src/HOL/Probability/Lebesgue_Measure.thy

-
+(*  Author: Robert Himmelmann, TU Muenchen *)
 header {* Lebsegue measure *}
-(*  Author:                     Robert Himmelmann, TU Muenchen *)
-
 theory Lebesgue_Measure
-  imports Gauge_Measure Measure Lebesgue_Integration
+  imports Product_Measure Gauge_Measure Complete_Measure
 begin
 
-subsection {* Various *}
-
-lemma seq_offset_iff:"f ----> l \<longleftrightarrow> (\<lambda>i. f (i + k)) ----> l"
-  using seq_offset_rev seq_offset[of f l k] by auto
-
-lemma has_integral_disjoint_union: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
-  assumes "(f has_integral i) s" "(f has_integral j) t" "s \<inter> t = {}"
-  shows "(f has_integral (i + j)) (s \<union> t)"
-  apply(rule has_integral_union[OF assms(1-2)]) unfolding assms by auto
-
-lemma lim_eq: assumes "\<forall>n>N. f n = g n" shows "(f ----> l) \<longleftrightarrow> (g ----> l)" using assms 
-proof(induct N arbitrary: f g) case 0
-  hence *:"\<And>n. f (Suc n) = g (Suc n)" by auto
-  show ?case apply(subst LIMSEQ_Suc_iff[THEN sym]) apply(subst(2) LIMSEQ_Suc_iff[THEN sym])
-    unfolding * ..
-next case (Suc n)
-  show ?case apply(subst LIMSEQ_Suc_iff[THEN sym]) apply(subst(2) LIMSEQ_Suc_iff[THEN sym])
-    apply(rule Suc(1)) using Suc(2) by auto
+lemma (in complete_lattice) SUP_pair:
+  "(SUP i:A. SUP j:B. f i j) = (SUP p:A\<times>B. (\<lambda> (i, j). f i j) p)" (is "?l = ?r")
+proof (intro antisym SUP_leI)
+  fix i j assume "i \<in> A" "j \<in> B"
+  then have "(case (i,j) of (i,j) \<Rightarrow> f i j) \<le> ?r"
+    by (intro SUPR_upper) auto
+  then show "f i j \<le> ?r" by auto
+next
+  fix p assume "p \<in> A \<times> B"
+  then obtain i j where "p = (i,j)" "i \<in> A" "j \<in> B" by auto
+  have "f i j \<le> (SUP j:B. f i j)" using `j \<in> B` by (intro SUPR_upper)
+  also have "(SUP j:B. f i j) \<le> ?l" using `i \<in> A` by (intro SUPR_upper)
+  finally show "(case p of (i, j) \<Rightarrow> f i j) \<le> ?l" using `p = (i,j)` by simp
+qed
+
+lemma (in complete_lattice) SUP_surj_compose:
+  assumes *: "f`A = B" shows "SUPR A (g \<circ> f) = SUPR B g"
+  unfolding SUPR_def unfolding *[symmetric]
+  by (simp add: image_compose)
+
+lemma (in complete_lattice) SUP_swap:
+  "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"
+proof -
+  have *: "(\<lambda>(i,j). (j,i)) ` (B \<times> A) = A \<times> B" by auto
+  show ?thesis
+    unfolding SUP_pair SUP_surj_compose[symmetric, OF *]
+    by (auto intro!: arg_cong[where f=Sup] image_eqI simp: comp_def SUPR_def)
+qed
+
+lemma SUP_\<omega>: "(SUP i:A. f i) = \<omega> \<longleftrightarrow> (\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)"
+proof
+  assume *: "(SUP i:A. f i) = \<omega>"
+  show "(\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)" unfolding *[symmetric]
+  proof (intro allI impI)
+    fix x assume "x < SUPR A f" then show "\<exists>i\<in>A. x < f i"
+      unfolding less_SUP_iff by auto
+  qed
+next
+  assume *: "\<forall>x<\<omega>. \<exists>i\<in>A. x < f i"
+  show "(SUP i:A. f i) = \<omega>"
+  proof (rule pinfreal_SUPI)
+    fix y assume **: "\<And>i. i \<in> A \<Longrightarrow> f i \<le> y"
+    show "\<omega> \<le> y"
+    proof cases
+      assume "y < \<omega>"
+      from *[THEN spec, THEN mp, OF this]
+      obtain i where "i \<in> A" "\<not> (f i \<le> y)" by auto
+      with ** show ?thesis by auto
+    qed auto
+  qed auto
+qed
+
+lemma psuminf_commute:
+  shows "(\<Sum>\<^isub>\<infinity> i j. f i j) = (\<Sum>\<^isub>\<infinity> j i. f i j)"
+proof -
+  have "(SUP n. \<Sum> i < n. SUP m. \<Sum> j < m. f i j) = (SUP n. SUP m. \<Sum> i < n. \<Sum> j < m. f i j)"
+    apply (subst SUPR_pinfreal_setsum)
+    by auto
+  also have "\<dots> = (SUP m n. \<Sum> j < m. \<Sum> i < n. f i j)"
+    apply (subst SUP_swap)
+    apply (subst setsum_commute)
+    by auto
+  also have "\<dots> = (SUP m. \<Sum> j < m. SUP n. \<Sum> i < n. f i j)"
+    apply (subst SUPR_pinfreal_setsum)
+    by auto
+  finally show ?thesis
+    unfolding psuminf_def by auto
+qed
+
+lemma psuminf_SUP_eq:
+  assumes "\<And>n i. f n i \<le> f (Suc n) i"
+  shows "(\<Sum>\<^isub>\<infinity> i. SUP n::nat. f n i) = (SUP n::nat. \<Sum>\<^isub>\<infinity> i. f n i)"
+proof -
+  { fix n :: nat
+    have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)"
+      using assms by (auto intro!: SUPR_pinfreal_setsum[symmetric]) }
+  note * = this
+  show ?thesis
+    unfolding psuminf_def
+    unfolding *
+    apply (subst SUP_swap) ..
