumes "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(f has_integral I) UNIV"
+  shows integrable_has_integral_nonneg: "integrable lborel f"
+    and integral_has_integral_nonneg: "integral\<^sup>L lborel f = I"
+  by (metis assms borel_measurable_lebesgueI integrable_has_integral_lebesgue_nonneg lebesgue_integral_eq_borel(1))
+     (metis assms borel_measurable_lebesgueI has_integral_lebesgue_integral has_integral_unique integrable_has_integral_lebesgue_nonneg lebesgue_integral_eq_borel(1))
 
 subsection {* Equivalence between product spaces and euclidean spaces *}
 
@@ -978,52 +1007,36 @@
   by (subst lborel_eq_lborel_space) (simp add: positive_integral_distr measurable_p2e f)
 
 lemma borel_fubini_integrable:
-  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
+  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> _::{banach, second_countable_topology}"
   shows "integrable lborel f \<longleftrightarrow> integrable (\<Pi>\<^sub>M (i::'a)\<in>Basis. lborel) (\<lambda>x. f (p2e x))"
-    (is "_ \<longleftrightarrow> integrable ?B ?f")
-proof
-  assume *: "integrable lborel f"
-  then have f: "f \<in> borel_measurable borel"
-    by auto
-  with measurable_p2e have "f \<circ> p2e \<in> borel_measurable ?B"
-    by (rule measurable_comp)
-  with * f show "integrable ?B ?f"
-    by (simp add: comp_def borel_fubini_positiv_integral integrable_def)
-next
-  assume *: "integrable ?B ?f"
-  then have "?f \<circ> e2p \<in> borel_measurable (borel::'a measure)"
-    by (auto intro!: measurable_e2p)
-  then have "f \<in> borel_measurable borel"
-    by (simp cong: measurable_cong)
-  with * show "integrable lborel f"
-    by (simp add: borel_fubini_positiv_integral integrable_def)
-qed
+  unfolding integrable_iff_bounded
+proof (intro conj_cong[symmetric])
+  show "((\<lambda>x. f (p2e x)) \<in> borel_measurable (Pi\<^sub>M Basis (\<lambda>i. lborel))) = (f \<in> borel_measurable lborel)"
+  proof
+    assume "((\<lambda>x. f (p2e x)) \<in> borel_measurable (Pi\<^sub>M Basis (\<lambda>i. lborel)))"
+    then have "(\<lambda>x. f (p2e (e2p x))) \<in> borel_measurable borel"
+      by measurable
+    then show "f \<in> borel_measurable lborel"
+      by simp
+  qed simp
+qed (simp add: borel_fubini_positiv_integral)
 
 lemma borel_fubini:
-  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
-  assumes f: "f \<in> borel_measurable borel"
-  shows "integral\<^sup>L lborel f = \<integral>x. f (p2e x) \<partial>((\<Pi>\<^sub>M (i::'a)\<in>Basis. lborel))"
-  using f by (simp add: borel_fubini_positiv_integral lebesgue_integral_def)
+  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> _::{banach, second_countable_topology}"
+  shows "f \<in> borel_measurable borel \<Longrightarrow>
+    integral\<^sup>L lborel f = \<integral>x. f (p2e x) \<partial>((\<Pi>\<^sub>M (i::'a)\<in>Basis. lborel))"
+  by (subst lborel_eq_lborel_space) (simp add: integral_distr)
 
 lemma integrable_on_borel_integrable:
   fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
-  assumes f_borel: "f \<in> borel_measurable borel" and nonneg: "\<And>x. 0 \<le> f x"
-  assumes f: "f integrable_on UNIV" 
-  shows "integrable lborel f"
-proof -
-  have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lborel) \<noteq> \<infinity>" 
-    using has_integral_iff_positive_integral_lborel[OF f_borel nonneg] f
-    by (auto simp: integrable_on_def)
-  moreover have "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>lborel) = 0"
-    using f_borel nonneg by (subst positive_integral_0_iff_AE) auto
-  ultimately show ?thesis
-    using f_borel by (auto simp: integrable_def)
-qed
+  shows "f \<in> borel_measurable borel \<Longrightarrow> (\<And>x. 0 \<le> f x) \<Longrightarrow> f integrable_on UNIV \<Longrightarrow> integrable lborel f"
+  by (metis borel_measurable_lebesgueI integrable_has_integral_nonneg integrable_on_def
+            lebesgue_integral_eq_borel(1))
 
 subsection {* Fundamental Theorem of Calculus for the Lebesgue integral *}
 
 lemma borel_integrable_atLeastAtMost:
-  fixes a b :: real
+  fixes f :: "real \<Rightarrow> real"
   assumes f: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
   shows "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is "integrable _ ?f")
 proof cases
@@ -1037,10 +1050,10 @@
   show ?thesis
   proof (rule integrable_bound)
     show "integrable lborel (\<lambda>x. max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x)"
-      by (rule integral_cmul_indicator) simp_all
-    show "AE x in lborel. \<bar>?f x\<bar> \<le> max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x"
+      by (intro integrable_mult_right integrable_real_indicator) simp_all
+    show "AE x in lborel. norm (?f x) \<le> norm (max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x)"
     proof (rule AE_I2)
-      fix x show "\<bar>?f x\<bar> \<le> max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x"
+      fix x show "norm (?f x) \<le> norm (max \<bar>M\<bar> \<bar>L\<bar> * indicator {a..b} x)"
         using bounds[of x] by (auto split: split_indicator)
     qed
 
@@ -1071,7 +1084,7 @@
   let ?f = "\<lambda>x. f x * indicator {a .. b} x"
   have "(?f has_integral (\<integral>x. ?f x \<partial>lborel)) UNIV"
     using borel_integrable_atLeastAtMost[OF f]
-    by (rule borel_integral_has_integral)
+    by (rule has_integral_lebesgue_integral)
   moreover
   have "(f has_integral F b - F a) {a .. b}"
     by (intro fundamental_theorem_of_calculus)
@@ -1091,10 +1104,12 @@
 
 *}
 
-lemma positive_integral_FTC_atLeastAtMost:
+lemma
+  fixes f :: "real \<Rightarrow> real"
   assumes f_borel: "f \<in> borel_measurable borel"
   assumes f: "\<And>x. x \<in> {a..b} \<Longrightarrow> DERIV F x :> f x" "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> f x" and "a \<le> b"
-  shows "(\<integral>\<^sup>+x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a"
+  shows integral_FTC_Icc_nonneg: "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?eq)
+    and integrable_FTC_Icc_nonneg: "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is ?int)
 proof -
   have i: "(f has_integral F b - F a) {a..b}"
     by (intro fundamental_theorem_of_calculus)
@@ -1104,12 +1119,24 @@
     by (rule has_integral_eq[OF _ i]) auto
   have i: "((\<lambda>x. f x * indicator {a..b} x) has_integral F b - F a) UNIV"
     by (rule has_integral_on_superset[OF _ _ i]) auto
-  then have "(\<integral>\<^sup>+x. ereal (f x * indicator {a .. b} x) \<partial>lborel) = F b - F a"
-    using f f_borel
-    by (subst has_integral_iff_positive_integral_lborel[symmetric]) (auto split: split_indicator)
-  also have "(\<integral>\<^sup>+x. ereal (f x * indicator {a .. b} x) \<partial>lborel) = (\<integral>\<^sup>+x. ereal (f x) * indicator {a .. b} x \<partial>lborel)"
-    by (auto intro!: positive_integral_cong simp: indicator_def)
-  finally show ?thesis by simp
+  from i f f_borel show ?eq
+    by (intro integral_has_integral_nonneg) (auto split: split_indicator)
+  from i f f_borel show ?int
+    by (intro integrable_has_integral_nonneg) (auto split: split_indicator)
+qed
+
+lemma positive_integral_FTC_atLeastAtMost:
+  assumes "f \<in> borel_measurable borel" "\<And>x. x \<in> {a..b} \<Longrightarrow> DERIV F x :> f x" "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> f x" "a \<le> b"
+  shows "(\<integral>\<^sup>+x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a"
+proof -
+  have "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)"
+    by (rule integrable_FTC_Icc_nonneg) fact+
+  then have "(\<integral>\<^sup>+x. f x * indicator {a .. b} x \<partial>lborel) = (\<integral>x. f x * indicator {a .. b} x \<partial>lborel)"
+    using assms by (intro positive_integral_eq_integral) (auto simp: indicator_def)
+  also have "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a"
+    by (rule integral_FTC_Icc_nonneg) fact+
+  finally show ?thesis
+    unfolding ereal_indicator[symmetric] by simp
 qed
 
 lemma positive_integral_FTC_atLeast:
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Measurable.thy
--- a/src/HOL/Probability/Measurable.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Measurable.thy	Mon May 19 12:04:45 2014 +0200
@@ -256,6 +256,76 @@
   shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)"
   unfolding measurable_def by (safe intro!: sets_Least) simp_all
 