 qed
 
 subsection {* Standard Cubes *}
 
-definition cube where
-  "cube (n::nat) \<equiv> {\<chi>\<chi> i. - real n .. (\<chi>\<chi> i. real n)::_::ordered_euclidean_space}"
-
-lemma cube_subset[intro]:"n\<le>N \<Longrightarrow> cube n \<subseteq> (cube N::'a::ordered_euclidean_space set)"
-  apply(auto simp: eucl_le[where 'a='a] cube_def) apply(erule_tac[!] x=i in allE)+ by auto
+definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where
+  "cube n \<equiv> {\<chi>\<chi> i. - real n .. \<chi>\<chi> i. real n}"
+
+lemma cube_closed[intro]: "closed (cube n)"
+  unfolding cube_def by auto
+
+lemma cube_subset[intro]: "n \<le> N \<Longrightarrow> cube n \<subseteq> cube N"
+  by (fastsimp simp: eucl_le[where 'a='a] cube_def)
+
+lemma cube_subset_iff:
+  "cube n \<subseteq> cube N \<longleftrightarrow> n \<le> N"
+proof
+  assume subset: "cube n \<subseteq> (cube N::'a set)"
+  then have "((\<chi>\<chi> i. real n)::'a) \<in> cube N"
+    using DIM_positive[where 'a='a]
+    by (fastsimp simp: cube_def eucl_le[where 'a='a])
+  then show "n \<le> N"
+    by (fastsimp simp: cube_def eucl_le[where 'a='a])
+next
+  assume "n \<le> N" then show "cube n \<subseteq> (cube N::'a set)" by (rule cube_subset)
+qed
 
 lemma ball_subset_cube:"ball (0::'a::ordered_euclidean_space) (real n) \<subseteq> cube n"
   unfolding cube_def subset_eq mem_interval apply safe unfolding euclidean_lambda_beta'
 proof- fix x::'a and i assume as:"x\<in>ball 0 (real n)" "i<DIM('a)"
   thus "- real n \<le> x $$ i" "real n \<ge> x $$ i"

 lemma gmeasure_le_inter_cube[intro]: fixes s::"'a::ordered_euclidean_space set"
   assumes "gmeasurable (s \<inter> cube n)" shows "gmeasure (s \<inter> cube n) \<le> gmeasure (cube n::'a set)"
   apply(rule has_gmeasure_subset[of "s\<inter>cube n" _ "cube n"])
   unfolding has_gmeasure_measure[THEN sym] using assms by auto
 
+lemma has_gmeasure_cube[intro]: "(cube n::('a::ordered_euclidean_space) set)
+  has_gmeasure ((2 * real n) ^ (DIM('a)))"
+proof-
+  have "content {\<chi>\<chi> i. - real n..(\<chi>\<chi> i. real n)::'a} = (2 * real n) ^ (DIM('a))"
+    apply(subst content_closed_interval) defer
+    by (auto simp add:setprod_constant)
+  thus ?thesis unfolding cube_def
+    using has_gmeasure_interval(1)[of "(\<chi>\<chi> i. - real n)::'a" "(\<chi>\<chi> i. real n)::'a"]
+    by auto
+qed
+
+lemma gmeasure_cube_eq[simp]:
+  "gmeasure (cube n::('a::ordered_euclidean_space) set) = (2 * real n) ^ DIM('a)"
+  by (intro measure_unique) auto
+
+lemma gmeasure_cube_ge_n: "gmeasure (cube n::('a::ordered_euclidean_space) set) \<ge> real n"
+proof cases
+  assume "n = 0" then show ?thesis by simp
+next
+  assume "n \<noteq> 0"
+  have "real n \<le> (2 * real n)^1" by simp
+  also have "\<dots> \<le> (2 * real n)^DIM('a)"
+    using DIM_positive[where 'a='a] `n \<noteq> 0`
+    by (intro power_increasing) auto
+  also have "\<dots> = gmeasure (cube n::'a set)" by simp
+  finally show ?thesis .
+qed
+
+lemma gmeasure_setsum:
+  assumes "finite A" and "\<And>s t. s \<in> A \<Longrightarrow> t \<in> A \<Longrightarrow> s \<noteq> t \<Longrightarrow> f s \<inter> f t = {}"
+    and "\<And>i. i \<in> A \<Longrightarrow> gmeasurable (f i)"
+  shows "gmeasure (\<Union>i\<in>A. f i) = (\<Sum>i\<in>A. gmeasure (f i))"
+proof -
+  have "gmeasure (\<Union>i\<in>A. f i) = gmeasure (\<Union>f ` A)" by auto
+  also have "\<dots> = setsum gmeasure (f ` A)" using assms
+  proof (intro measure_negligible_unions)
+    fix X Y assume "X \<in> f`A" "Y \<in> f`A" "X \<noteq> Y"
+    then have "X \<inter> Y = {}" using assms by auto
+    then show "negligible (X \<inter> Y)" by auto
+  qed auto
+  also have "\<dots> = setsum gmeasure (f ` A - {{}})"
+    using assms by (intro setsum_mono_zero_cong_right) auto
+  also have "\<dots> = (\<Sum>i\<in>A - {i. f i = {}}. gmeasure (f i))"
+  proof (intro setsum_reindex_cong inj_onI)
+    fix s t assume *: "s \<in> A - {i. f i = {}}" "t \<in> A - {i. f i = {}}" "f s = f t"
+    show "s = t"
+    proof (rule ccontr)
+      assume "s \<noteq> t" with assms(2)[of s t] * show False by auto
+    qed
+  qed auto
+  also have "\<dots> = (\<Sum>i\<in>A. gmeasure (f i))"
+    using assms by (intro setsum_mono_zero_cong_left) auto
+  finally show ?thesis .
+qed
+
+lemma gmeasurable_finite_UNION[intro]:
+  assumes "\<And>i. i \<in> S \<Longrightarrow> gmeasurable (A i)" "finite S"
+  shows "gmeasurable (\<Union>i\<in>S. A i)"
+  unfolding UNION_eq_Union_image using assms
+  by (intro gmeasurable_finite_unions) auto
+
+lemma gmeasurable_countable_UNION[intro]:
+  fixes A :: "nat \<Rightarrow> ('a::ordered_euclidean_space) set"
+  assumes measurable: "\<And>i. gmeasurable (A i)"
+    and finite: "\<And>n. gmeasure (UNION {.. n} A) \<le> B"
+  shows "gmeasurable (\<Union>i. A i)"
+proof -
+  have *: "\<And>n. \<Union>{A k |k. k \<le> n} = (\<Union>i\<le>n. A i)"
+    "(\<Union>{A n |n. n \<in> UNIV}) = (\<Union>i. A i)" by auto
+  show ?thesis
+    by (rule gmeasurable_countable_unions_strong[of A B, unfolded *, OF assms])
+qed
 
 subsection {* Measurability *}
 
-definition lmeasurable :: "('a::ordered_euclidean_space) set => bool" where
-  "lmeasurable s \<equiv> (\<forall>n::nat. gmeasurable (s \<inter> cube n))"
-
-lemma lmeasurableD[dest]:assumes "lmeasurable s"
-  shows "\<And>n. gmeasurable (s \<inter> cube n)"
-  using assms unfolding lmeasurable_def by auto
-
-lemma measurable_imp_lmeasurable: assumes"gmeasurable s"
-  shows "lmeasurable s" unfolding lmeasurable_def cube_def 
+definition lebesgue :: "'a::ordered_euclidean_space algebra" where
+  "lebesgue = \<lparr> space = UNIV, sets = {A. \<forall>n. gmeasurable (A \<inter> cube n)} \<rparr>"
+
+lemma space_lebesgue[simp]:"space lebesgue = UNIV"
+  unfolding lebesgue_def by auto
+
+lemma lebesgueD[dest]: assumes "S \<in> sets lebesgue"
+  shows "\<And>n. gmeasurable (S \<inter> cube n)"
+  using assms unfolding lebesgue_def by auto
+
+lemma lebesgueI[intro]: assumes "gmeasurable S"
+  shows "S \<in> sets lebesgue" unfolding lebesgue_def cube_def
   using assms gmeasurable_interval by auto
 
-lemma lmeasurable_empty[intro]: "lmeasurable {}"
-  using gmeasurable_empty apply- apply(drule_tac measurable_imp_lmeasurable) .
-
-lemma lmeasurable_union[intro]: assumes "lmeasurable s" "lmeasurable t"
-  shows "lmeasurable (s \<union> t)"
-  using assms unfolding lmeasurable_def Int_Un_distrib2 
-  by(auto intro:gmeasurable_union)
-
-lemma lmeasurable_countable_unions_strong:
-  fixes s::"nat => 'a::ordered_euclidean_space set"
-  assumes "\<And>n::nat. lmeasurable(s n)"
-  shows "lmeasurable(\<Union>{ s(n) |n. n \<in> UNIV })"
-  unfolding lmeasurable_def
-proof fix n::nat
-  have *:"\<Union>{s n |n. n \<in> UNIV} \<inter> cube n = \<Union>{s k \<inter> cube n |k. k \<in> UNIV}" by auto
-  show "gmeasurable (\<Union>{s n |n. n \<in> UNIV} \<inter> cube n)" unfolding *
-    apply(rule gmeasurable_countable_unions_strong)
-    apply(rule assms[unfolded lmeasurable_def,rule_format])
-  proof- fix k::nat
-    show "gmeasure (\<Union>{s ka \<inter> cube n |ka. ka \<le> k}) \<le> gmeasure (cube n::'a set)"
-      apply(rule measure_subset) apply(rule gmeasurable_finite_unions)
-      using assms(1)[unfolded lmeasurable_def] by auto
-  qed
-qed
-
-lemma lmeasurable_inter[intro]: fixes s::"'a :: ordered_euclidean_space set"
-  assumes "lmeasurable s" "lmeasurable t" shows "lmeasurable (s \<inter> t)"
-  unfolding lmeasurable_def
-proof fix n::nat
-  have *:"s \<inter> t \<inter> cube n = (s \<inter> cube n) \<inter> (t \<inter> cube n)" by auto
-  show "gmeasurable (s \<inter> t \<inter> cube n)"
-    using assms unfolding lmeasurable_def *
-    using gmeasurable_inter[of "s \<inter> cube n" "t \<inter> cube n"] by auto
-qed
-
-lemma lmeasurable_complement[intro]: assumes "lmeasurable s"
-  shows "lmeasurable (UNIV - s)"
-  unfolding lmeasurable_def
-proof fix n::nat
-  have *:"(UNIV - s) \<inter> cube n = cube n - (s \<inter> cube n)" by auto
-  show "gmeasurable ((UNIV - s) \<inter> cube n)" unfolding * 
-    apply(rule gmeasurable_diff) using assms unfolding lmeasurable_def by auto
-qed
-
-lemma lmeasurable_finite_unions:
-  assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> lmeasurable s"
-  shows "lmeasurable (\<Union> f)" unfolding lmeasurable_def
-proof fix n::nat have *:"(\<Union>f \<inter> cube n) = \<Union>{x \<inter> cube n | x . x\<in>f}" by auto
-  show "gmeasurable (\<Union>f \<inter> cube n)" unfolding *
-    apply(rule gmeasurable_finite_unions) unfolding simple_image 
-    using assms unfolding lmeasurable_def by auto
-qed
-
-lemma negligible_imp_lmeasurable[dest]: fixes s::"'a::ordered_euclidean_space set"
-  assumes "negligible s" shows "lmeasurable s"
-  unfolding lmeasurable_def
-proof case goal1
+lemma lebesgueI2: "(\<And>n. gmeasurable (S \<inter> cube n)) \<Longrightarrow> S \<in> sets lebesgue"
+  using assms unfolding lebesgue_def by auto
+
+interpretation lebesgue: sigma_algebra lebesgue
+proof
+  show "sets lebesgue \<subseteq> Pow (space lebesgue)"
+    unfolding lebesgue_def by auto
+  show "{} \<in> sets lebesgue"
+    using gmeasurable_empty by auto
+  { fix A B :: "'a set" assume "A \<in> sets lebesgue" "B \<in> sets lebesgue"
+    then show "A \<union> B \<in> sets lebesgue"
+      by (auto intro: gmeasurable_union simp: lebesgue_def Int_Un_distrib2) }
+  { fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets lebesgue"
+    show "(\<Union>i. A i) \<in> sets lebesgue"
+    proof (rule lebesgueI2)
+      fix n show "gmeasurable ((\<Union>i. A i) \<inter> cube n)" unfolding UN_extend_simps
+        using A
+        by (intro gmeasurable_countable_UNION[where B="gmeasure (cube n::'a set)"])
+           (auto intro!: measure_subset gmeasure_setsum simp: UN_extend_simps simp del: gmeasure_cube_eq UN_simps)
+    qed }
+  { fix A assume A: "A \<in> sets lebesgue" show "space lebesgue - A \<in> sets lebesgue"