+lemma measurable_Max_nat[measurable (raw)]: 
+  fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
+  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
+  shows "(\<lambda>x. Max {i. P i x}) \<in> measurable M (count_space UNIV)"
+  unfolding measurable_count_space_eq2_countable
+proof safe
+  fix n
+
+  { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
+    then have "infinite {i. P i x}"
+      unfolding infinite_nat_iff_unbounded_le by auto
+    then have "Max {i. P i x} = the None"
+      by (rule Max.infinite) }
+  note 1 = this
+
+  { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
+    then have "finite {i. P i x}"
+      by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
+    with `P i x` have "P (Max {i. P i x}) x" "i \<le> Max {i. P i x}" "finite {i. P i x}"
+      using Max_in[of "{i. P i x}"] by auto }
+  note 2 = this
+
+  have "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Max {i. P i x} = n}"
+    by auto
+  also have "\<dots> = 
+    {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else 
+      if (\<exists>i. P i x) then P n x \<and> (\<forall>i>n. \<not> P i x)
+      else Max {} = n}"
+    by (intro arg_cong[where f=Collect] ext conj_cong)
+       (auto simp add: 1 2 not_le[symmetric] intro!: Max_eqI)
+  also have "\<dots> \<in> sets M"
+    by measurable
+  finally show "(\<lambda>x. Max {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
+qed simp
+
+lemma measurable_Min_nat[measurable (raw)]: 
+  fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"
+  assumes [measurable]: "\<And>i. Measurable.pred M (P i)"
+  shows "(\<lambda>x. Min {i. P i x}) \<in> measurable M (count_space UNIV)"
+  unfolding measurable_count_space_eq2_countable
+proof safe
+  fix n
+
+  { fix x assume "\<forall>i. \<exists>n\<ge>i. P n x"
+    then have "infinite {i. P i x}"
+      unfolding infinite_nat_iff_unbounded_le by auto
+    then have "Min {i. P i x} = the None"
+      by (rule Min.infinite) }
+  note 1 = this
+
+  { fix x i j assume "P i x" "\<forall>n\<ge>j. \<not> P n x"
+    then have "finite {i. P i x}"
+      by (auto simp: subset_eq not_le[symmetric] finite_nat_iff_bounded)
+    with `P i x` have "P (Min {i. P i x}) x" "Min {i. P i x} \<le> i" "finite {i. P i x}"
+      using Min_in[of "{i. P i x}"] by auto }
+  note 2 = this
+
+  have "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M = {x\<in>space M. Min {i. P i x} = n}"
+    by auto
+  also have "\<dots> = 
+    {x\<in>space M. if (\<forall>i. \<exists>n\<ge>i. P n x) then the None = n else 
+      if (\<exists>i. P i x) then P n x \<and> (\<forall>i<n. \<not> P i x)
+      else Min {} = n}"
+    by (intro arg_cong[where f=Collect] ext conj_cong)
+       (auto simp add: 1 2 not_le[symmetric] intro!: Min_eqI)
+  also have "\<dots> \<in> sets M"
+    by measurable
+  finally show "(\<lambda>x. Min {i. P i x}) -` {n} \<inter> space M \<in> sets M" .
+qed simp
+
 lemma measurable_count_space_insert[measurable (raw)]:
   "s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)"
   by simp
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy	Mon May 19 12:04:45 2014 +0200
@@ -0,0 +1,2039 @@
+(*  Title:      HOL/Probability/Nonnegative_Lebesgue_Integration.thy
+    Author:     Johannes Hölzl, TU München
+    Author:     Armin Heller, TU München
+*)
+
+header {* Lebesgue Integration for Nonnegative Functions *}
+
+theory Nonnegative_Lebesgue_Integration
+  imports Measure_Space Borel_Space
+begin
+
+lemma indicator_less_ereal[simp]:
+  "indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)"
+  by (simp add: indicator_def not_le)
+
+section "Simple function"
+
+text {*
+
+Our simple functions are not restricted to positive real numbers. Instead
+they are just functions with a finite range and are measurable when singleton
+sets are measurable.
+
+*}
+
+definition "simple_function M g \<longleftrightarrow>
+    finite (g ` space M) \<and>
+    (\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
+
+lemma simple_functionD:
+  assumes "simple_function M g"
+  shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
+proof -
+  show "finite (g ` space M)"
+    using assms unfolding simple_function_def by auto
+  have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
+  also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
+  finally show "g -` X \<inter> space M \<in> sets M" using assms
+    by (auto simp del: UN_simps simp: simple_function_def)
+qed
+
+lemma measurable_simple_function[measurable_dest]:
+  "simple_function M f \<Longrightarrow> f \<in> measurable M (count_space UNIV)"
+  unfolding simple_function_def measurable_def
+proof safe
+  fix A assume "finite (f ` space M)" "\<forall>x\<in>f ` space M. f -` {x} \<inter> space M \<in> sets M"
+  then have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) \<in> sets M"
+    by (intro sets.finite_UN) auto
+  also have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) = f -` A \<inter> space M"
+    by (auto split: split_if_asm)
+  finally show "f -` A \<inter> space M \<in> sets M" .
+qed simp
+
+lemma borel_measurable_simple_function:
+  "simple_function M f \<Longrightarrow> f \<in> borel_measurable M"
+  by (auto dest!: measurable_simple_function simp: measurable_def)
+
+lemma simple_function_measurable2[intro]:
+  assumes "simple_function M f" "simple_function M g"
+  shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
+proof -
+  have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
+    by auto
+  then show ?thesis using assms[THEN simple_functionD(2)] by auto
+qed
+
+lemma simple_function_indicator_representation:
+  fixes f ::"'a \<Rightarrow> ereal"
+  assumes f: "simple_function M f" and x: "x \<in> space M"
+  shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
+  (is "?l = ?r")
+proof -
+  have "?r = (\<Sum>y \<in> f ` space M.
+    (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
+    by (auto intro!: setsum_cong2)
+  also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
+    using assms by (auto dest: simple_functionD simp: setsum_delta)
+  also have "... = f x" using x by (auto simp: indicator_def)
+  finally show ?thesis by auto
+qed
+
+lemma simple_function_notspace:
+  "simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h")
+proof -
+  have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
+  hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
+  have "?h -` {0} \<inter> space M = space M" by auto
+  thus ?thesis unfolding simple_function_def by auto
+qed
+
+lemma simple_function_cong:
+  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
+  shows "simple_function M f \<longleftrightarrow> simple_function M g"
+proof -
+  have "f ` space M = g ` space M"
+    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
+    using assms by (auto intro!: image_eqI)
+  thus ?thesis unfolding simple_function_def using assms by simp
+qed
+
+lemma simple_function_cong_algebra:
+  assumes "sets N = sets M" "space N = space M"
+  shows "simple_function M f \<longleftrightarrow> simple_function N f"
+  unfolding simple_function_def assms ..
+
+lemma simple_function_borel_measurable:
+  fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
+  assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
+  shows "simple_function M f"
+  using assms unfolding simple_function_def
+  by (auto intro: borel_measurable_vimage)
+
+lemma simple_function_eq_measurable:
+  fixes f :: "'a \<Rightarrow> ereal"
+  shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> measurable M (count_space UNIV)"
+  using simple_function_borel_measurable[of f] measurable_simple_function[of M f]
+  by (fastforce simp: simple_function_def)
+
+lemma simple_function_const[intro, simp]:
+  "simple_function M (\<lambda>x. c)"
+  by (auto intro: finite_subset simp: simple_function_def)
+lemma simple_function_compose[intro, simp]:
+  assumes "simple_function M f"
+  shows "simple_function M (g \<circ> f)"
+  unfolding simple_function_def
+proof safe
+  show "finite ((g \<circ> f) ` space M)"
+    using assms unfolding simple_function_def by (auto simp: image_comp [symmetric])
+next
+  fix x assume "x \<in> space M"
+  let ?G = "g -` {g (f x)} \<inter> (f`space M)"
+  have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
+    (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
+  show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
+    using assms unfolding simple_function_def *
+    by (rule_tac sets.finite_UN) auto
+qed
+
+lemma simple_function_indicator[intro, simp]:
+  assumes "A \<in> sets M"
+  shows "simple_function M (indicator A)"
+proof -
+  have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
+    by (auto simp: indicator_def)
+  hence "finite ?S" by (rule finite_subset) simp
+  moreover have "- A \<inter> space M = space M - A" by auto
+  ultimately show ?thesis unfolding simple_function_def
+    using assms by (auto simp: indicator_def [abs_def])
+qed
+
+lemma simple_function_Pair[intro, simp]:
+  assumes "simple_function M f"
+  assumes "simple_function M g"
+  shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
+  unfolding simple_function_def
+proof safe
+  show "finite (?p ` space M)"
+    using assms unfolding simple_function_def
+    by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
+next
+  fix x assume "x \<in> space M"
+  have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
+      (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
+    by auto
+  with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
+    using assms unfolding simple_function_def by auto
+qed
+
+lemma simple_function_compose1:
+  assumes "simple_function M f"
+  shows "simple_function M (\<lambda>x. g (f x))"
+  using simple_function_compose[OF assms, of g]
+  by (simp add: comp_def)
+
+lemma simple_function_compose2:
+  assumes "simple_function M f" and "simple_function M g"
+  shows "simple_function M (\<lambda>x. h (f x) (g x))"
+proof -
+  have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
+    using assms by auto
+  thus ?thesis by (simp_all add: comp_def)
+qed
+
+lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
+  and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
+  and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
+  and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
+  and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
+  and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
+  and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
+
+lemma simple_function_setsum[intro, simp]:
+  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
+  shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
+proof cases
+  assume "finite P" from this assms show ?thesis by induct auto
+qed auto
+
+lemma simple_function_ereal[intro, simp]: 
+  fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
+  shows "simple_function M (\<lambda>x. ereal (f x))"
+  by (auto intro!: simple_function_compose1[OF sf])
+
+lemma simple_function_real_of_nat[intro, simp]: 
+  fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
+  shows "simple_function M (\<lambda>x. real (f x))"
+  by (auto intro!: simple_function_compose1[OF sf])
+
+lemma borel_measurable_implies_simple_function_sequence:
+  fixes u :: "'a \<Rightarrow> ereal"
+  assumes u: "u \<in> borel_measurable M"
+  shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
+             (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
+proof -
+  def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)"
+  { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
+    proof (split split_if, intro conjI impI)
+      assume "\<not> real j \<le> u x"
+      then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
+         by (cases "u x") (auto intro!: natfloor_mono)
+      moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
+        by (intro real_natfloor_le) auto
+      ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
+        unfolding real_of_nat_le_iff by auto
+    qed auto }
+  note f_upper = this
+
+  have real_f:
+    "\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (natfloor (real (u x) * 2 ^ i)))"
+    unfolding f_def by auto
+
+  let ?g = "\<lambda>j x. real (f x j) / 2^j :: ereal"
+  show ?thesis
+  proof (intro exI[of _ ?g] conjI allI ballI)
+    fix i
+    have "simple_function M (\<lambda>x. real (f x i))"
+    proof (intro simple_function_borel_measurable)
+      show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
+        using u by (auto simp: real_f)
+      have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
+        using f_upper[of _ i] by auto
+      then show "finite ((\<lambda>x. real (f x i))`space M)"
+        by (rule finite_subset) auto
+    qed
+    then show "simple_function M (?g i)"
+      by (auto intro: simple_function_ereal simple_function_div)
+  next
+    show "incseq ?g"
+    proof (intro incseq_ereal incseq_SucI le_funI)
+      fix x and i :: nat
+      have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
+      proof ((split split_if)+, intro conjI impI)
+        assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
+        then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)"
+          by (cases "u x") (auto intro!: le_natfloor)
+      next
+        assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x"
+        then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i"
+          by (cases "u x") auto
+      next
+        assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
+        have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
+          by simp
+        also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
+        proof cases
+          assume "0 \<le> u x" then show ?thesis
+            by (intro le_mult_natfloor) 
+        next
+          assume "\<not> 0 \<le> u x" then show ?thesis
+            by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
+        qed
+        also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
+          by (simp add: ac_simps)
+        finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" .
+      qed simp
+      then show "?g i x \<le> ?g (Suc i) x"
+        by (auto simp: field_simps)
+    qed
+  next
+    fix x show "(SUP i. ?g i x) = max 0 (u x)"
+    proof (rule SUP_eqI)
+      fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
+        by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
+                                     mult_nonpos_nonneg)
+    next
+      fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
+      have "\<And>i. 0 \<le> ?g i x" by auto
+      from order_trans[OF this *] have "0 \<le> y" by simp
+      show "max 0 (u x) \<le> y"
+      proof (cases y)
+        case (real r)
+        with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
+        from reals_Archimedean2[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
+        then have "\<exists>p. max 0 (u x) = ereal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
+        then guess p .. note ux = this
+        obtain m :: nat where m: "p < real m" using reals_Archimedean2 ..
+        have "p \<le> r"
+        proof (rule ccontr)
+          assume "\<not> p \<le> r"
+          with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
+          obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: field_simps)
+          then have "r * 2^max N m < p * 2^max N m - 1" by simp
+          moreover
+          have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m"
+            using *[of "max N m"] m unfolding real_f using ux
+            by (cases "0 \<le> u x") (simp_all add: max_def split: split_if_asm)
+          then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
+            by (metis real_natfloor_gt_diff_one less_le_trans)
+          ultimately show False by auto
+        qed
+        then show "max 0 (u x) \<le> y" using real ux by simp
+      qed (insert `0 \<le> y`, auto)
+    qed
+  qed auto
+qed
+
+lemma borel_measurable_implies_simple_function_sequence':
+  fixes u :: "'a \<Rightarrow> ereal"
+  assumes u: "u \<in> borel_measurable M"
+  obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
+    "\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
+  using borel_measurable_implies_simple_function_sequence[OF u] by auto
+
+lemma simple_function_induct[consumes 1, case_names cong set mult add, induct set: simple_function]:
+  fixes u :: "'a \<Rightarrow> ereal"
+  assumes u: "simple_function M u"
+  assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
+  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
+  assumes mult: "\<And>u c. P u \<Longrightarrow> P (\<lambda>x. c * u x)"
+  assumes add: "\<And>u v. P u \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
+  shows "P u"
+proof (rule cong)
+  from AE_space show "AE x in M. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x"
+  proof eventually_elim
+    fix x assume x: "x \<in> space M"
+    from simple_function_indicator_representation[OF u x]
+    show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
+  qed
+next
+  from u have "finite (u ` space M)"
+    unfolding simple_function_def by auto
+  then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
+  proof induct
+    case empty show ?case
+      using set[of "{}"] by (simp add: indicator_def[abs_def])
+  qed (auto intro!: add mult set simple_functionD u)
+next
+  show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
+    apply (subst simple_function_cong)
+    apply (rule simple_function_indicator_representation[symmetric])
+    apply (auto intro: u)
+    done
+qed fact
+
+lemma simple_function_induct_nn[consumes 2, case_names cong set mult add]:
+  fixes u :: "'a \<Rightarrow> ereal"
+  assumes u: "simple_function M u" and nn: "\<And>x. 0 \<le> u x"
+  assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
+  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
+  assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
+  assumes add: "\<And>u v. simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> simple_function M v \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
+  shows "P u"
+proof -
+  show ?thesis
+  proof (rule cong)
+    fix x assume x: "x \<in> space M"
+    from simple_function_indicator_representation[OF u x]
+    show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
+  next
+    show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
+      apply (subst simple_function_cong)
+      apply (rule simple_function_indicator_representation[symmetric])
+      apply (auto intro: u)
+      done
+  next
+    
+    from u nn have "finite (u ` space M)" "\<And>x. x \<in> u ` space M \<Longrightarrow> 0 \<le> x"
+      unfolding simple_function_def by auto
+    then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
+    proof induct
+      case empty show ?case
+        using set[of "{}"] by (simp add: indicator_def[abs_def])
+    next
+      case (insert x S)
+      { fix z have "(\<Sum>y\<in>S. y * indicator (u -` {y} \<inter> space M) z) = 0 \<or>
+          x * indicator (u -` {x} \<inter> space M) z = 0"
+          using insert by (subst setsum_ereal_0) (auto simp: indicator_def) }
+      note disj = this
+      from insert show ?case
+        by (auto intro!: add mult set simple_functionD u setsum_nonneg simple_function_setsum disj)
+    qed
+  qed fact
+qed
+
+lemma borel_measurable_induct[consumes 2, case_names cong set mult add seq, induct set: borel_measurable]:
+  fixes u :: "'a \<Rightarrow> ereal"
+  assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x"
+  assumes cong: "\<And>f g. f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P g \<Longrightarrow> P f"
+  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
+  assumes mult': "\<And>u c. 0 \<le> c \<Longrightarrow> c < \<infinity> \<Longrightarrow> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x < \<infinity>) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
+  assumes add: "\<And>u v. u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x < \<infinity>) \<Longrightarrow> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> v x < \<infinity>) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