+    proof (rule lebesgueI2)
+      fix n
+      have *: "(space lebesgue - A) \<inter> cube n = cube n - (A \<inter> cube n)"
+        unfolding lebesgue_def by auto
+      show "gmeasurable ((space lebesgue - A) \<inter> cube n)" unfolding *
+        using A by (auto intro!: gmeasurable_diff)
+    qed }
+qed
+
+lemma lebesgueI_borel[intro, simp]: fixes s::"'a::ordered_euclidean_space set"
+  assumes "s \<in> sets borel" shows "s \<in> sets lebesgue"
+proof- let ?S = "range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)})"
+  have *:"?S \<subseteq> sets lebesgue" by auto
+  have "s \<in> sigma_sets UNIV ?S" using assms
+    unfolding borel_eq_atLeastAtMost by (simp add: sigma_def)
+  thus ?thesis
+    using lebesgue.sigma_subset[of "\<lparr> space = UNIV, sets = ?S\<rparr>", simplified, OF *]
+    by (auto simp: sigma_def)
+qed
+
+lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set"
+  assumes "negligible s" shows "s \<in> sets lebesgue"
+proof (rule lebesgueI2)
+  fix n
   have *:"\<And>x. (if x \<in> cube n then indicator s x else 0) = (if x \<in> s \<inter> cube n then 1 else 0)"
     unfolding indicator_def_raw by auto
   note assms[unfolded negligible_def,rule_format,of "(\<chi>\<chi> i. - real n)::'a" "\<chi>\<chi> i. real n"]
-  thus ?case apply-apply(rule gmeasurableI[of _ 0]) unfolding has_gmeasure_def
+  thus "gmeasurable (s \<inter> cube n)" apply-apply(rule gmeasurableI[of _ 0]) unfolding has_gmeasure_def
     apply(subst(asm) has_integral_restrict_univ[THEN sym]) unfolding cube_def[symmetric]
     apply(subst has_integral_restrict_univ[THEN sym]) unfolding * .
 qed
 
-
 section {* The Lebesgue Measure *}
 
-definition has_lmeasure (infixr "has'_lmeasure" 80) where
-  "s has_lmeasure m \<equiv> lmeasurable s \<and> ((\<lambda>n. Real (gmeasure (s \<inter> cube n))) ---> m) sequentially"
-
-lemma has_lmeasureD: assumes "s has_lmeasure m"
-  shows "lmeasurable s" "gmeasurable (s \<inter> cube n)"
-  "((\<lambda>n. Real (gmeasure (s \<inter> cube n))) ---> m) sequentially"
-  using assms unfolding has_lmeasure_def lmeasurable_def by auto
-
-lemma has_lmeasureI: assumes "lmeasurable s" "((\<lambda>n. Real (gmeasure (s \<inter> cube n))) ---> m) sequentially"
-  shows "s has_lmeasure m" using assms unfolding has_lmeasure_def by auto
-
-definition lmeasure where
-  "lmeasure s \<equiv> SOME m. s has_lmeasure m"
-
-lemma has_lmeasure_has_gmeasure: assumes "s has_lmeasure (Real m)" "m\<ge>0"
+definition "lmeasure A = (SUP n. Real (gmeasure (A \<inter> cube n)))"
+
+lemma lmeasure_eq_0: assumes "negligible S" shows "lmeasure S = 0"
+proof -
+  from lebesgueI_negligible[OF assms]
+  have "\<And>n. gmeasurable (S \<inter> cube n)" by auto
+  from gmeasurable_measure_eq_0[OF this]
+  have "\<And>n. gmeasure (S \<inter> cube n) = 0" using assms by auto
+  then show ?thesis unfolding lmeasure_def by simp
+qed
+
+lemma lmeasure_iff_LIMSEQ:
+  assumes "A \<in> sets lebesgue" "0 \<le> m"
+  shows "lmeasure A = Real m \<longleftrightarrow> (\<lambda>n. (gmeasure (A \<inter> cube n))) ----> m"
+  unfolding lmeasure_def using assms cube_subset[where 'a='a]
+  by (intro SUP_eq_LIMSEQ monoI measure_subset) force+
+
+interpretation lebesgue: measure_space lebesgue lmeasure
+proof
+  show "lmeasure {} = 0"
+    by (auto intro!: lmeasure_eq_0)
+  show "countably_additive lebesgue lmeasure"
+  proof (unfold countably_additive_def, intro allI impI conjI)
+    fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets lebesgue" "disjoint_family A"
+    then have A: "\<And>i. A i \<in> sets lebesgue" by auto
+    show "(\<Sum>\<^isub>\<infinity>n. lmeasure (A n)) = lmeasure (\<Union>i. A i)" unfolding lmeasure_def
+    proof (subst psuminf_SUP_eq)
+      { fix i n
+        have "gmeasure (A i \<inter> cube n) \<le> gmeasure (A i \<inter> cube (Suc n))"
+          using A cube_subset[of n "Suc n"] by (auto intro!: measure_subset)
+        then show "Real (gmeasure (A i \<inter> cube n)) \<le> Real (gmeasure (A i \<inter> cube (Suc n)))"
+          by auto }
+      show "(SUP n. \<Sum>\<^isub>\<infinity>i. Real (gmeasure (A i \<inter> cube n))) = (SUP n. Real (gmeasure ((\<Union>i. A i) \<inter> cube n)))"
+      proof (intro arg_cong[where f="SUPR UNIV"] ext)
+        fix n
+        have sums: "(\<lambda>i. gmeasure (A i \<inter> cube n)) sums gmeasure (\<Union>{A i \<inter> cube n |i. i \<in> UNIV})"
+        proof (rule has_gmeasure_countable_negligible_unions(2))
+          fix i show "gmeasurable (A i \<inter> cube n)" using A by auto
+        next
+          fix i m :: nat assume "m \<noteq> i"
+          then have "A m \<inter> cube n \<inter> (A i \<inter> cube n) = {}"
+            using `disjoint_family A` unfolding disjoint_family_on_def by auto
+          then show "negligible (A m \<inter> cube n \<inter> (A i \<inter> cube n))" by auto
+        next
+          fix i
+          have "(\<Sum>k = 0..i. gmeasure (A k \<inter> cube n)) = gmeasure (\<Union>k\<le>i . A k \<inter> cube n)"
+            unfolding atLeast0AtMost using A
+          proof (intro gmeasure_setsum[symmetric])
+            fix s t :: nat assume "s \<noteq> t" then have "A t \<inter> A s = {}"
+              using `disjoint_family A` unfolding disjoint_family_on_def by auto
+            then show "A s \<inter> cube n \<inter> (A t \<inter> cube n) = {}" by auto
+          qed auto
+          also have "\<dots> \<le> gmeasure (cube n :: 'b set)" using A
+            by (intro measure_subset gmeasurable_finite_UNION) auto
+          finally show "(\<Sum>k = 0..i. gmeasure (A k \<inter> cube n)) \<le> gmeasure (cube n :: 'b set)" .