+  assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow> (\<And>i x. 0 \<le> U i x) \<Longrightarrow> (\<And>i x. x \<in> space M \<Longrightarrow> U i x < \<infinity>) \<Longrightarrow>  (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> u = (SUP i. U i) \<Longrightarrow> P (SUP i. U i)"
+  shows "P u"
+  using u
+proof (induct rule: borel_measurable_implies_simple_function_sequence')
+  fix U assume U: "\<And>i. simple_function M (U i)" "incseq U" "\<And>i. \<infinity> \<notin> range (U i)" and
+    sup: "\<And>x. (SUP i. U i x) = max 0 (u x)" and nn: "\<And>i x. 0 \<le> U i x"
+  have u_eq: "u = (SUP i. U i)"
+    using nn u sup by (auto simp: max_def)
+
+  have not_inf: "\<And>x i. x \<in> space M \<Longrightarrow> U i x < \<infinity>"
+    using U by (auto simp: image_iff eq_commute)
+  
+  from U have "\<And>i. U i \<in> borel_measurable M"
+    by (simp add: borel_measurable_simple_function)
+
+  show "P u"
+    unfolding u_eq
+  proof (rule seq)
+    fix i show "P (U i)"
+      using `simple_function M (U i)` nn[of i] not_inf[of _ i]
+    proof (induct rule: simple_function_induct_nn)
+      case (mult u c)
+      show ?case
+      proof cases
+        assume "c = 0 \<or> space M = {} \<or> (\<forall>x\<in>space M. u x = 0)"
+        with mult(2) show ?thesis
+          by (intro cong[of "\<lambda>x. c * u x" "indicator {}"] set)
+             (auto dest!: borel_measurable_simple_function)
+      next
+        assume "\<not> (c = 0 \<or> space M = {} \<or> (\<forall>x\<in>space M. u x = 0))"
+        with mult obtain x where u_fin: "\<And>x. x \<in> space M \<Longrightarrow> u x < \<infinity>"
+          and x: "x \<in> space M" "u x \<noteq> 0" "c \<noteq> 0"
+          by auto
+        with mult have "P u"
+          by auto
+        from x mult(5)[OF `x \<in> space M`] mult(1) mult(3)[of x] have "c < \<infinity>"
+          by auto
+        with u_fin mult
+        show ?thesis
+          by (intro mult') (auto dest!: borel_measurable_simple_function)
+      qed
+    qed (auto intro: cong intro!: set add dest!: borel_measurable_simple_function)
+  qed fact+
+qed
+
+lemma simple_function_If_set:
+  assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
+  shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
+proof -
+  def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
+  show ?thesis unfolding simple_function_def
+  proof safe
+    have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
+    from finite_subset[OF this] assms
+    show "finite (?IF ` space M)" unfolding simple_function_def by auto
+  next
+    fix x assume "x \<in> space M"
+    then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
+      then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
+      else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
+      using sets.sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
+    have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
+      unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
+    show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
+  qed
+qed
+
+lemma simple_function_If:
+  assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
+  shows "simple_function M (\<lambda>x. if P x then f x else g x)"
+proof -
+  have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
+  with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
+qed
+
+lemma simple_function_subalgebra:
+  assumes "simple_function N f"
+  and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
+  shows "simple_function M f"
+  using assms unfolding simple_function_def by auto
+
+lemma simple_function_comp:
+  assumes T: "T \<in> measurable M M'"
+    and f: "simple_function M' f"
+  shows "simple_function M (\<lambda>x. f (T x))"
+proof (intro simple_function_def[THEN iffD2] conjI ballI)
+  have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
+    using T unfolding measurable_def by auto
+  then show "finite ((\<lambda>x. f (T x)) ` space M)"
+    using f unfolding simple_function_def by (auto intro: finite_subset)
+  fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
+  then have "i \<in> f ` space M'"
+    using T unfolding measurable_def by auto
+  then have "f -` {i} \<inter> space M' \<in> sets M'"
+    using f unfolding simple_function_def by auto
+  then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
+    using T unfolding measurable_def by auto
+  also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
+    using T unfolding measurable_def by auto
+  finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" .
+qed
+
+section "Simple integral"
+
+definition simple_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>S") where
+  "integral\<^sup>S M f = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M))"
+
+syntax
+  "_simple_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>S _. _ \<partial>_" [60,61] 110)
+
+translations
+  "\<integral>\<^sup>S x. f \<partial>M" == "CONST simple_integral M (%x. f)"
+
+lemma simple_integral_cong:
+  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
+  shows "integral\<^sup>S M f = integral\<^sup>S M g"
+proof -
+  have "f ` space M = g ` space M"
+    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
+    using assms by (auto intro!: image_eqI)
+  thus ?thesis unfolding simple_integral_def by simp
+qed
+
+lemma simple_integral_const[simp]:
+  "(\<integral>\<^sup>Sx. c \<partial>M) = c * (emeasure M) (space M)"
+proof (cases "space M = {}")
+  case True thus ?thesis unfolding simple_integral_def by simp
+next
+  case False hence "(\<lambda>x. c) ` space M = {c}" by auto
+  thus ?thesis unfolding simple_integral_def by simp
+qed
+
+lemma simple_function_partition:
+  assumes f: "simple_function M f" and g: "simple_function M g"
+  assumes sub: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow> g x = g y \<Longrightarrow> f x = f y"
+  assumes v: "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)"
+  shows "integral\<^sup>S M f = (\<Sum>y\<in>g ` space M. v y * emeasure M {x\<in>space M. g x = y})"
+    (is "_ = ?r")
+proof -
+  from f g have [simp]: "finite (f`space M)" "finite (g`space M)"
+    by (auto simp: simple_function_def)
+  from f g have [measurable]: "f \<in> measurable M (count_space UNIV)" "g \<in> measurable M (count_space UNIV)"
+    by (auto intro: measurable_simple_function)
+
+  { fix y assume "y \<in> space M"
+    then have "f ` space M \<inter> {i. \<exists>x\<in>space M. i = f x \<and> g y = g x} = {v (g y)}"
+      by (auto cong: sub simp: v[symmetric]) }
+  note eq = this
+
+  have "integral\<^sup>S M f =
+    (\<Sum>y\<in>f`space M. y * (\<Sum>z\<in>g`space M. 
+      if \<exists>x\<in>space M. y = f x \<and> z = g x then emeasure M {x\<in>space M. g x = z} else 0))"
+    unfolding simple_integral_def
+  proof (safe intro!: setsum_cong ereal_left_mult_cong)
+    fix y assume y: "y \<in> space M" "f y \<noteq> 0"
+    have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} = 
+        {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
+      by auto
+    have eq:"(\<Union>i\<in>{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}. {x \<in> space M. g x = i}) =
+        f -` {f y} \<inter> space M"
+      by (auto simp: eq_commute cong: sub rev_conj_cong)
+    have "finite (g`space M)" by simp
+    then have "finite {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
+      by (rule rev_finite_subset) auto
+    then show "emeasure M (f -` {f y} \<inter> space M) =
+      (\<Sum>z\<in>g ` space M. if \<exists>x\<in>space M. f y = f x \<and> z = g x then emeasure M {x \<in> space M. g x = z} else 0)"
+      apply (simp add: setsum_cases)
+      apply (subst setsum_emeasure)
+      apply (auto simp: disjoint_family_on_def eq)
+      done
+  qed
+  also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M. 
+      if \<exists>x\<in>space M. y = f x \<and> z = g x then y * emeasure M {x\<in>space M. g x = z} else 0))"
+    by (auto intro!: setsum_cong simp: setsum_ereal_right_distrib emeasure_nonneg)
+  also have "\<dots> = ?r"
+    by (subst setsum_commute)
+       (auto intro!: setsum_cong simp: setsum_cases scaleR_setsum_right[symmetric] eq)
+  finally show "integral\<^sup>S M f = ?r" .
+qed
+
+lemma simple_integral_add[simp]:
+  assumes f: "simple_function M f" and "\<And>x. 0 \<le> f x" and g: "simple_function M g" and "\<And>x. 0 \<le> g x"
+  shows "(\<integral>\<^sup>Sx. f x + g x \<partial>M) = integral\<^sup>S M f + integral\<^sup>S M g"
+proof -
+  have "(\<integral>\<^sup>Sx. f x + g x \<partial>M) =
+    (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. (fst y + snd y) * emeasure M {x\<in>space M. (f x, g x) = y})"
+    by (intro simple_function_partition) (auto intro: f g)
+  also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * emeasure M {x\<in>space M. (f x, g x) = y}) +
+    (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y})"
+    using assms(2,4) by (auto intro!: setsum_cong ereal_left_distrib simp: setsum_addf[symmetric])
+  also have "(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * emeasure M {x\<in>space M. (f x, g x) = y}) = (\<integral>\<^sup>Sx. f x \<partial>M)"
+    by (intro simple_function_partition[symmetric]) (auto intro: f g)
+  also have "(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y}) = (\<integral>\<^sup>Sx. g x \<partial>M)"
+    by (intro simple_function_partition[symmetric]) (auto intro: f g)
+  finally show ?thesis .
+qed
+
+lemma simple_integral_setsum[simp]:
+  assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
+  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
+  shows "(\<integral>\<^sup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>S M (f i))"
+proof cases
+  assume "finite P"
+  from this assms show ?thesis
+    by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
+qed auto
+
+lemma simple_integral_mult[simp]:
+  assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
+  shows "(\<integral>\<^sup>Sx. c * f x \<partial>M) = c * integral\<^sup>S M f"
+proof -
+  have "(\<integral>\<^sup>Sx. c * f x \<partial>M) = (\<Sum>y\<in>f ` space M. (c * y) * emeasure M {x\<in>space M. f x = y})"
+    using f by (intro simple_function_partition) auto
+  also have "\<dots> = c * integral\<^sup>S M f"
+    using f unfolding simple_integral_def
+    by (subst setsum_ereal_right_distrib) (auto simp: emeasure_nonneg mult_assoc Int_def conj_commute)
+  finally show ?thesis .
+qed
+
+lemma simple_integral_mono_AE:
+  assumes f[measurable]: "simple_function M f" and g[measurable]: "simple_function M g"
+  and mono: "AE x in M. f x \<le> g x"
+  shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
+proof -
+  let ?\<mu> = "\<lambda>P. emeasure M {x\<in>space M. P x}"
+  have "integral\<^sup>S M f = (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * ?\<mu> (\<lambda>x. (f x, g x) = y))"
+    using f g by (intro simple_function_partition) auto
+  also have "\<dots> \<le> (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * ?\<mu> (\<lambda>x. (f x, g x) = y))"
+  proof (clarsimp intro!: setsum_mono)
+    fix x assume "x \<in> space M"
+    let ?M = "?\<mu> (\<lambda>y. f y = f x \<and> g y = g x)"
+    show "f x * ?M \<le> g x * ?M"
+    proof cases
+      assume "?M \<noteq> 0"
+      then have "0 < ?M"
+        by (simp add: less_le emeasure_nonneg)
+      also have "\<dots> \<le> ?\<mu> (\<lambda>y. f x \<le> g x)"
+        using mono by (intro emeasure_mono_AE) auto
+      finally have "\<not> \<not> f x \<le> g x"
+        by (intro notI) auto
+      then show ?thesis
+        by (intro ereal_mult_right_mono) auto
+    qed simp
+  qed
+  also have "\<dots> = integral\<^sup>S M g"
+    using f g by (intro simple_function_partition[symmetric]) auto
+  finally show ?thesis .
+qed
+
+lemma simple_integral_mono:
+  assumes "simple_function M f" and "simple_function M g"
+  and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
+  shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
+  using assms by (intro simple_integral_mono_AE) auto
+
+lemma simple_integral_cong_AE:
+  assumes "simple_function M f" and "simple_function M g"
+  and "AE x in M. f x = g x"
+  shows "integral\<^sup>S M f = integral\<^sup>S M g"
+  using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
+
+lemma simple_integral_cong':
+  assumes sf: "simple_function M f" "simple_function M g"
+  and mea: "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0"
+  shows "integral\<^sup>S M f = integral\<^sup>S M g"
+proof (intro simple_integral_cong_AE sf AE_I)
+  show "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" by fact
+  show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
+    using sf[THEN borel_measurable_simple_function] by auto
+qed simp
+
+lemma simple_integral_indicator:
+  assumes A: "A \<in> sets M"
+  assumes f: "simple_function M f"
+  shows "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
+    (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))"
+proof -
+  have eq: "(\<lambda>x. (f x, indicator A x)) ` space M \<inter> {x. snd x = 1} = (\<lambda>x. (f x, 1::ereal))`A"
+    using A[THEN sets.sets_into_space] by (auto simp: indicator_def image_iff split: split_if_asm)
+  have eq2: "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
+    by (auto simp: image_iff)
+
+  have "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
+    (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x))`space M. (fst y * snd y) * emeasure M {x\<in>space M. (f x, indicator A x) = y})"
+    using assms by (intro simple_function_partition) auto
+  also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x::ereal))`space M.
+    if snd y = 1 then fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A) else 0)"
+    by (auto simp: indicator_def split: split_if_asm intro!: arg_cong2[where f="op *"] arg_cong2[where f=emeasure] setsum_cong)
+  also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, 1::ereal))`A. fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A))"
+    using assms by (subst setsum_cases) (auto intro!: simple_functionD(1) simp: eq)
+  also have "\<dots> = (\<Sum>y\<in>fst`(\<lambda>x. (f x, 1::ereal))`A. y * emeasure M (f -` {y} \<inter> space M \<inter> A))"
+    by (subst setsum_reindex[where f=fst]) (auto simp: inj_on_def)
+  also have "\<dots> = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))"
+    using A[THEN sets.sets_into_space]
+    by (intro setsum_mono_zero_cong_left simple_functionD f) (auto simp: image_comp comp_def eq2)
+  finally show ?thesis .
+qed
+
+lemma simple_integral_indicator_only[simp]:
+  assumes "A \<in> sets M"
+  shows "integral\<^sup>S M (indicator A) = emeasure M A"
+  using simple_integral_indicator[OF assms, of "\<lambda>x. 1"] sets.sets_into_space[OF assms]
+  by (simp_all add: image_constant_conv Int_absorb1 split: split_if_asm)
+
+lemma simple_integral_null_set:
+  assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets M"
+  shows "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = 0"
+proof -
+  have "AE x in M. indicator N x = (0 :: ereal)"
+    using `N \<in> null_sets M` by (auto simp: indicator_def intro!: AE_I[of _ _ N])
+  then have "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^sup>Sx. 0 \<partial>M)"
+    using assms apply (intro simple_integral_cong_AE) by auto
+  then show ?thesis by simp
+qed
+
+lemma simple_integral_cong_AE_mult_indicator:
+  assumes sf: "simple_function M f" and eq: "AE x in M. x \<in> S" and "S \<in> sets M"
+  shows "integral\<^sup>S M f = (\<integral>\<^sup>Sx. f x * indicator S x \<partial>M)"
+  using assms by (intro simple_integral_cong_AE) auto
+
+lemma simple_integral_cmult_indicator:
+  assumes A: "A \<in> sets M"
+  shows "(\<integral>\<^sup>Sx. c * indicator A x \<partial>M) = c * emeasure M A"
+  using simple_integral_mult[OF simple_function_indicator[OF A]]
+  unfolding simple_integral_indicator_only[OF A] by simp
+
+lemma simple_integral_positive:
+  assumes f: "simple_function M f" and ae: "AE x in M. 0 \<le> f x"
+  shows "0 \<le> integral\<^sup>S M f"
+proof -
+  have "integral\<^sup>S M (\<lambda>x. 0) \<le> integral\<^sup>S M f"
+    using simple_integral_mono_AE[OF _ f ae] by auto
+  then show ?thesis by simp
+qed
+
+section "Continuous positive integration"
+
+definition positive_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>P") where
+  "integral\<^sup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^sup>S M g)"
+
+syntax
+  "_positive_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>+ _. _ \<partial>_" [60,61] 110)
+
+translations
+  "\<integral>\<^sup>+ x. f \<partial>M" == "CONST positive_integral M (%x. f)"
+
+lemma positive_integral_positive:
+  "0 \<le> integral\<^sup>P M f"
+  by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
+
+lemma positive_integral_not_MInfty[simp]: "integral\<^sup>P M f \<noteq> -\<infinity>"
+  using positive_integral_positive[of M f] by auto
+
+lemma positive_integral_def_finite:
+  "integral\<^sup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^sup>S M g)"
+    (is "_ = SUPREMUM ?A ?f")
+  unfolding positive_integral_def
+proof (safe intro!: antisym SUP_least)
+  fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
+  let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
+  note gM = g(1)[THEN borel_measurable_simple_function]
+  have \<mu>_G_pos: "0 \<le> (emeasure M) ?G" using gM by auto
+  let ?g = "\<lambda>y x. if g x = \<infinity> then y else max 0 (g x)"
+  from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
+    apply (safe intro!: simple_function_max simple_function_If)
+    apply (force simp: max_def le_fun_def split: split_if_asm)+
+    done
+  show "integral\<^sup>S M g \<le> SUPREMUM ?A ?f"
+  proof cases
+    have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
+    assume "(emeasure M) ?G = 0"
+    with gM have "AE x in M. x \<notin> ?G"
+      by (auto simp add: AE_iff_null intro!: null_setsI)
+    with gM g show ?thesis
+      by (intro SUP_upper2[OF g0] simple_integral_mono_AE)
+         (auto simp: max_def intro!: simple_function_If)
+  next
+    assume \<mu>_G: "(emeasure M) ?G \<noteq> 0"
+    have "SUPREMUM ?A (integral\<^sup>S M) = \<infinity>"
+    proof (intro SUP_PInfty)
+      fix n :: nat
+      let ?y = "ereal (real n) / (if (emeasure M) ?G = \<infinity> then 1 else (emeasure M) ?G)"
+      have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>_G \<mu>_G_pos by (auto simp: ereal_divide_eq)
+      then have "?g ?y \<in> ?A" by (rule g_in_A)
+      have "real n \<le> ?y * (emeasure M) ?G"
+        using \<mu>_G \<mu>_G_pos by (cases "(emeasure M) ?G") (auto simp: field_simps)
+      also have "\<dots> = (\<integral>\<^sup>Sx. ?y * indicator ?G x \<partial>M)"
+        using `0 \<le> ?y` `?g ?y \<in> ?A` gM
+        by (subst simple_integral_cmult_indicator) auto
+      also have "\<dots> \<le> integral\<^sup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
+        by (intro simple_integral_mono) auto
+      finally show "\<exists>i\<in>?A. real n \<le> integral\<^sup>S M i"
+        using `?g ?y \<in> ?A` by blast
+    qed
+    then show ?thesis by simp
+  qed
+qed (auto intro: SUP_upper)
+
+lemma positive_integral_mono_AE:
+  assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^sup>P M u \<le> integral\<^sup>P M v"
+  unfolding positive_integral_def
+proof (safe intro!: SUP_mono)
+  fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
+  from ae[THEN AE_E] guess N . note N = this
+  then have ae_N: "AE x in M. x \<notin> N" by (auto intro: AE_not_in)
+  let ?n = "\<lambda>x. n x * indicator (space M - N) x"
+  have "AE x in M. n x \<le> ?n x" "simple_function M ?n"
+    using n N ae_N by auto
+  moreover
+  { fix x have "?n x \<le> max 0 (v x)"
+    proof cases
+      assume x: "x \<in> space M - N"
+      with N have "u x \<le> v x" by auto
+      with n(2)[THEN le_funD, of x] x show ?thesis
+        by (auto simp: max_def split: split_if_asm)
+    qed simp }
+  then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
+  moreover have "integral\<^sup>S M n \<le> integral\<^sup>S M ?n"
+    using ae_N N n by (auto intro!: simple_integral_mono_AE)
+  ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^sup>S M n \<le> integral\<^sup>S M m"