+        qed
+        show "(\<Sum>\<^isub>\<infinity>i. Real (gmeasure (A i \<inter> cube n))) = Real (gmeasure ((\<Union>i. A i) \<inter> cube n))"
+          unfolding psuminf_def
+          apply (subst setsum_Real)
+          apply (simp add: measure_pos_le)
+        proof (rule SUP_eq_LIMSEQ[THEN iffD2])
+          have "(\<Union>{A i \<inter> cube n |i. i \<in> UNIV}) = (\<Union>i. A i) \<inter> cube n" by auto
+          with sums show "(\<lambda>i. \<Sum>k<i. gmeasure (A k \<inter> cube n)) ----> gmeasure ((\<Union>i. A i) \<inter> cube n)"
+            unfolding sums_def atLeast0LessThan by simp
+        qed (auto intro!: monoI setsum_nonneg setsum_mono2)
+      qed
+    qed
+  qed
+qed
+
+lemma lmeasure_finite_has_gmeasure: assumes "s \<in> sets lebesgue" "lmeasure s = Real m" "0 \<le> m"
   shows "s has_gmeasure m"
-proof- note s = has_lmeasureD[OF assms(1)]
+proof-
   have *:"(\<lambda>n. (gmeasure (s \<inter> cube n))) ----> m"
-    using s(3) apply(subst (asm) lim_Real) using s(2) assms(2) by auto
-
+    using `lmeasure s = Real m` unfolding lmeasure_iff_LIMSEQ[OF `s \<in> sets lebesgue` `0 \<le> m`] .
+  have s: "\<And>n. gmeasurable (s \<inter> cube n)" using assms by auto
   have "(\<lambda>x. if x \<in> s then 1 else (0::real)) integrable_on UNIV \<and>
     (\<lambda>k. integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)))
     ----> integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)"
   proof(rule monotone_convergence_increasing)
-    have "\<forall>n. gmeasure (s \<inter> cube n) \<le> m" apply(rule ccontr) unfolding not_all not_le
-    proof(erule exE) fix k assume k:"m < gmeasure (s \<inter> cube k)"
-      hence "gmeasure (s \<inter> cube k) - m > 0" by auto
-      from *[unfolded Lim_sequentially,rule_format,OF this] guess N ..
-      note this[unfolded dist_real_def,rule_format,of "N + k"]
-      moreover have "gmeasure (s \<inter> cube (N + k)) \<ge> gmeasure (s \<inter> cube k)" apply-
-        apply(rule measure_subset) prefer 3 using s(2) 
-        using cube_subset[of k "N + k"] by auto
-      ultimately show False by auto
-    qed
-    thus *:"bounded {integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)) |k. True}" 
-      unfolding integral_measure_univ[OF s(2)] bounded_def apply-
+    have "lmeasure s \<le> Real m" using `lmeasure s = Real m` by simp
+    then have "\<forall>n. gmeasure (s \<inter> cube n) \<le> m"
+      unfolding lmeasure_def complete_lattice_class.SUP_le_iff
+      using `0 \<le> m` by (auto simp: measure_pos_le)
+    thus *:"bounded {integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)) |k. True}"
+      unfolding integral_measure_univ[OF s] bounded_def apply-
       apply(rule_tac x=0 in exI,rule_tac x=m in exI) unfolding dist_real_def
       by (auto simp: measure_pos_le)
-
     show "\<forall>k. (\<lambda>x. if x \<in> s \<inter> cube k then (1::real) else 0) integrable_on UNIV"
       unfolding integrable_restrict_univ
-      using s(2) unfolding gmeasurable_def has_gmeasure_def by auto
+      using s unfolding gmeasurable_def has_gmeasure_def by auto
     have *:"\<And>n. n \<le> Suc n" by auto
     show "\<forall>k. \<forall>x\<in>UNIV. (if x \<in> s \<inter> cube k then 1 else 0) \<le> (if x \<in> s \<inter> cube (Suc k) then 1 else (0::real))"
       using cube_subset[OF *] by fastsimp
     show "\<forall>x\<in>UNIV. (\<lambda>k. if x \<in> s \<inter> cube k then 1 else 0) ----> (if x \<in> s then 1 else (0::real))"
-      unfolding Lim_sequentially 
+      unfolding Lim_sequentially
     proof safe case goal1 from real_arch_lt[of "norm x"] guess N .. note N = this
       show ?case apply(rule_tac x=N in exI)
       proof safe case goal1
         have "x \<in> cube n" using cube_subset[OF goal1] N
-          using ball_subset_cube[of N] by(auto simp: dist_norm) 
+          using ball_subset_cube[of N] by(auto simp: dist_norm)
         thus ?case using `e>0` by auto
       qed
     qed
   qed note ** = conjunctD2[OF this]
   hence *:"m = integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)" apply-
-    apply(rule LIMSEQ_unique[OF _ **(2)]) unfolding measure_integral_univ[THEN sym,OF s(2)] using * .