+    by force
+qed
+
+lemma positive_integral_mono:
+  "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^sup>P M u \<le> integral\<^sup>P M v"
+  by (auto intro: positive_integral_mono_AE)
+
+lemma positive_integral_cong_AE:
+  "AE x in M. u x = v x \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P M v"
+  by (auto simp: eq_iff intro!: positive_integral_mono_AE)
+
+lemma positive_integral_cong:
+  "(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P M v"
+  by (auto intro: positive_integral_cong_AE)
+
+lemma positive_integral_cong_strong:
+  "M = N \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P N v"
+  by (auto intro: positive_integral_cong)
+
+lemma positive_integral_eq_simple_integral:
+  assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^sup>P M f = integral\<^sup>S M f"
+proof -
+  let ?f = "\<lambda>x. f x * indicator (space M) x"
+  have f': "simple_function M ?f" using f by auto
+  with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
+    by (auto simp: fun_eq_iff max_def split: split_indicator)
+  have "integral\<^sup>P M ?f \<le> integral\<^sup>S M ?f" using f'
+    by (force intro!: SUP_least simple_integral_mono simp: le_fun_def positive_integral_def)
+  moreover have "integral\<^sup>S M ?f \<le> integral\<^sup>P M ?f"
+    unfolding positive_integral_def
+    using f' by (auto intro!: SUP_upper)
+  ultimately show ?thesis
+    by (simp cong: positive_integral_cong simple_integral_cong)
+qed
+
+lemma positive_integral_eq_simple_integral_AE:
+  assumes f: "simple_function M f" "AE x in M. 0 \<le> f x" shows "integral\<^sup>P M f = integral\<^sup>S M f"
+proof -
+  have "AE x in M. f x = max 0 (f x)" using f by (auto split: split_max)
+  with f have "integral\<^sup>P M f = integral\<^sup>S M (\<lambda>x. max 0 (f x))"
+    by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
+             add: positive_integral_eq_simple_integral)
+  with assms show ?thesis
+    by (auto intro!: simple_integral_cong_AE split: split_max)
+qed
+
+lemma positive_integral_SUP_approx:
+  assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
+  and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
+  shows "integral\<^sup>S M u \<le> (SUP i. integral\<^sup>P M (f i))" (is "_ \<le> ?S")
+proof (rule ereal_le_mult_one_interval)
+  have "0 \<le> (SUP i. integral\<^sup>P M (f i))"
+    using f(3) by (auto intro!: SUP_upper2 positive_integral_positive)
+  then show "(SUP i. integral\<^sup>P M (f i)) \<noteq> -\<infinity>" by auto
+  have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
+    using u(3) by auto
+  fix a :: ereal assume "0 < a" "a < 1"
+  hence "a \<noteq> 0" by auto
+  let ?B = "\<lambda>i. {x \<in> space M. a * u x \<le> f i x}"
+  have B: "\<And>i. ?B i \<in> sets M"
+    using f `simple_function M u`[THEN borel_measurable_simple_function] by auto
+
+  let ?uB = "\<lambda>i x. u x * indicator (?B i) x"
+
+  { fix i have "?B i \<subseteq> ?B (Suc i)"
+    proof safe
+      fix i x assume "a * u x \<le> f i x"
+      also have "\<dots> \<le> f (Suc i) x"
+        using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
+      finally show "a * u x \<le> f (Suc i) x" .
+    qed }
+  note B_mono = this
+
+  note B_u = sets.Int[OF u(1)[THEN simple_functionD(2)] B]
+
+  let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n"
+  have measure_conv: "\<And>i. (emeasure M) (u -` {i} \<inter> space M) = (SUP n. (emeasure M) (?B' i n))"
+  proof -
+    fix i
+    have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
+    have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
+    have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
+    proof safe
+      fix x i assume x: "x \<in> space M"
+      show "x \<in> (\<Union>i. ?B' (u x) i)"
+      proof cases
+        assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
+      next
+        assume "u x \<noteq> 0"
+        with `a < 1` u_range[OF `x \<in> space M`]
+        have "a * u x < 1 * u x"
+          by (intro ereal_mult_strict_right_mono) (auto simp: image_iff)
+        also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def)
+        finally obtain i where "a * u x < f i x" unfolding SUP_def
+          by (auto simp add: less_SUP_iff)
+        hence "a * u x \<le> f i x" by auto
+        thus ?thesis using `x \<in> space M` by auto
+      qed
+    qed
+    then show "?thesis i" using SUP_emeasure_incseq[OF 1 2] by simp
+  qed
+
+  have "integral\<^sup>S M u = (SUP i. integral\<^sup>S M (?uB i))"
+    unfolding simple_integral_indicator[OF B `simple_function M u`]
+  proof (subst SUP_ereal_setsum, safe)
+    fix x n assume "x \<in> space M"
+    with u_range show "incseq (\<lambda>i. u x * (emeasure M) (?B' (u x) i))" "\<And>i. 0 \<le> u x * (emeasure M) (?B' (u x) i)"
+      using B_mono B_u by (auto intro!: emeasure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff)
+  next
+    show "integral\<^sup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * (emeasure M) (?B' i n))"
+      using measure_conv u_range B_u unfolding simple_integral_def
+      by (auto intro!: setsum_cong SUP_ereal_cmult [symmetric])
+  qed
+  moreover
+  have "a * (SUP i. integral\<^sup>S M (?uB i)) \<le> ?S"
+    apply (subst SUP_ereal_cmult [symmetric])
+  proof (safe intro!: SUP_mono bexI)
+    fix i
+    have "a * integral\<^sup>S M (?uB i) = (\<integral>\<^sup>Sx. a * ?uB i x \<partial>M)"
+      using B `simple_function M u` u_range
+      by (subst simple_integral_mult) (auto split: split_indicator)
+    also have "\<dots> \<le> integral\<^sup>P M (f i)"
+    proof -
+      have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
+      show ?thesis using f(3) * u_range `0 < a`
+        by (subst positive_integral_eq_simple_integral[symmetric])
+           (auto intro!: positive_integral_mono split: split_indicator)
+    qed
+    finally show "a * integral\<^sup>S M (?uB i) \<le> integral\<^sup>P M (f i)"
+      by auto
+  next
+    fix i show "0 \<le> \<integral>\<^sup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
+      by (intro simple_integral_positive) (auto split: split_indicator)
+  qed (insert `0 < a`, auto)
+  ultimately show "a * integral\<^sup>S M u \<le> ?S" by simp
+qed
+
+lemma incseq_positive_integral:
+  assumes "incseq f" shows "incseq (\<lambda>i. integral\<^sup>P M (f i))"
+proof -
+  have "\<And>i x. f i x \<le> f (Suc i) x"
+    using assms by (auto dest!: incseq_SucD simp: le_fun_def)
+  then show ?thesis
+    by (auto intro!: incseq_SucI positive_integral_mono)
+qed
+
+text {* Beppo-Levi monotone convergence theorem *}
+lemma positive_integral_monotone_convergence_SUP:
+  assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
+  shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
+proof (rule antisym)
+  show "(SUP j. integral\<^sup>P M (f j)) \<le> (\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M)"
+    by (auto intro!: SUP_least SUP_upper positive_integral_mono)
+next
+  show "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^sup>P M (f j))"
+    unfolding positive_integral_def_finite[of _ "\<lambda>x. SUP i. f i x"]
+  proof (safe intro!: SUP_least)
+    fix g assume g: "simple_function M g"
+      and *: "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
+    then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}"
+      using f by (auto intro!: SUP_upper2)
+    with * show "integral\<^sup>S M g \<le> (SUP j. integral\<^sup>P M (f j))"
+      by (intro  positive_integral_SUP_approx[OF f g _ g'])
+         (auto simp: le_fun_def max_def)
+  qed
+qed
+
+lemma positive_integral_monotone_convergence_SUP_AE:
+  assumes f: "\<And>i. AE x in M. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
+  shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
+proof -
+  from f have "AE x in M. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
+    by (simp add: AE_all_countable)
+  from this[THEN AE_E] guess N . note N = this
+  let ?f = "\<lambda>i x. if x \<in> space M - N then f i x else 0"
+  have f_eq: "AE x in M. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N])
+  then have "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^sup>+ x. (SUP i. ?f i x) \<partial>M)"
+    by (auto intro!: positive_integral_cong_AE)
+  also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. ?f i x \<partial>M))"
+  proof (rule positive_integral_monotone_convergence_SUP)
+    show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
+    { fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
+        using f N(3) by (intro measurable_If_set) auto
+      fix x show "0 \<le> ?f i x"
+        using N(1) by auto }
+  qed
+  also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. f i x \<partial>M))"
+    using f_eq by (force intro!: arg_cong[where f="SUPREMUM UNIV"] positive_integral_cong_AE ext)
+  finally show ?thesis .
+qed
+
+lemma positive_integral_monotone_convergence_SUP_AE_incseq:
+  assumes f: "incseq f" "\<And>i. AE x in M. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
+  shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
+  using f[unfolded incseq_Suc_iff le_fun_def]
+  by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
+     auto
+
+lemma positive_integral_monotone_convergence_simple:
+  assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
+  shows "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)"
+  using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
+    f(3)[THEN borel_measurable_simple_function] f(2)]
+  by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPREMUM UNIV"] ext)
+
+lemma positive_integral_max_0:
+  "(\<integral>\<^sup>+x. max 0 (f x) \<partial>M) = integral\<^sup>P M f"
+  by (simp add: le_fun_def positive_integral_def)
+
+lemma positive_integral_cong_pos:
+  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
+  shows "integral\<^sup>P M f = integral\<^sup>P M g"
+proof -
+  have "integral\<^sup>P M (\<lambda>x. max 0 (f x)) = integral\<^sup>P M (\<lambda>x. max 0 (g x))"
+  proof (intro positive_integral_cong)
+    fix x assume "x \<in> space M"
+    from assms[OF this] show "max 0 (f x) = max 0 (g x)"
+      by (auto split: split_max)
+  qed
+  then show ?thesis by (simp add: positive_integral_max_0)
+qed
+
+lemma SUP_simple_integral_sequences:
+  assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
+  and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
+  and eq: "AE x in M. (SUP i. f i x) = (SUP i. g i x)"
+  shows "(SUP i. integral\<^sup>S M (f i)) = (SUP i. integral\<^sup>S M (g i))"
+    (is "SUPREMUM _ ?F = SUPREMUM _ ?G")
+proof -
+  have "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)"
+    using f by (rule positive_integral_monotone_convergence_simple)
+  also have "\<dots> = (\<integral>\<^sup>+x. (SUP i. g i x) \<partial>M)"
+    unfolding eq[THEN positive_integral_cong_AE] ..
+  also have "\<dots> = (SUP i. ?G i)"
+    using g by (rule positive_integral_monotone_convergence_simple[symmetric])
+  finally show ?thesis by simp
+qed
+
+lemma positive_integral_const[simp]:
+  "0 \<le> c \<Longrightarrow> (\<integral>\<^sup>+ x. c \<partial>M) = c * (emeasure M) (space M)"
+  by (subst positive_integral_eq_simple_integral) auto
+
+lemma positive_integral_linear:
+  assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
+  and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
+  shows "(\<integral>\<^sup>+ x. a * f x + g x \<partial>M) = a * integral\<^sup>P M f + integral\<^sup>P M g"
+    (is "integral\<^sup>P M ?L = _")
+proof -
+  from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
+  note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
+  from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
+  note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
+  let ?L' = "\<lambda>i x. a * u i x + v i x"
+
+  have "?L \<in> borel_measurable M" using assms by auto
+  from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
+  note l = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
+
+  have inc: "incseq (\<lambda>i. a * integral\<^sup>S M (u i))" "incseq (\<lambda>i. integral\<^sup>S M (v i))"
+    using u v `0 \<le> a`
+    by (auto simp: incseq_Suc_iff le_fun_def
+             intro!: add_mono ereal_mult_left_mono simple_integral_mono)
+  have pos: "\<And>i. 0 \<le> integral\<^sup>S M (u i)" "\<And>i. 0 \<le> integral\<^sup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^sup>S M (u i)"
+    using u v `0 \<le> a` by (auto simp: simple_integral_positive)
+  { fix i from pos[of i] have "a * integral\<^sup>S M (u i) \<noteq> -\<infinity>" "integral\<^sup>S M (v i) \<noteq> -\<infinity>"
+      by (auto split: split_if_asm) }
+  note not_MInf = this
+
+  have l': "(SUP i. integral\<^sup>S M (l i)) = (SUP i. integral\<^sup>S M (?L' i))"
+  proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
+    show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
+      using u v  `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
+      by (auto intro!: add_mono ereal_mult_left_mono)
+    { fix x
+      { fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using `0 \<le> a` u(6)[of i x] v(6)[of i x]
+          by auto }
+      then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
+        using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
+        by (subst SUP_ereal_cmult [symmetric, OF u(6) `0 \<le> a`])
+           (auto intro!: SUP_ereal_add
+                 simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono) }
+    then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)"
+      unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
+      by (intro AE_I2) (auto split: split_max)
+  qed
+  also have "\<dots> = (SUP i. a * integral\<^sup>S M (u i) + integral\<^sup>S M (v i))"
+    using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPREMUM UNIV"] ext)
+  finally have "(\<integral>\<^sup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^sup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+x. max 0 (g x) \<partial>M)"
+    unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
+    unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
+    apply (subst SUP_ereal_cmult [symmetric, OF pos(1) `0 \<le> a`])
+    apply (subst SUP_ereal_add [symmetric, OF inc not_MInf]) .
+  then show ?thesis by (simp add: positive_integral_max_0)
+qed
+
+lemma positive_integral_cmult:
+  assumes f: "f \<in> borel_measurable M" "0 \<le> c"
+  shows "(\<integral>\<^sup>+ x. c * f x \<partial>M) = c * integral\<^sup>P M f"
+proof -
+  have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
+    by (auto split: split_max simp: ereal_zero_le_0_iff)
+  have "(\<integral>\<^sup>+ x. c * f x \<partial>M) = (\<integral>\<^sup>+ x. c * max 0 (f x) \<partial>M)"
+    by (simp add: positive_integral_max_0)
+  then show ?thesis
+    using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f
+    by (auto simp: positive_integral_max_0)
+qed
+
+lemma positive_integral_multc:
+  assumes "f \<in> borel_measurable M" "0 \<le> c"
+  shows "(\<integral>\<^sup>+ x. f x * c \<partial>M) = integral\<^sup>P M f * c"
+  unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
+
+lemma positive_integral_indicator[simp]:
+  "A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. indicator A x\<partial>M) = (emeasure M) A"
+  by (subst positive_integral_eq_simple_integral)
+     (auto simp: simple_integral_indicator)
+
+lemma positive_integral_cmult_indicator:
+  "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. c * indicator A x \<partial>M) = c * (emeasure M) A"
+  by (subst positive_integral_eq_simple_integral)
+     (auto simp: simple_function_indicator simple_integral_indicator)
+
+lemma positive_integral_indicator':
+  assumes [measurable]: "A \<inter> space M \<in> sets M"
+  shows "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = emeasure M (A \<inter> space M)"
+proof -
+  have "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = (\<integral>\<^sup>+ x. indicator (A \<inter> space M) x \<partial>M)"
+    by (intro positive_integral_cong) (simp split: split_indicator)
+  also have "\<dots> = emeasure M (A \<inter> space M)"
+    by simp
+  finally show ?thesis .
+qed
+
+lemma positive_integral_add:
+  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+  and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
+  shows "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = integral\<^sup>P M f + integral\<^sup>P M g"
+proof -
+  have ae: "AE x in M. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
+    using assms by (auto split: split_max)
+  have "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = (\<integral>\<^sup>+ x. max 0 (f x + g x) \<partial>M)"
+    by (simp add: positive_integral_max_0)
+  also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
+    unfolding ae[THEN positive_integral_cong_AE] ..
+  also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+ x. max 0 (g x) \<partial>M)"
+    using positive_integral_linear[of "\<lambda>x. max 0 (f x)" _ 1 "\<lambda>x. max 0 (g x)"] f g
+    by auto
+  finally show ?thesis
+    by (simp add: positive_integral_max_0)
+qed
+
+lemma positive_integral_setsum:
+  assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x in M. 0 \<le> f i x"
+  shows "(\<integral>\<^sup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>P M (f i))"
+proof cases
+  assume f: "finite P"
+  from assms have "AE x in M. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
+  from f this assms(1) show ?thesis
+  proof induct
+    case (insert i P)
+    then have "f i \<in> borel_measurable M" "AE x in M. 0 \<le> f i x"
+      "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x in M. 0 \<le> (\<Sum>i\<in>P. f i x)"
+      by (auto intro!: setsum_nonneg)
+    from positive_integral_add[OF this]
+    show ?case using insert by auto
+  qed simp
+qed simp
+
+lemma positive_integral_Markov_inequality:
+  assumes u: "u \<in> borel_measurable M" "AE x in M. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c"
+  shows "(emeasure M) ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)"
+    (is "(emeasure M) ?A \<le> _ * ?PI")
+proof -
+  have "?A \<in> sets M"
+    using `A \<in> sets M` u by auto
+  hence "(emeasure M) ?A = (\<integral>\<^sup>+ x. indicator ?A x \<partial>M)"
+    using positive_integral_indicator by simp
+  also have "\<dots> \<le> (\<integral>\<^sup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
+    by (auto intro!: positive_integral_mono_AE
+      simp: indicator_def ereal_zero_le_0_iff)
+  also have "\<dots> = c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)"
+    using assms
+    by (auto intro!: positive_integral_cmult simp: ereal_zero_le_0_iff)
+  finally show ?thesis .
+qed
+
+lemma positive_integral_noteq_infinite:
+  assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
+  and "integral\<^sup>P M g \<noteq> \<infinity>"
+  shows "AE x in M. g x \<noteq> \<infinity>"
+proof (rule ccontr)
+  assume c: "\<not> (AE x in M. g x \<noteq> \<infinity>)"
+  have "(emeasure M) {x\<in>space M. g x = \<infinity>} \<noteq> 0"
+    using c g by (auto simp add: AE_iff_null)
+  moreover have "0 \<le> (emeasure M) {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
+  ultimately have "0 < (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
+  then have "\<infinity> = \<infinity> * (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
+  also have "\<dots> \<le> (\<integral>\<^sup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
+    using g by (subst positive_integral_cmult_indicator) auto
+  also have "\<dots> \<le> integral\<^sup>P M g"
+    using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
+  finally show False using `integral\<^sup>P M g \<noteq> \<infinity>` by auto
+qed
+
+lemma positive_integral_PInf:
+  assumes f: "f \<in> borel_measurable M"
+  and not_Inf: "integral\<^sup>P M f \<noteq> \<infinity>"
+  shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
+proof -
+  have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^sup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
+    using f by (subst positive_integral_cmult_indicator) (auto simp: measurable_sets)
+  also have "\<dots> \<le> integral\<^sup>P M (\<lambda>x. max 0 (f x))"
+    by (auto intro!: positive_integral_mono simp: indicator_def max_def)
+  finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^sup>P M f"
+    by (simp add: positive_integral_max_0)
+  moreover have "0 \<le> (emeasure M) (f -` {\<infinity>} \<inter> space M)"
+    by (rule emeasure_nonneg)
+  ultimately show ?thesis
+    using assms by (auto split: split_if_asm)
+qed
+
+lemma positive_integral_PInf_AE:
+  assumes "f \<in> borel_measurable M" "integral\<^sup>P M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>"
+proof (rule AE_I)
+  show "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
+    by (rule positive_integral_PInf[OF assms])
+  show "f -` {\<infinity>} \<inter> space M \<in> sets M"
+    using assms by (auto intro: borel_measurable_vimage)
+qed auto
+
+lemma simple_integral_PInf:
+  assumes "simple_function M f" "\<And>x. 0 \<le> f x"
+  and "integral\<^sup>S M f \<noteq> \<infinity>"
+  shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
+proof (rule positive_integral_PInf)
+  show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
+  show "integral\<^sup>P M f \<noteq> \<infinity>"
+    using assms by (simp add: positive_integral_eq_simple_integral)
+qed
+
+lemma positive_integral_diff:
+  assumes f: "f \<in> borel_measurable M"
+  and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
+  and fin: "integral\<^sup>P M g \<noteq> \<infinity>"
+  and mono: "AE x in M. g x \<le> f x"
+  shows "(\<integral>\<^sup>+ x. f x - g x \<partial>M) = integral\<^sup>P M f - integral\<^sup>P M g"
+proof -
+  have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x in M. 0 \<le> f x - g x"
+    using assms by (auto intro: ereal_diff_positive)
+  have pos_f: "AE x in M. 0 \<le> f x" using mono g by auto
+  { fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
+      by (cases rule: ereal2_cases[of a b]) auto }
+  note * = this
+  then have "AE x in M. f x = f x - g x + g x"
+    using mono positive_integral_noteq_infinite[OF g fin] assms by auto
+  then have **: "integral\<^sup>P M f = (\<integral>\<^sup>+x. f x - g x \<partial>M) + integral\<^sup>P M g"
+    unfolding positive_integral_add[OF diff g, symmetric]
+    by (rule positive_integral_cong_AE)
+  show ?thesis unfolding **
+    using fin positive_integral_positive[of M g]
+    by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. f x - g x \<partial>M" "integral\<^sup>P M g"]) auto
+qed
+
+lemma positive_integral_suminf:
+  assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x"
+  shows "(\<integral>\<^sup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^sup>P M (f i))"
+proof -
+  have all_pos: "AE x in M. \<forall>i. 0 \<le> f i x"
+    using assms by (auto simp: AE_all_countable)
+  have "(\<Sum>i. integral\<^sup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^sup>P M (f i))"
+    using positive_integral_positive by (rule suminf_ereal_eq_SUP)
+  also have "\<dots> = (SUP n. \<integral>\<^sup>+x. (\<Sum>i<n. f i x) \<partial>M)"
+    unfolding positive_integral_setsum[OF f] ..
+  also have "\<dots> = \<integral>\<^sup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
+    by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
+       (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
+  also have "\<dots> = \<integral>\<^sup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
+    by (intro positive_integral_cong_AE) (auto simp: suminf_ereal_eq_SUP)
+  finally show ?thesis by simp
+qed
+
+lemma positive_integral_mult_bounded_inf:
+  assumes f: "f \<in> borel_measurable M" "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>"
+    and c: "0 \<le> c" "c \<noteq> \<infinity>" and ae: "AE x in M. g x \<le> c * f x"
+  shows "(\<integral>\<^sup>+x. g x \<partial>M) < \<infinity>"
+proof -
+  have "(\<integral>\<^sup>+x. g x \<partial>M) \<le> (\<integral>\<^sup>+x. c * f x \<partial>M)"
+    by (intro positive_integral_mono_AE ae)
+  also have "(\<integral>\<^sup>+x. c * f x \<partial>M) < \<infinity>"
+    using c f by (subst positive_integral_cmult) auto
+  finally show ?thesis .
+qed
+
+text {* Fatou's lemma: convergence theorem on limes inferior *}
+
+lemma positive_integral_liminf:
+  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
+  assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> u i x"
+  shows "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>P M (u n))"
+proof -
+  have pos: "AE x in M. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
+  have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
+    (SUP n. \<integral>\<^sup>+ x. (INF i:{n..}. u i x) \<partial>M)"
+    unfolding liminf_SUP_INF using pos u
+    by (intro positive_integral_monotone_convergence_SUP_AE)
+       (elim AE_mp, auto intro!: AE_I2 intro: INF_greatest INF_superset_mono)
+  also have "\<dots> \<le> liminf (\<lambda>n. integral\<^sup>P M (u n))"
+    unfolding liminf_SUP_INF
+    by (auto intro!: SUP_mono exI INF_greatest positive_integral_mono INF_lower)
+  finally show ?thesis .
+qed
+
+lemma le_Limsup:
+  "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. c \<le> g x) F \<Longrightarrow> c \<le> Limsup F g"
+  using Limsup_mono[of "\<lambda>_. c" g F] by (simp add: Limsup_const)
+
+lemma Limsup_le:
+  "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. f x \<le> c) F \<Longrightarrow> Limsup F f \<le> c"
+  using Limsup_mono[of f "\<lambda>_. c" F] by (simp add: Limsup_const)
+
+lemma ereal_mono_minus_cancel:
+  fixes a b c :: ereal
+  shows "c - a \<le> c - b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c < \<infinity> \<Longrightarrow> b \<le> a"
+  by (cases a b c rule: ereal3_cases) auto
+
+lemma positive_integral_limsup:
+  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
+  assumes [measurable]: "\<And>i. u i \<in> borel_measurable M" "w \<in> borel_measurable M"
+  assumes bounds: "\<And>i. AE x in M. 0 \<le> u i x" "\<And>i. AE x in M. u i x \<le> w x" and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
+  shows "limsup (\<lambda>n. integral\<^sup>P M (u n)) \<le> (\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M)"
+proof -
+  have bnd: "AE x in M. \<forall>i. 0 \<le> u i x \<and> u i x \<le> w x"
+    using bounds by (auto simp: AE_all_countable)
+
+  from bounds[of 0] have w_nonneg: "AE x in M. 0 \<le> w x"
+    by auto
+
+  have "(\<integral>\<^sup>+x. w x \<partial>M) - (\<integral>\<^sup>+x. limsup (\<lambda>n. u n x) \<partial>M) = (\<integral>\<^sup>+x. w x - limsup (\<lambda>n. u n x) \<partial>M)"
+  proof (intro positive_integral_diff[symmetric])
+    show "AE x in M. 0 \<le> limsup (\<lambda>n. u n x)"
+      using bnd by (auto intro!: le_Limsup)
+    show "AE x in M. limsup (\<lambda>n. u n x) \<le> w x"
+      using bnd by (auto intro!: Limsup_le)
+    then have "(\<integral>\<^sup>+x. limsup (\<lambda>n. u n x) \<partial>M) < \<infinity>"
+      by (intro positive_integral_mult_bounded_inf[OF _ w, of 1]) auto
+    then show "(\<integral>\<^sup>+x. limsup (\<lambda>n. u n x) \<partial>M) \<noteq> \<infinity>"
+      by simp
+  qed auto
+  also have "\<dots> = (\<integral>\<^sup>+x. liminf (\<lambda>n. w x - u n x) \<partial>M)"
+    using w_nonneg
+    by (intro positive_integral_cong_AE, eventually_elim)
+       (auto intro!: liminf_ereal_cminus[symmetric])
+  also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^sup>+x. w x - u n x \<partial>M)"
+  proof (rule positive_integral_liminf)
+    fix i show "AE x in M. 0 \<le> w x - u i x"
+      using bounds[of i] by eventually_elim (auto intro: ereal_diff_positive)
+  qed simp
+  also have "(\<lambda>n. \<integral>\<^sup>+x. w x - u n x \<partial>M) = (\<lambda>n. (\<integral>\<^sup>+x. w x \<partial>M) - (\<integral>\<^sup>+x. u n x \<partial>M))"
+  proof (intro ext positive_integral_diff)
+    fix i have "(\<integral>\<^sup>+x. u i x \<partial>M) < \<infinity>"
+      using bounds by (intro positive_integral_mult_bounded_inf[OF _ w, of 1]) auto
+    then show "(\<integral>\<^sup>+x. u i x \<partial>M) \<noteq> \<infinity>" by simp
+  qed (insert bounds, auto)
+  also have "liminf (\<lambda>n. (\<integral>\<^sup>+x. w x \<partial>M) - (\<integral>\<^sup>+x. u n x \<partial>M)) = (\<integral>\<^sup>+x. w x \<partial>M) - limsup (\<lambda>n. \<integral>\<^sup>+x. u n x \<partial>M)"
+    using w by (intro liminf_ereal_cminus) auto
+  finally show ?thesis
+    by (rule ereal_mono_minus_cancel) (intro w positive_integral_positive)+
+qed
+
+lemma positive_integral_dominated_convergence:
+  assumes [measurable]:
+       "\<And>i. u i \<in> borel_measurable M" "u' \<in> borel_measurable M" "w \<in> borel_measurable M"
+    and bound: "\<And>j. AE x in M. 0 \<le> u j x" "\<And>j. AE x in M. u j x \<le> w x"
+    and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
+    and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
+  shows "(\<lambda>i. (\<integral>\<^sup>+x. u i x \<partial>M)) ----> (\<integral>\<^sup>+x. u' x \<partial>M)"
+proof -
+  have "limsup (\<lambda>n. integral\<^sup>P M (u n)) \<le> (\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M)"
+    by (intro positive_integral_limsup[OF _ _ bound w]) auto
+  moreover have "(\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M) = (\<integral>\<^sup>+ x. u' x \<partial>M)"
+    using u' by (intro positive_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot)
+  moreover have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) = (\<integral>\<^sup>+ x. u' x \<partial>M)"
+    using u' by (intro positive_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot)
+  moreover have "(\<integral>\<^sup>+x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>P M (u n))"
+    by (intro positive_integral_liminf[OF _ bound(1)]) auto
+  moreover have "liminf (\<lambda>n. integral\<^sup>P M (u n)) \<le> limsup (\<lambda>n. integral\<^sup>P M (u n))"
+    by (intro Liminf_le_Limsup sequentially_bot)
+  ultimately show ?thesis
+    by (intro Liminf_eq_Limsup) auto
+qed
+
+lemma positive_integral_null_set:
+  assumes "N \<in> null_sets M" shows "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = 0"
+proof -
+  have "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)"
+  proof (intro positive_integral_cong_AE AE_I)
+    show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
+      by (auto simp: indicator_def)
+    show "(emeasure M) N = 0" "N \<in> sets M"
+      using assms by auto
+  qed
+  then show ?thesis by simp
+qed
+
+lemma positive_integral_0_iff:
+  assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x"
+  shows "integral\<^sup>P M u = 0 \<longleftrightarrow> emeasure M {x\<in>space M. u x \<noteq> 0} = 0"
+    (is "_ \<longleftrightarrow> (emeasure M) ?A = 0")
+proof -
+  have u_eq: "(\<integral>\<^sup>+ x. u x * indicator ?A x \<partial>M) = integral\<^sup>P M u"
+    by (auto intro!: positive_integral_cong simp: indicator_def)
+  show ?thesis
+  proof
+    assume "(emeasure M) ?A = 0"
+    with positive_integral_null_set[of ?A M u] u
+    show "integral\<^sup>P M u = 0" by (simp add: u_eq null_sets_def)
+  next
+    { fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r"
+      then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def)
+      then have "0 \<le> r" by (auto simp add: ereal_zero_less_0_iff) }
+    note gt_1 = this
+    assume *: "integral\<^sup>P M u = 0"
+    let ?M = "\<lambda>n. {x \<in> space M. 1 \<le> real (n::nat) * u x}"
+    have "0 = (SUP n. (emeasure M) (?M n \<inter> ?A))"
+    proof -
+      { fix n :: nat
+        from positive_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"]
+        have "(emeasure M) (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
+        moreover have "0 \<le> (emeasure M) (?M n \<inter> ?A)" using u by auto
+        ultimately have "(emeasure M) (?M n \<inter> ?A) = 0" by auto }
+      thus ?thesis by simp
+    qed
+    also have "\<dots> = (emeasure M) (\<Union>n. ?M n \<inter> ?A)"
+    proof (safe intro!: SUP_emeasure_incseq)
+      fix n show "?M n \<inter> ?A \<in> sets M"
+        using u by (auto intro!: sets.Int)
+    next
+      show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
+      proof (safe intro!: incseq_SucI)
+        fix n :: nat and x
+        assume *: "1 \<le> real n * u x"
+        also from gt_1[OF *] have "real n * u x \<le> real (Suc n) * u x"
+          using `0 \<le> u x` by (auto intro!: ereal_mult_right_mono)
+        finally show "1 \<le> real (Suc n) * u x" by auto
+      qed
+    qed
+    also have "\<dots> = (emeasure M) {x\<in>space M. 0 < u x}"
+    proof (safe intro!: arg_cong[where f="(emeasure M)"] dest!: gt_1)
+      fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
+      show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
+      proof (cases "u x")
+        case (real r) with `0 < u x` have "0 < r" by auto
+        obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
+        hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
+        hence "1 \<le> real j * r" using real `0 < r` by auto
+        thus ?thesis using `0 < r` real by (auto simp: one_ereal_def)
+      qed (insert `0 < u x`, auto)
+    qed auto
+    finally have "(emeasure M) {x\<in>space M. 0 < u x} = 0" by simp
+    moreover
+    from pos have "AE x in M. \<not> (u x < 0)" by auto
+    then have "(emeasure M) {x\<in>space M. u x < 0} = 0"
+      using AE_iff_null[of M] u by auto
+    moreover have "(emeasure M) {x\<in>space M. u x \<noteq> 0} = (emeasure M) {x\<in>space M. u x < 0} + (emeasure M) {x\<in>space M. 0 < u x}"
+      using u by (subst plus_emeasure) (auto intro!: arg_cong[where f="emeasure M"])
+    ultimately show "(emeasure M) ?A = 0" by simp
+  qed
+qed
+
+lemma positive_integral_0_iff_AE:
+  assumes u: "u \<in> borel_measurable M"
+  shows "integral\<^sup>P M u = 0 \<longleftrightarrow> (AE x in M. u x \<le> 0)"
+proof -
+  have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
+    using u by auto
+  from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
+  have "integral\<^sup>P M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)"
+    unfolding positive_integral_max_0
+    using AE_iff_null[OF sets] u by auto
+  also have "\<dots> \<longleftrightarrow> (AE x in M. u x \<le> 0)" by (auto split: split_max)
+  finally show ?thesis .
+qed
+
+lemma AE_iff_positive_integral: 
+  "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> integral\<^sup>P M (indicator {x. \<not> P x}) = 0"
+  by (subst positive_integral_0_iff_AE) (auto simp: one_ereal_def zero_ereal_def
+    sets.sets_Collect_neg indicator_def[abs_def] measurable_If)
+
+lemma positive_integral_const_If:
+  "(\<integral>\<^sup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)"
+  by (auto intro!: positive_integral_0_iff_AE[THEN iffD2])
+
+lemma positive_integral_subalgebra:
+  assumes f: "f \<in> borel_measurable N" "\<And>x. 0 \<le> f x"
+  and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
+  shows "integral\<^sup>P N f = integral\<^sup>P M f"
+proof -
+  have [simp]: "\<And>f :: 'a \<Rightarrow> ereal. f \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable M"
+    using N by (auto simp: measurable_def)
+  have [simp]: "\<And>P. (AE x in N. P x) \<Longrightarrow> (AE x in M. P x)"
+    using N by (auto simp add: eventually_ae_filter null_sets_def)
+  have [simp]: "\<And>A. A \<in> sets N \<Longrightarrow> A \<in> sets M"
+    using N by auto
+  from f show ?thesis
+    apply induct
+    apply (simp_all add: positive_integral_add positive_integral_cmult positive_integral_monotone_convergence_SUP N)
+    apply (auto intro!: positive_integral_cong cong: positive_integral_cong simp: N(2)[symmetric])
+    done
+qed
+
+lemma positive_integral_nat_function:
+  fixes f :: "'a \<Rightarrow> nat"
+  assumes "f \<in> measurable M (count_space UNIV)"
+  shows "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<Sum>t. emeasure M {x\<in>space M. t < f x})"
+proof -
+  def F \<equiv> "\<lambda>i. {x\<in>space M. i < f x}"
+  with assms have [measurable]: "\<And>i. F i \<in> sets M"
+    by auto
+
+  { fix x assume "x \<in> space M"
+    have "(\<lambda>i. if i < f x then 1 else 0) sums (of_nat (f x)::real)"
+      using sums_If_finite[of "\<lambda>i. i < f x" "\<lambda>_. 1::real"] by simp
+    then have "(\<lambda>i. ereal(if i < f x then 1 else 0)) sums (ereal(of_nat(f x)))"
+      unfolding sums_ereal .
+    moreover have "\<And>i. ereal (if i < f x then 1 else 0) = indicator (F i) x"
+      using `x \<in> space M` by (simp add: one_ereal_def F_def)
+    ultimately have "ereal(of_nat(f x)) = (\<Sum>i. indicator (F i) x)"
+      by (simp add: sums_iff) }
+  then have "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. indicator (F i) x) \<partial>M)"
+    by (simp cong: positive_integral_cong)
+  also have "\<dots> = (\<Sum>i. emeasure M (F i))"
+    by (simp add: positive_integral_suminf)
+  finally show ?thesis
+    by (simp add: F_def)
+qed
+
+section {* Distributions *}
+
+lemma positive_integral_distr':
+  assumes T: "T \<in> measurable M M'"
+  and f: "f \<in> borel_measurable (distr M M' T)" "\<And>x. 0 \<le> f x"
+  shows "integral\<^sup>P (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
+  using f 
+proof induct
+  case (cong f g)
+  with T show ?case
+    apply (subst positive_integral_cong[of _ f g])
+    apply simp
+    apply (subst positive_integral_cong[of _ "\<lambda>x. f (T x)" "\<lambda>x. g (T x)"])
+    apply (simp add: measurable_def Pi_iff)
+    apply simp
+    done
+next
+  case (set A)
+  then have eq: "\<And>x. x \<in> space M \<Longrightarrow> indicator A (T x) = indicator (T -` A \<inter> space M) x"
+    by (auto simp: indicator_def)
+  from set T show ?case
+    by (subst positive_integral_cong[OF eq])
+       (auto simp add: emeasure_distr intro!: positive_integral_indicator[symmetric] measurable_sets)
+qed (simp_all add: measurable_compose[OF T] T positive_integral_cmult positive_integral_add
+                   positive_integral_monotone_convergence_SUP le_fun_def incseq_def)
+
+lemma positive_integral_distr:
+  "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^sup>P (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
+  by (subst (1 2) positive_integral_max_0[symmetric])
+     (simp add: positive_integral_distr')
+
+section {* Lebesgue integration on @{const count_space} *}
+
+lemma simple_function_count_space[simp]:
+  "simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)"
+  unfolding simple_function_def by simp
+
+lemma positive_integral_count_space:
+  assumes A: "finite {a\<in>A. 0 < f a}"
+  shows "integral\<^sup>P (count_space A) f = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
+proof -
+  have *: "(\<integral>\<^sup>+x. max 0 (f x) \<partial>count_space A) =
+    (\<integral>\<^sup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)"
+    by (auto intro!: positive_integral_cong
+             simp add: indicator_def if_distrib setsum_cases[OF A] max_def le_less)
+  also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. \<integral>\<^sup>+ x. f a * indicator {a} x \<partial>count_space A)"
+    by (subst positive_integral_setsum)
+       (simp_all add: AE_count_space ereal_zero_le_0_iff less_imp_le)
+  also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
+    by (auto intro!: setsum_cong simp: positive_integral_cmult_indicator one_ereal_def[symmetric])
+  finally show ?thesis by (simp add: positive_integral_max_0)
+qed
+
+lemma positive_integral_count_space_finite:
+    "finite A \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. max 0 (f a))"
+  by (subst positive_integral_max_0[symmetric])
+     (auto intro!: setsum_mono_zero_left simp: positive_integral_count_space less_le)
+
+lemma emeasure_UN_countable:
+  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M" and I: "countable I" 
+  assumes disj: "disjoint_family_on X I"
+  shows "emeasure M (UNION I X) = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)"
+proof cases
+  assume "finite I" with sets disj show ?thesis
+    by (subst setsum_emeasure[symmetric])
+       (auto intro!: setsum_cong simp add: max_def subset_eq positive_integral_count_space_finite emeasure_nonneg)
+next
+  assume f: "\<not> finite I"