+    apply(rule LIMSEQ_unique[OF _ **(2)]) unfolding measure_integral_univ[THEN sym,OF s] using * .
   show ?thesis unfolding has_gmeasure * apply(rule integrable_integral) using ** by auto
 qed
 
-lemma has_lmeasure_unique: "s has_lmeasure m1 \<Longrightarrow> s has_lmeasure m2 \<Longrightarrow> m1 = m2"
-  unfolding has_lmeasure_def apply(rule Lim_unique) using trivial_limit_sequentially by auto
-
-lemma lmeasure_unique[intro]: assumes "A has_lmeasure m" shows "lmeasure A = m"
-  using assms unfolding lmeasure_def lmeasurable_def apply-
-  apply(rule some_equality) defer apply(rule has_lmeasure_unique) by auto
-
-lemma glmeasurable_finite: assumes "lmeasurable s" "lmeasure s \<noteq> \<omega>" 
+lemma lmeasure_finite_gmeasurable: assumes "s \<in> sets lebesgue" "lmeasure s \<noteq> \<omega>"
   shows "gmeasurable s"
-proof-  have "\<exists>B. \<forall>n. gmeasure (s \<inter> cube n) \<le> B"
-  proof(rule ccontr) case goal1
-    note as = this[unfolded not_ex not_all not_le]
-    have "s has_lmeasure \<omega>" apply- apply(rule has_lmeasureI[OF assms(1)])
-      unfolding Lim_omega
-    proof fix B::real
-      from as[rule_format,of B] guess N .. note N = this
-      have "\<And>n. N \<le> n \<Longrightarrow> B \<le> gmeasure (s \<inter> cube n)"
-        apply(rule order_trans[where y="gmeasure (s \<inter> cube N)"]) defer
-        apply(rule measure_subset) prefer 3
-        using cube_subset N assms(1)[unfolded lmeasurable_def] by auto
-      thus "\<exists>N. \<forall>n\<ge>N. Real B \<le> Real (gmeasure (s \<inter> cube n))" apply-
-        apply(subst Real_max') apply(rule_tac x=N in exI,safe)
-        unfolding pinfreal_less_eq apply(subst if_P) by auto
-    qed note lmeasure_unique[OF this]
-    thus False using assms(2) by auto
-  qed then guess B .. note B=this
-
-  show ?thesis apply(rule gmeasurable_nested_unions[of "\<lambda>n. s \<inter> cube n",
-    unfolded Union_inter_cube,THEN conjunct1, where B1=B])
-  proof- fix n::nat
-    show " gmeasurable (s \<inter> cube n)" using assms by auto
-    show "gmeasure (s \<inter> cube n) \<le> B" using B by auto
-    show "s \<inter> cube n \<subseteq> s \<inter> cube (Suc n)"
-      by (rule Int_mono) (simp_all add: cube_subset)
-  qed
-qed
-
-lemma lmeasure_empty[intro]:"lmeasure {} = 0"
-  apply(rule lmeasure_unique)
-  unfolding has_lmeasure_def by auto
-
-lemma lmeasurableI[dest]:"s has_lmeasure m \<Longrightarrow> lmeasurable s"
-  unfolding has_lmeasure_def by auto
-
-lemma has_gmeasure_has_lmeasure: assumes "s has_gmeasure m"
-  shows "s has_lmeasure (Real m)"
-proof- have gmea:"gmeasurable s" using assms by auto
+proof (cases "lmeasure s")
+  case (preal m) from lmeasure_finite_has_gmeasure[OF `s \<in> sets lebesgue` this]
+  show ?thesis unfolding gmeasurable_def by auto
+qed (insert assms, auto)
+
+lemma has_gmeasure_lmeasure: assumes "s has_gmeasure m"
+  shows "lmeasure s = Real m"
+proof-
+  have gmea:"gmeasurable s" using assms by auto
+  then have s: "s \<in> sets lebesgue" by auto
   have m:"m \<ge> 0" using assms by auto
   have *:"m = gmeasure (\<Union>{s \<inter> cube n |n. n \<in> UNIV})" unfolding Union_inter_cube
     using assms by(rule measure_unique[THEN sym])
-  show ?thesis unfolding has_lmeasure_def
-    apply(rule,rule measurable_imp_lmeasurable[OF gmea])
-    apply(subst lim_Real) apply(rule,rule,rule m) unfolding *
+  show ?thesis
+    unfolding lmeasure_iff_LIMSEQ[OF s `0 \<le> m`] unfolding *
     apply(rule gmeasurable_nested_unions[THEN conjunct2, where B1="gmeasure s"])
   proof- fix n::nat show *:"gmeasurable (s \<inter> cube n)"
       using gmeasurable_inter[OF gmea gmeasurable_cube] .