+  then have [intro]: "I \<noteq> {}" by auto
+  from from_nat_into_inj_infinite[OF I f] from_nat_into[OF this] disj
+  have disj2: "disjoint_family (\<lambda>i. X (from_nat_into I i))"
+    unfolding disjoint_family_on_def by metis
+
+  from f have "bij_betw (from_nat_into I) UNIV I"
+    using bij_betw_from_nat_into[OF I] by simp
+  then have "(\<Union>i\<in>I. X i) = (\<Union>i. (X \<circ> from_nat_into I) i)"
+    unfolding SUP_def image_comp [symmetric] by (simp add: bij_betw_def)
+  then have "emeasure M (UNION I X) = emeasure M (\<Union>i. X (from_nat_into I i))"
+    by simp
+  also have "\<dots> = (\<Sum>i. emeasure M (X (from_nat_into I i)))"
+    by (intro suminf_emeasure[symmetric] disj disj2) (auto intro!: sets from_nat_into[OF `I \<noteq> {}`])
+  also have "\<dots> = (\<Sum>n. \<integral>\<^sup>+i. emeasure M (X i) * indicator {from_nat_into I n} i \<partial>count_space I)"
+  proof (intro arg_cong[where f=suminf] ext)
+    fix i
+    have eq: "{a \<in> I. 0 < emeasure M (X a) * indicator {from_nat_into I i} a}
+     = (if 0 < emeasure M (X (from_nat_into I i)) then {from_nat_into I i} else {})"
+     using ereal_0_less_1
+     by (auto simp: ereal_zero_less_0_iff indicator_def from_nat_into `I \<noteq> {}` simp del: ereal_0_less_1)
+    have "(\<integral>\<^sup>+ ia. emeasure M (X ia) * indicator {from_nat_into I i} ia \<partial>count_space I) =
+      (if 0 < emeasure M (X (from_nat_into I i)) then emeasure M (X (from_nat_into I i)) else 0)"
+      by (subst positive_integral_count_space) (simp_all add: eq)
+    also have "\<dots> = emeasure M (X (from_nat_into I i))"
+      by (simp add: less_le emeasure_nonneg)
+    finally show "emeasure M (X (from_nat_into I i)) =
+         \<integral>\<^sup>+ ia. emeasure M (X ia) * indicator {from_nat_into I i} ia \<partial>count_space I" ..
+  qed
+  also have "\<dots> = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)"
+    apply (subst positive_integral_suminf[symmetric])
+    apply (auto simp: emeasure_nonneg intro!: positive_integral_cong)
+  proof -
+    fix x assume "x \<in> I"
+    then have "(\<Sum>i. emeasure M (X x) * indicator {from_nat_into I i} x) = (\<Sum>i\<in>{to_nat_on I x}. emeasure M (X x) * indicator {from_nat_into I i} x)"
+      by (intro suminf_finite) (auto simp: indicator_def I f)
+    also have "\<dots> = emeasure M (X x)"
+      by (simp add: I f `x\<in>I`)
+    finally show "(\<Sum>i. emeasure M (X x) * indicator {from_nat_into I i} x) = emeasure M (X x)" .
+  qed
+  finally show ?thesis .
+qed
+
+section {* Measures with Restricted Space *}
+
+lemma positive_integral_restrict_space:
+  assumes \<Omega>: "\<Omega> \<in> sets M" and f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "\<And>x. x \<in> space M - \<Omega> \<Longrightarrow> f x = 0"
+  shows "positive_integral (restrict_space M \<Omega>) f = positive_integral M f"
+using f proof (induct rule: borel_measurable_induct)
+  case (cong f g) then show ?case
+    using positive_integral_cong[of M f g] positive_integral_cong[of "restrict_space M \<Omega>" f g]
+      sets.sets_into_space[OF `\<Omega> \<in> sets M`]
+    by (simp add: subset_eq space_restrict_space)
+next
+  case (set A)
+  then have "A \<subseteq> \<Omega>"
+    unfolding indicator_eq_0_iff by (auto dest: sets.sets_into_space)
+  with set `\<Omega> \<in> sets M` sets.sets_into_space[OF `\<Omega> \<in> sets M`] show ?case
+    by (subst positive_integral_indicator')
+       (auto simp add: sets_restrict_space_iff space_restrict_space
+                  emeasure_restrict_space Int_absorb2
+                dest: sets.sets_into_space)
+next
+  case (mult f c) then show ?case
+    by (cases "c = 0") (simp_all add: measurable_restrict_space1 \<Omega> positive_integral_cmult)
+next
+  case (add f g) then show ?case
+    by (simp add: measurable_restrict_space1 \<Omega> positive_integral_add ereal_add_nonneg_eq_0_iff)
+next
+  case (seq F) then show ?case
+    by (auto simp add: SUP_eq_iff measurable_restrict_space1 \<Omega> positive_integral_monotone_convergence_SUP)
+qed
+
+section {* Measure spaces with an associated density *}
+
+definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
+  "density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
+
+lemma 
+  shows sets_density[simp]: "sets (density M f) = sets M"
+    and space_density[simp]: "space (density M f) = space M"
+  by (auto simp: density_def)
+
+(* FIXME: add conversion to simplify space, sets and measurable *)
+lemma space_density_imp[measurable_dest]:
+  "\<And>x M f. x \<in> space (density M f) \<Longrightarrow> x \<in> space M" by auto
+
+lemma 
+  shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'"
+    and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'"
+    and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u"
+  unfolding measurable_def simple_function_def by simp_all
+
+lemma density_cong: "f \<in> borel_measurable M \<Longrightarrow> f' \<in> borel_measurable M \<Longrightarrow>
+  (AE x in M. f x = f' x) \<Longrightarrow> density M f = density M f'"
+  unfolding density_def by (auto intro!: measure_of_eq positive_integral_cong_AE sets.space_closed)
+
+lemma density_max_0: "density M f = density M (\<lambda>x. max 0 (f x))"
+proof -
+  have "\<And>x A. max 0 (f x) * indicator A x = max 0 (f x * indicator A x)"
+    by (auto simp: indicator_def)
+  then show ?thesis
+    unfolding density_def by (simp add: positive_integral_max_0)
+qed
+
+lemma density_ereal_max_0: "density M (\<lambda>x. ereal (f x)) = density M (\<lambda>x. ereal (max 0 (f x)))"
+  by (subst density_max_0) (auto intro!: arg_cong[where f="density M"] split: split_max)
+
+lemma emeasure_density:
+  assumes f[measurable]: "f \<in> borel_measurable M" and A[measurable]: "A \<in> sets M"
+  shows "emeasure (density M f) A = (\<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
+    (is "_ = ?\<mu> A")
+  unfolding density_def
+proof (rule emeasure_measure_of_sigma)
+  show "sigma_algebra (space M) (sets M)" ..
+  show "positive (sets M) ?\<mu>"
+    using f by (auto simp: positive_def intro!: positive_integral_positive)
+  have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^sup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'")
+    apply (subst positive_integral_max_0[symmetric])
+    apply (intro ext positive_integral_cong_AE AE_I2)
+    apply (auto simp: indicator_def)
+    done
+  show "countably_additive (sets M) ?\<mu>"
+    unfolding \<mu>_eq
+  proof (intro countably_additiveI)
+    fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M"
+    then have "\<And>i. A i \<in> sets M" by auto
+    then have *: "\<And>i. (\<lambda>x. max 0 (f x) * indicator (A i) x) \<in> borel_measurable M"
+      by (auto simp: set_eq_iff)
+    assume disj: "disjoint_family A"
+    have "(\<Sum>n. ?\<mu>' (A n)) = (\<integral>\<^sup>+ x. (\<Sum>n. max 0 (f x) * indicator (A n) x) \<partial>M)"
+      using f * by (simp add: positive_integral_suminf)
+    also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * (\<Sum>n. indicator (A n) x) \<partial>M)" using f
+      by (auto intro!: suminf_cmult_ereal positive_integral_cong_AE)
+    also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * indicator (\<Union>n. A n) x \<partial>M)"
+      unfolding suminf_indicator[OF disj] ..
+    finally show "(\<Sum>n. ?\<mu>' (A n)) = ?\<mu>' (\<Union>x. A x)" by simp
+  qed
+qed fact
+
+lemma null_sets_density_iff:
+  assumes f: "f \<in> borel_measurable M"
+  shows "A \<in> null_sets (density M f) \<longleftrightarrow> A \<in> sets M \<and> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
+proof -
+  { assume "A \<in> sets M"
+    have eq: "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. max 0 (f x) * indicator A x \<partial>M)"
+      apply (subst positive_integral_max_0[symmetric])
+      apply (intro positive_integral_cong)
+      apply (auto simp: indicator_def)
+      done
+    have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> 
+      emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0"
+      unfolding eq
+      using f `A \<in> sets M`
+      by (intro positive_integral_0_iff) auto
+    also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)"
+      using f `A \<in> sets M`
+      by (intro AE_iff_measurable[OF _ refl, symmetric]) auto
+    also have "(AE x in M. max 0 (f x) * indicator A x = 0) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
+      by (auto simp add: indicator_def max_def split: split_if_asm)
+    finally have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)" . }
+  with f show ?thesis
+    by (simp add: null_sets_def emeasure_density cong: conj_cong)
+qed
+
+lemma AE_density:
+  assumes f: "f \<in> borel_measurable M"
+  shows "(AE x in density M f. P x) \<longleftrightarrow> (AE x in M. 0 < f x \<longrightarrow> P x)"
+proof
+  assume "AE x in density M f. P x"
+  with f obtain N where "{x \<in> space M. \<not> P x} \<subseteq> N" "N \<in> sets M" and ae: "AE x in M. x \<in> N \<longrightarrow> f x \<le> 0"
+    by (auto simp: eventually_ae_filter null_sets_density_iff)
+  then have "AE x in M. x \<notin> N \<longrightarrow> P x" by auto
+  with ae show "AE x in M. 0 < f x \<longrightarrow> P x"
+    by (rule eventually_elim2) auto
+next
+  fix N assume ae: "AE x in M. 0 < f x \<longrightarrow> P x"
+  then obtain N where "{x \<in> space M. \<not> (0 < f x \<longrightarrow> P x)} \<subseteq> N" "N \<in> null_sets M"
+    by (auto simp: eventually_ae_filter)
+  then have *: "{x \<in> space (density M f). \<not> P x} \<subseteq> N \<union> {x\<in>space M. \<not> 0 < f x}"
+    "N \<union> {x\<in>space M. \<not> 0 < f x} \<in> sets M" and ae2: "AE x in M. x \<notin> N"
+    using f by (auto simp: subset_eq intro!: sets.sets_Collect_neg AE_not_in)
+  show "AE x in density M f. P x"
+    using ae2
+    unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f]
+    by (intro exI[of _ "N \<union> {x\<in>space M. \<not> 0 < f x}"] conjI *)
+       (auto elim: eventually_elim2)
+qed
+
+lemma positive_integral_density':
+  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+  assumes g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
+  shows "integral\<^sup>P (density M f) g = (\<integral>\<^sup>+ x. f x * g x \<partial>M)"
+using g proof induct
+  case (cong u v)
+  then show ?case
+    apply (subst positive_integral_cong[OF cong(3)])
+    apply (simp_all cong: positive_integral_cong)
+    done
+next
+  case (set A) then show ?case
+    by (simp add: emeasure_density f)
+next
+  case (mult u c)
+  moreover have "\<And>x. f x * (c * u x) = c * (f x * u x)" by (simp add: field_simps)
+  ultimately show ?case
+    using f by (simp add: positive_integral_cmult)
+next
+  case (add u v)
+  then have "\<And>x. f x * (v x + u x) = f x * v x + f x * u x"
+    by (simp add: ereal_right_distrib)
+  with add f show ?case
+    by (auto simp add: positive_integral_add ereal_zero_le_0_iff intro!: positive_integral_add[symmetric])
+next
+  case (seq U)
+  from f(2) have eq: "AE x in M. f x * (SUP i. U i x) = (SUP i. f x * U i x)"
+    by eventually_elim (simp add: SUP_ereal_cmult seq)
+  from seq f show ?case
+    apply (simp add: positive_integral_monotone_convergence_SUP)
+    apply (subst positive_integral_cong_AE[OF eq])
+    apply (subst positive_integral_monotone_convergence_SUP_AE)
+    apply (auto simp: incseq_def le_fun_def intro!: ereal_mult_left_mono)
+    done
+qed
+
+lemma positive_integral_density:
+  "f \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> g' \<in> borel_measurable M \<Longrightarrow> 
+    integral\<^sup>P (density M f) g' = (\<integral>\<^sup>+ x. f x * g' x \<partial>M)"
+  by (subst (1 2) positive_integral_max_0[symmetric])
+     (auto intro!: positive_integral_cong_AE
+           simp: measurable_If max_def ereal_zero_le_0_iff positive_integral_density')
+
+lemma emeasure_restricted:
+  assumes S: "S \<in> sets M" and X: "X \<in> sets M"
+  shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)"
+proof -
+  have "emeasure (density M (indicator S)) X = (\<integral>\<^sup>+x. indicator S x * indicator X x \<partial>M)"
+    using S X by (simp add: emeasure_density)
+  also have "\<dots> = (\<integral>\<^sup>+x. indicator (S \<inter> X) x \<partial>M)"
+    by (auto intro!: positive_integral_cong simp: indicator_def)
+  also have "\<dots> = emeasure M (S \<inter> X)"
+    using S X by (simp add: sets.Int)
+  finally show ?thesis .
+qed
+
+lemma measure_restricted:
+  "S \<in> sets M \<Longrightarrow> X \<in> sets M \<Longrightarrow> measure (density M (indicator S)) X = measure M (S \<inter> X)"
+  by (simp add: emeasure_restricted measure_def)
+
+lemma (in finite_measure) finite_measure_restricted:
+  "S \<in> sets M \<Longrightarrow> finite_measure (density M (indicator S))"
+  by default (simp add: emeasure_restricted)
+
+lemma emeasure_density_const:
+  "A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> emeasure (density M (\<lambda>_. c)) A = c * emeasure M A"
+  by (auto simp: positive_integral_cmult_indicator emeasure_density)
+
+lemma measure_density_const:
+  "A \<in> sets M \<Longrightarrow> 0 < c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real c * measure M A"
+  by (auto simp: emeasure_density_const measure_def)
+
+lemma density_density_eq:
+   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow>
+   density (density M f) g = density M (\<lambda>x. f x * g x)"
+  by (auto intro!: measure_eqI simp: emeasure_density positive_integral_density ac_simps)
+
+lemma distr_density_distr:
+  assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
+    and inv: "\<forall>x\<in>space M. T' (T x) = x"
+  assumes f: "f \<in> borel_measurable M'"
+  shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)" (is "?R = ?L")
+proof (rule measure_eqI)
+  fix A assume A: "A \<in> sets ?R"
+  { fix x assume "x \<in> space M"
+    with sets.sets_into_space[OF A]
+    have "indicator (T' -` A \<inter> space M') (T x) = (indicator A x :: ereal)"
+      using T inv by (auto simp: indicator_def measurable_space) }
+  with A T T' f show "emeasure ?R A = emeasure ?L A"
+    by (simp add: measurable_comp emeasure_density emeasure_distr
+                  positive_integral_distr measurable_sets cong: positive_integral_cong)
+qed simp
+
+lemma density_density_divide:
+  fixes f g :: "'a \<Rightarrow> real"
+  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+  assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
+  assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
+  shows "density (density M f) (\<lambda>x. g x / f x) = density M g"
+proof -
+  have "density M g = density M (\<lambda>x. f x * (g x / f x))"
+    using f g ac by (auto intro!: density_cong measurable_If)
+  then show ?thesis
+    using f g by (subst density_density_eq) auto
+qed
+
+section {* Point measure *}
+
+definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
+  "point_measure A f = density (count_space A) f"
+
+lemma
+  shows space_point_measure: "space (point_measure A f) = A"
+    and sets_point_measure: "sets (point_measure A f) = Pow A"
+  by (auto simp: point_measure_def)
+
+lemma measurable_point_measure_eq1[simp]:
+  "g \<in> measurable (point_measure A f) M \<longleftrightarrow> g \<in> A \<rightarrow> space M"
+  unfolding point_measure_def by simp
+
+lemma measurable_point_measure_eq2_finite[simp]:
+  "finite A \<Longrightarrow>
+   g \<in> measurable M (point_measure A f) \<longleftrightarrow>
+    (g \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. g -` {a} \<inter> space M \<in> sets M))"
+  unfolding point_measure_def by (simp add: measurable_count_space_eq2)
+
+lemma simple_function_point_measure[simp]:
+  "simple_function (point_measure A f) g \<longleftrightarrow> finite (g ` A)"
+  by (simp add: point_measure_def)
+
+lemma emeasure_point_measure:
+  assumes A: "finite {a\<in>X. 0 < f a}" "X \<subseteq> A"
+  shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)"
+proof -
+  have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}"
+    using `X \<subseteq> A` by auto
+  with A show ?thesis
+    by (simp add: emeasure_density positive_integral_count_space ereal_zero_le_0_iff
+                  point_measure_def indicator_def)
+qed
+
+lemma emeasure_point_measure_finite:
+  "finite A \<Longrightarrow> (\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
+  by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
+
+lemma emeasure_point_measure_finite2:
+  "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> (\<And>i. i \<in> X \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
+  by (subst emeasure_point_measure)
+     (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
+
+lemma null_sets_point_measure_iff:
+  "X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x \<le> 0)"
+ by (auto simp: AE_count_space null_sets_density_iff point_measure_def)
+
+lemma AE_point_measure:
+  "(AE x in point_measure A f. P x) \<longleftrightarrow> (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)"
+  unfolding point_measure_def
+  by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def)
+
+lemma positive_integral_point_measure:
+  "finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow>
+    integral\<^sup>P (point_measure A f) g = (\<Sum>a|a\<in>A \<and> 0 < f a \<and> 0 < g a. f a * g a)"
+  unfolding point_measure_def
+  apply (subst density_max_0)
+  apply (subst positive_integral_density)
+  apply (simp_all add: AE_count_space positive_integral_density)
+  apply (subst positive_integral_count_space )
+  apply (auto intro!: setsum_cong simp: max_def ereal_zero_less_0_iff)
+  apply (rule finite_subset)
+  prefer 2
+  apply assumption
+  apply auto
+  done
+
+lemma positive_integral_point_measure_finite:
+  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> g a) \<Longrightarrow>
+    integral\<^sup>P (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
+  by (subst positive_integral_point_measure) (auto intro!: setsum_mono_zero_left simp: less_le)
+
+section {* Uniform measure *}
+
+definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
+
+lemma
+  shows sets_uniform_measure[simp]: "sets (uniform_measure M A) = sets M"
+    and space_uniform_measure[simp]: "space (uniform_measure M A) = space M"
+  by (auto simp: uniform_measure_def)
+
+lemma emeasure_uniform_measure[simp]:
+  assumes A: "A \<in> sets M" and B: "B \<in> sets M"
+  shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A"
+proof -
+  from A B have "emeasure (uniform_measure M A) B = (\<integral>\<^sup>+x. (1 / emeasure M A) * indicator (A \<inter> B) x \<partial>M)"
+    by (auto simp add: uniform_measure_def emeasure_density split: split_indicator
+             intro!: positive_integral_cong)
+  also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A"
+    using A B
+    by (subst positive_integral_cmult_indicator) (simp_all add: sets.Int emeasure_nonneg)
+  finally show ?thesis .
+qed
+
+lemma measure_uniform_measure[simp]:
+  assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>" and B: "B \<in> sets M"
+  shows "measure (uniform_measure M A) B = measure M (A \<inter> B) / measure M A"
+  using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A
+  by (cases "emeasure M A" "emeasure M (A \<inter> B)" rule: ereal2_cases) (simp_all add: measure_def)
+
+section {* Uniform count measure *}
+
+definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)"
+ 
+lemma 
+  shows space_uniform_count_measure: "space (uniform_count_measure A) = A"
+    and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A"
+    unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure)
+ 
+lemma emeasure_uniform_count_measure:
+  "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A"
+  by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def)
+ 
+lemma measure_uniform_count_measure:
+  "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A"
+  by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def)
+
+end
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Probability_Measure.thy
--- a/src/HOL/Probability/Probability_Measure.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Probability_Measure.thy	Mon May 19 12:04:45 2014 +0200
@@ -112,6 +112,7 @@
 qed
 