     show "gmeasure (s \<inter> cube n) \<le> gmeasure s" apply(rule measure_subset)
       apply(rule * gmea)+ by auto
     show "s \<inter> cube n \<subseteq> s \<inter> cube (Suc n)" using cube_subset[of n "Suc n"] by auto
   qed
-qed    
-    
+qed
+
+lemma has_gmeasure_iff_lmeasure:
+  "A has_gmeasure m \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m)"
+proof
+  assume "A has_gmeasure m"
+  with has_gmeasure_lmeasure[OF this]
+  have "gmeasurable A" "0 \<le> m" "lmeasure A = Real m" by auto
+  then show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m" by auto
+next
+  assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m"
+  then show "A has_gmeasure m" by (intro lmeasure_finite_has_gmeasure) auto
+qed
+
 lemma gmeasure_lmeasure: assumes "gmeasurable s" shows "lmeasure s = Real (gmeasure s)"
-proof- note has_gmeasure_measureI[OF assms]
-  note has_gmeasure_has_lmeasure[OF this]
-  thus ?thesis by(rule lmeasure_unique)
-qed
-
-lemma has_lmeasure_lmeasure: "lmeasurable s \<longleftrightarrow> s has_lmeasure (lmeasure s)" (is "?l = ?r")
-proof assume ?l let ?f = "\<lambda>n. Real (gmeasure (s \<inter> cube n))"
-  have "\<forall>n m. n\<ge>m \<longrightarrow> ?f n \<ge> ?f m" unfolding pinfreal_less_eq apply safe
-    apply(subst if_P) defer apply(rule measure_subset) prefer 3
-    apply(drule cube_subset) using `?l` by auto
-  from lim_pinfreal_increasing[OF this] guess l . note l=this
-  hence "s has_lmeasure l" using `?l` apply-apply(rule has_lmeasureI) by auto
-  thus ?r using lmeasure_unique by auto
-next assume ?r thus ?l unfolding has_lmeasure_def by auto
-qed
-
-lemma lmeasure_subset[dest]: assumes "lmeasurable s" "lmeasurable t" "s \<subseteq> t"
-  shows "lmeasure s \<le> lmeasure t"
-proof(cases "lmeasure t = \<omega>")
-  case False have som:"lmeasure s \<noteq> \<omega>"
-  proof(rule ccontr,unfold not_not) assume as:"lmeasure s = \<omega>"
-    have "t has_lmeasure \<omega>" using assms(2) apply(rule has_lmeasureI)
-      unfolding Lim_omega
-    proof case goal1
-      note assms(1)[unfolded has_lmeasure_lmeasure]
-      note has_lmeasureD(3)[OF this,unfolded as Lim_omega,rule_format,of B]
-      then guess N .. note N = this
-      show ?case apply(rule_tac x=N in exI) apply safe
-        apply(rule order_trans) apply(rule N[rule_format],assumption)
-        unfolding pinfreal_less_eq apply(subst if_P)defer
-        apply(rule measure_subset) using assms by auto
-    qed
-    thus False using lmeasure_unique False by auto
-  qed
-
-  note assms(1)[unfolded has_lmeasure_lmeasure] note has_lmeasureD(3)[OF this]
-  hence "(\<lambda>n. Real (gmeasure (s \<inter> cube n))) ----> Real (real (lmeasure s))"
-    unfolding Real_real'[OF som] .
-  hence l1:"(\<lambda>n. gmeasure (s \<inter> cube n)) ----> real (lmeasure s)"
-    apply-apply(subst(asm) lim_Real) by auto
-
-  note assms(2)[unfolded has_lmeasure_lmeasure] note has_lmeasureD(3)[OF this]
-  hence "(\<lambda>n. Real (gmeasure (t \<inter> cube n))) ----> Real (real (lmeasure t))"
-    unfolding Real_real'[OF False] .
-  hence l2:"(\<lambda>n. gmeasure (t \<inter> cube n)) ----> real (lmeasure t)"
-    apply-apply(subst(asm) lim_Real) by auto
-
-  have "real (lmeasure s) \<le> real (lmeasure t)" apply(rule LIMSEQ_le[OF l1 l2])
-    apply(rule_tac x=0 in exI,safe) apply(rule measure_subset) using assms by auto
-  hence "Real (real (lmeasure s)) \<le> Real (real (lmeasure t))"
-    unfolding pinfreal_less_eq by auto
-  thus ?thesis unfolding Real_real'[OF som] Real_real'[OF False] .
-qed auto
-
-lemma has_lmeasure_negligible_unions_image:
-  assumes "finite s" "\<And>x. x \<in> s ==> lmeasurable(f x)"
-  "\<And>x y. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x \<noteq> y \<Longrightarrow> negligible((f x) \<inter> (f y))"
-  shows "(\<Union> (f ` s)) has_lmeasure (setsum (\<lambda>x. lmeasure(f x)) s)"
-  unfolding has_lmeasure_def
-proof show lmeaf:"lmeasurable (\<Union>f ` s)" apply(rule lmeasurable_finite_unions)
-    using assms(1-2) by auto
-  show "(\<lambda>n. Real (gmeasure (\<Union>f ` s \<inter> cube n))) ----> (\<Sum>x\<in>s. lmeasure (f x))" (is ?l)
-  proof(cases "\<exists>x\<in>s. lmeasure (f x) = \<omega>")
-    case False hence *:"(\<Sum>x\<in>s. lmeasure (f x)) \<noteq> \<omega>" apply-
-      apply(rule setsum_neq_omega) using assms(1) by auto
-    have gmea:"\<And>x. x\<in>s \<Longrightarrow> gmeasurable (f x)" apply(rule glmeasurable_finite) using False assms(2) by auto
-    have "(\<Sum>x\<in>s. lmeasure (f x)) = (\<Sum>x\<in>s. Real (gmeasure (f x)))" apply(rule setsum_cong2)
-      apply(rule gmeasure_lmeasure) using False assms(2) gmea by auto
-    also have "... = Real (\<Sum>x\<in>s. (gmeasure (f x)))" apply(rule setsum_Real) by auto
-    finally have sum:"(\<Sum>x\<in>s. lmeasure (f x)) = Real (\<Sum>x\<in>s. gmeasure (f x))" .