 lemma (in prob_space) expectation_less:
+  fixes X :: "_ \<Rightarrow> real"
   assumes [simp]: "integrable M X"
   assumes gt: "AE x in M. X x < b"
   shows "expectation X < b"
@@ -123,6 +124,7 @@
 qed
 
 lemma (in prob_space) expectation_greater:
+  fixes X :: "_ \<Rightarrow> real"
   assumes [simp]: "integrable M X"
   assumes gt: "AE x in M. a < X x"
   shows "a < expectation X"
@@ -134,7 +136,7 @@
 qed
 
 lemma (in prob_space) jensens_inequality:
-  fixes a b :: real
+  fixes q :: "real \<Rightarrow> real"
   assumes X: "integrable M X" "AE x in M. X x \<in> I"
   assumes I: "I = {a <..< b} \<or> I = {a <..} \<or> I = {..< b} \<or> I = UNIV"
   assumes q: "integrable M (\<lambda>x. q (X x))" "convex_on I q"
@@ -173,8 +175,8 @@
       using prob_space by (simp add: X)
     also have "\<dots> \<le> expectation (\<lambda>w. q (X w))"
       using `x \<in> I` `open I` X(2)
-      apply (intro integral_mono_AE integral_add integral_cmult integral_diff
-                lebesgue_integral_const X q)
+      apply (intro integral_mono_AE integrable_add integrable_mult_right integrable_diff
+                integrable_const X q)
       apply (elim eventually_elim1)
       apply (intro convex_le_Inf_differential)
       apply (auto simp: interior_open q)
@@ -452,7 +454,7 @@
 lemma distributed_integral:
   "distributed M N X f \<Longrightarrow> g \<in> borel_measurable N \<Longrightarrow> (\<integral>x. f x * g x \<partial>N) = (\<integral>x. g (X x) \<partial>M)"
   by (auto simp: distributed_real_AE
-                 distributed_distr_eq_density[symmetric] integral_density[symmetric] integral_distr)
+                 distributed_distr_eq_density[symmetric] integral_real_density[symmetric] integral_distr)
   
 lemma distributed_transform_integral:
   assumes Px: "distributed M N X Px"
@@ -520,7 +522,7 @@
     also have "\<dots> = (\<integral>\<^sup>+x. (\<integral>\<^sup>+y. (f (x, y) * indicator B y) * indicator A x \<partial>T) \<partial>S)"
       using f by (auto simp add: eq positive_integral_multc intro!: positive_integral_cong)
     also have "\<dots> = emeasure ?R E"
-      by (auto simp add: emeasure_density T.positive_integral_fst_measurable(2)[symmetric]
+      by (auto simp add: emeasure_density T.positive_integral_fst[symmetric]
                intro!: positive_integral_cong split: split_indicator)
     finally show "emeasure ?L E = emeasure ?R E" .
   qed
@@ -588,7 +590,7 @@
   show X: "X \<in> measurable M S" by simp
 
   show borel: "Px \<in> borel_measurable S"
-    by (auto intro!: T.positive_integral_fst_measurable simp: Px_def)
+    by (auto intro!: T.positive_integral_fst simp: Px_def)
 
   interpret Pxy: prob_space "distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
     by (intro prob_space_distr) simp
@@ -606,7 +608,7 @@
       using Pxy by (simp add: distributed_def)
     also have "\<dots> = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. Pxy (x, y) * indicator (A \<times> space T) (x, y) \<partial>T \<partial>S"
       using A borel Pxy
-      by (simp add: emeasure_density T.positive_integral_fst_measurable(2)[symmetric])
+      by (simp add: emeasure_density T.positive_integral_fst[symmetric])
     also have "\<dots> = \<integral>\<^sup>+ x. Px x * indicator A x \<partial>S"
       apply (rule positive_integral_cong_AE)
       using Pxy[THEN distributed_AE, THEN ST.AE_pair] AE_space
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Radon_Nikodym.thy
--- a/src/HOL/Probability/Radon_Nikodym.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Radon_Nikodym.thy	Mon May 19 12:04:45 2014 +0200
@@ -5,7 +5,7 @@
 header {*Radon-Nikod{\'y}m derivative*}
 
 theory Radon_Nikodym
-imports Lebesgue_Integration
+imports Bochner_Integration
 begin
 
 definition "diff_measure M N =
@@ -1062,35 +1062,53 @@
 
 section "Radon-Nikodym derivative"
 
-definition
-  "RN_deriv M N \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N"
+definition RN_deriv :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a \<Rightarrow> ereal" where
+  "RN_deriv M N =
+    (if \<exists>f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N
+       then SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N
+       else (\<lambda>_. 0))"
 
-lemma
-  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
-  shows borel_measurable_RN_deriv_density: "RN_deriv M (density M f) \<in> borel_measurable M" (is ?borel)
-    and density_RN_deriv_density: "density M (RN_deriv M (density M f)) = density M f" (is ?density)
-    and RN_deriv_density_nonneg: "0 \<le> RN_deriv M (density M f) x" (is ?pos)
+lemma RN_derivI: 
+  assumes "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "density M f = N"
+  shows "density M (RN_deriv M N) = N"
 proof -
-  let ?f = "\<lambda>x. max 0 (f x)"
-  let ?P = "\<lambda>g. g \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> g x) \<and> density M g = density M f"
-  from f have "?P ?f" using f by (auto intro!: density_cong simp: split: split_max)
-  then have "?P (RN_deriv M (density M f))"
-    unfolding RN_deriv_def by (rule someI[where P="?P"])
-  then show ?borel ?density ?pos by auto
+  have "\<exists>f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N"
+    using assms by auto
+  moreover then have "density M (SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N) = N"
+    by (rule someI2_ex) auto
+  ultimately show ?thesis
+    by (auto simp: RN_deriv_def)
 qed
 
-lemma (in sigma_finite_measure) RN_deriv:
-  assumes "absolutely_continuous M N" "sets N = sets M"
-  shows borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M" (is ?borel)
-    and density_RN_deriv: "density M (RN_deriv M N) = N" (is ?density)
-    and RN_deriv_nonneg: "0 \<le> RN_deriv M N x" (is ?pos)
+lemma 
+  shows borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M" (is ?m)
+    and RN_deriv_nonneg: "0 \<le> RN_deriv M N x" (is ?nn)
 proof -
-  note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
-  from Ex show ?borel unfolding RN_deriv_def by (rule someI2_ex) simp
-  from Ex show ?density unfolding RN_deriv_def by (rule someI2_ex) simp
-  from Ex show ?pos unfolding RN_deriv_def by (rule someI2_ex) simp
+  { assume ex: "\<exists>f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N"
+    have 1: "(SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N) \<in> borel_measurable M"
+      using ex by (rule someI2_ex) auto
+    moreover
+    have 2: "0 \<le> (SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N) x"
+      using ex by (rule someI2_ex) auto
+    note 1 2 }
+  from this show ?m ?nn
+    by (auto simp: RN_deriv_def)
 qed
 
+lemma density_RN_deriv_density:
+  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+  shows "density M (RN_deriv M (density M f)) = density M f"
+proof (rule RN_derivI)
+  show "(\<lambda>x. max 0 (f x)) \<in> borel_measurable M" "\<And>x. 0 \<le> max 0 (f x)"
+    using f by auto
+  show "density M (\<lambda>x. max 0 (f x)) = density M f"
+    using f by (intro density_cong) (auto simp: max_def)
+qed
+
+lemma (in sigma_finite_measure) density_RN_deriv:
+  "absolutely_continuous M N \<Longrightarrow> sets N = sets M \<Longrightarrow> density M (RN_deriv M N) = N"
+  by (metis RN_derivI Radon_Nikodym)
+
 lemma (in sigma_finite_measure) RN_deriv_positive_integral:
   assumes N: "absolutely_continuous M N" "sets N = sets M"
     and f: "f \<in> borel_measurable M"
@@ -1099,7 +1117,7 @@
   have "integral\<^sup>P N f = integral\<^sup>P (density M (RN_deriv M N)) f"
     using N by (simp add: density_RN_deriv)
   also have "\<dots> = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)"
-    using RN_deriv(1,3)[OF N] f by (simp add: positive_integral_density)
+    using f by (simp add: positive_integral_density RN_deriv_nonneg)
   finally show ?thesis by simp
 qed
 
@@ -1111,16 +1129,16 @@
   and eq: "density M f = N"
   shows "AE x in M. f x = RN_deriv M N x"
   unfolding eq[symmetric]
-  by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv_density
-            RN_deriv_density_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric])
+  by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv
+            RN_deriv_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric])
 
 lemma RN_deriv_unique_sigma_finite:
   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
   and eq: "density M f = N" and fin: "sigma_finite_measure N"
   shows "AE x in M. f x = RN_deriv M N x"
   using fin unfolding eq[symmetric]
-  by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv_density
-            RN_deriv_density_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric])
+  by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv
+            RN_deriv_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric])
 
 lemma (in sigma_finite_measure) RN_deriv_distr:
   fixes T :: "'a \<Rightarrow> 'b"
@@ -1165,7 +1183,7 @@
     using T ac by measurable
   then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M"
     by (simp add: comp_def)
-  show "AE x in M. 0 \<le> RN_deriv ?M' ?N' (T x)" using M'.RN_deriv_nonneg[OF ac] by auto
+  show "AE x in M. 0 \<le> RN_deriv ?M' ?N' (T x)" by (auto intro: RN_deriv_nonneg)
 
   have "N = distr N M (T' \<circ> T)"
     by (subst measure_of_of_measure[of N, symmetric])
@@ -1175,7 +1193,7 @@
   also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'"
     using ac by (simp add: M'.density_RN_deriv)
   also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)"
-    using M'.borel_measurable_RN_deriv[OF ac] by (simp add: distr_density_distr[OF T T', OF inv])
+    by (simp add: distr_density_distr[OF T T', OF inv])
   finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N"
     by (simp add: comp_def)
 qed
@@ -1186,7 +1204,9 @@
 proof -
   interpret N: sigma_finite_measure N by fact
   from N show ?thesis
-    using sigma_finite_iff_density_finite[OF RN_deriv(1)[OF ac]] RN_deriv(2,3)[OF ac] by simp
+    using sigma_finite_iff_density_finite[OF borel_measurable_RN_deriv, of N]
+      density_RN_deriv[OF ac]
+    by (simp add: RN_deriv_nonneg)
 qed
 
 lemma (in sigma_finite_measure)
@@ -1194,29 +1214,31 @@
     and f: "f \<in> borel_measurable M"
   shows RN_deriv_integrable: "integrable N f \<longleftrightarrow>
       integrable M (\<lambda>x. real (RN_deriv M N x) * f x)" (is ?integrable)
-    and RN_deriv_integral: "integral\<^sup>L N f =
-      (\<integral>x. real (RN_deriv M N x) * f x \<partial>M)" (is ?integral)
+    and RN_deriv_integral: "integral\<^sup>L N f = (\<integral>x. real (RN_deriv M N x) * f x \<partial>M)" (is ?integral)
 proof -
   note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp]
   interpret N: sigma_finite_measure N by fact
-  have minus_cong: "\<And>A B A' B'::ereal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A - real B = real A' - real B'" by simp
-  have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
-  have Nf: "f \<in> borel_measurable N" using f by (simp add: measurable_def)
-  { fix f :: "'a \<Rightarrow> real"
-    { fix x assume *: "RN_deriv M N x \<noteq> \<infinity>"
-      have "ereal (real (RN_deriv M N x)) * ereal (f x) = ereal (real (RN_deriv M N x) * f x)"
-        by (simp add: mult_le_0_iff)
-      then have "RN_deriv M N x * ereal (f x) = ereal (real (RN_deriv M N x) * f x)"
-        using RN_deriv(3)[OF ac] * by (auto simp add: ereal_real split: split_if_asm) }
-    then have "(\<integral>\<^sup>+x. ereal (real (RN_deriv M N x) * f x) \<partial>M) = (\<integral>\<^sup>+x. RN_deriv M N x * ereal (f x) \<partial>M)"
-              "(\<integral>\<^sup>+x. ereal (- (real (RN_deriv M N x) * f x)) \<partial>M) = (\<integral>\<^sup>+x. RN_deriv M N x * ereal (- f x) \<partial>M)"
-      using RN_deriv_finite[OF N ac] unfolding ereal_mult_minus_right uminus_ereal.simps(1)[symmetric]
-      by (auto intro!: positive_integral_cong_AE) }
-  note * = this
-  show ?integral ?integrable
-    unfolding lebesgue_integral_def integrable_def *
-    using Nf f RN_deriv(1)[OF ac]
-    by (auto simp: RN_deriv_positive_integral[OF ac])
+
+  have eq: "density M (RN_deriv M N) = density M (\<lambda>x. real (RN_deriv M N x))"
+  proof (rule density_cong)
+    from RN_deriv_finite[OF assms(1,2,3)]
+    show "AE x in M. RN_deriv M N x = ereal (real (RN_deriv M N x))"
+      by eventually_elim (insert RN_deriv_nonneg[of M N], auto simp: ereal_real)
+  qed (insert ac, auto)
+
+  show ?integrable
+    apply (subst density_RN_deriv[OF ac, symmetric])
+    unfolding eq
+    apply (intro integrable_real_density f AE_I2 real_of_ereal_pos RN_deriv_nonneg)
+    apply (insert ac, auto)
+    done
+
+  show ?integral
+    apply (subst density_RN_deriv[OF ac, symmetric])
+    unfolding eq
+    apply (intro integral_real_density f AE_I2 real_of_ereal_pos RN_deriv_nonneg)
+    apply (insert ac, auto)
+    done
 qed
 
 lemma (in sigma_finite_measure) real_RN_deriv:
@@ -1229,7 +1251,7 @@
 proof
   interpret N: finite_measure N by fact
   
-  note RN = RN_deriv[OF ac]
+  note RN = borel_measurable_RN_deriv density_RN_deriv[OF ac] RN_deriv_nonneg[of M N]
 
   let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}"
 
@@ -1277,11 +1299,10 @@
   and x: "{x} \<in> sets M"
   shows "N {x} = RN_deriv M N x * emeasure M {x}"
 proof -
-  note deriv = RN_deriv[OF ac]
-  from deriv(1,3) `{x} \<in> sets M`
+  from `{x} \<in> sets M`
   have "density M (RN_deriv M N) {x} = (\<integral>\<^sup>+w. RN_deriv M N x * indicator {x} w \<partial>M)"
     by (auto simp: indicator_def emeasure_density intro!: positive_integral_cong)
-  with x deriv show ?thesis
+  with x density_RN_deriv[OF ac] RN_deriv_nonneg[of M N] show ?thesis
     by (auto simp: positive_integral_cmult_indicator)
 qed
 
diff -r d0e5225601d3 -r e5366291d6aa src/HOL/Probability/Sigma_Algebra.thy
--- a/src/HOL/Probability/Sigma_Algebra.thy	Mon May 19 11:27:02 2014 +0200
+++ b/src/HOL/Probability/Sigma_Algebra.thy	Mon May 19 12:04:45 2014 +0200
@@ -1391,6 +1391,11 @@
   unfolding measurable_def using assms
   by (simp cong: vimage_inter_cong Pi_cong)
 
+lemma measurable_cong_strong:
+  "M = N \<Longrightarrow> M' = N' \<Longrightarrow> (\<And>w. w \<in> space M \<Longrightarrow> f w = g w) \<Longrightarrow>
+    f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable N N'"
+  by (metis measurable_cong)
+
 lemma measurable_eqI:
      "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
         sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
@@ -1540,6 +1545,21 @@
     unfolding measurable_def by auto
 qed
 
+lemma measurable_count_space_eq2_countable:
+  fixes f :: "'a => 'c::countable"
+  shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
+proof -
+  { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
+    assume *: "\<And>a. a\<in>A \<Longrightarrow> f -` {a} \<inter> space M \<in> sets M"
+    have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)"
+      by auto
+    also have "\<dots> \<in> sets M"
+      using * `X \<subseteq> A` by (intro sets.countable_UN) auto
+    finally have "f -` X \<inter> space M \<in> sets M" . }
+  then show ?thesis
+    unfolding measurable_def by auto
+qed
+
 lemma measurable_compose_countable:
   assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"
   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"