-    have sum_0:"(\<Sum>x\<in>s. gmeasure (f x)) \<ge> 0" apply(rule setsum_nonneg) by auto
-    have int_un:"\<Union>f ` s has_gmeasure (\<Sum>x\<in>s. gmeasure (f x))"
-      apply(rule has_gmeasure_negligible_unions_image) using assms gmea by auto
-
-    have unun:"\<Union>{\<Union>f ` s \<inter> cube n |n. n \<in> UNIV} = \<Union>f ` s" unfolding simple_image 
-    proof safe fix x y assume as:"x \<in> f y" "y \<in> s"
-      from mem_big_cube[of x] guess n . note n=this
-      thus "x \<in> \<Union>range (\<lambda>n. \<Union>f ` s \<inter> cube n)" unfolding Union_iff
-        apply-apply(rule_tac x="\<Union>f ` s \<inter> cube n" in bexI) using as by auto
-    qed
-    show ?l apply(subst Real_real'[OF *,THEN sym])apply(subst lim_Real) 
-      apply rule apply rule unfolding sum real_Real if_P[OF sum_0] apply(rule sum_0)
-      unfolding measure_unique[OF int_un,THEN sym] apply(subst(2) unun[THEN sym])
-      apply(rule has_gmeasure_nested_unions[THEN conjunct2])
-    proof- fix n::nat
-      show *:"gmeasurable (\<Union>f ` s \<inter> cube n)" using lmeaf unfolding lmeasurable_def by auto
-      thus "gmeasure (\<Union>f ` s \<inter> cube n) \<le> gmeasure (\<Union>f ` s)"
-        apply(rule measure_subset) using int_un by auto
-      show "\<Union>f ` s \<inter> cube n \<subseteq> \<Union>f ` s \<inter> cube (Suc n)"
-        using cube_subset[of n "Suc n"] by auto
-    qed
-
-  next case True then guess X .. note X=this
-    hence sum:"(\<Sum>x\<in>s. lmeasure (f x)) = \<omega>" using setsum_\<omega>[THEN iffD2, of s] assms by fastsimp
-    show ?l unfolding sum Lim_omega
-    proof fix B::real
-      have Xm:"(f X) has_lmeasure \<omega>" using X by (metis assms(2) has_lmeasure_lmeasure)
-      note this[unfolded has_lmeasure_def,THEN conjunct2, unfolded Lim_omega]
-      from this[rule_format,of B] guess N .. note N=this[rule_format]
-      show "\<exists>N. \<forall>n\<ge>N. Real B \<le> Real (gmeasure (\<Union>f ` s \<inter> cube n))"
-        apply(rule_tac x=N in exI)
-      proof safe case goal1 show ?case apply(rule order_trans[OF N[OF goal1]])
-          unfolding pinfreal_less_eq apply(subst if_P) defer
-          apply(rule measure_subset) using has_lmeasureD(2)[OF Xm]
-          using lmeaf unfolding lmeasurable_def using X(1) by auto
-      qed qed qed qed
-
-lemma has_lmeasure_negligible_unions:
-  assumes"finite f" "\<And>s. s \<in> f ==> s has_lmeasure (m s)"
-  "\<And>s t. s \<in> f \<Longrightarrow> t \<in> f \<Longrightarrow> s \<noteq> t ==> negligible (s\<inter>t)"
-  shows "(\<Union> f) has_lmeasure (setsum m f)"
-proof- have *:"setsum m f = setsum lmeasure f" apply(rule setsum_cong2)
-    apply(subst lmeasure_unique[OF assms(2)]) by auto
-  show ?thesis unfolding *
-    apply(rule has_lmeasure_negligible_unions_image[where s=f and f=id,unfolded image_id id_apply])
-    using assms by auto
-qed
-
-lemma has_lmeasure_disjoint_unions:
-  assumes"finite f" "\<And>s. s \<in> f ==> s has_lmeasure (m s)"
-  "\<And>s t. s \<in> f \<Longrightarrow> t \<in> f \<Longrightarrow> s \<noteq> t ==> s \<inter> t = {}"
-  shows "(\<Union> f) has_lmeasure (setsum m f)"
-proof- have *:"setsum m f = setsum lmeasure f" apply(rule setsum_cong2)
-    apply(subst lmeasure_unique[OF assms(2)]) by auto
-  show ?thesis unfolding *
-    apply(rule has_lmeasure_negligible_unions_image[where s=f and f=id,unfolded image_id id_apply])
-    using assms by auto
-qed
-
-lemma has_lmeasure_nested_unions:
-  assumes "\<And>n. lmeasurable(s n)" "\<And>n. s(n) \<subseteq> s(Suc n)"
-  shows "lmeasurable(\<Union> { s n | n. n \<in> UNIV }) \<and>
-  (\<lambda>n. lmeasure(s n)) ----> lmeasure(\<Union> { s(n) | n. n \<in> UNIV })" (is "?mea \<and> ?lim")
-proof- have cube:"\<And>m. \<Union> { s(n) | n. n \<in> UNIV } \<inter> cube m = \<Union> { s(n) \<inter> cube m | n. n \<in> UNIV }" by blast
-  have 3:"\<And>n. \<forall>m\<ge>n. s n \<subseteq> s m" apply(rule transitive_stepwise_le) using assms(2) by auto
-  have mea:"?mea" unfolding lmeasurable_def cube apply rule 
-    apply(rule_tac B1="gmeasure (cube n)" in has_gmeasure_nested_unions[THEN conjunct1])
-    prefer 3 apply rule using assms(1) unfolding lmeasurable_def
-    by(auto intro!:assms(2)[unfolded subset_eq,rule_format])
-  show ?thesis apply(rule,rule mea)
-  proof(cases "lmeasure(\<Union> { s(n) | n. n \<in> UNIV }) = \<omega>")
-    case True show ?lim unfolding True Lim_omega
-    proof(rule ccontr) case goal1 note this[unfolded not_all not_ex]
-      hence "\<exists>B. \<forall>n. \<exists>m\<ge>n. Real B > lmeasure (s m)" by(auto simp add:not_le)
-      from this guess B .. note B=this[rule_format]
-
-      have "\<forall>n. gmeasurable (s n) \<and> gmeasure (s n) \<le> max B 0" 
-      proof safe fix n::nat from B[of n] guess m .. note m=this
-        hence *:"lmeasure (s n) < Real B" apply-apply(rule le_less_trans)
-          apply(rule lmeasure_subset[OF assms(1,1)]) apply(rule 3[rule_format]) by auto
-        thus **:"gmeasurable (s n)" apply-apply(rule glmeasurable_finite[OF assms(1)]) by auto
-        thus "gmeasure (s n) \<le> max B 0"  using * unfolding gmeasure_lmeasure[OF **] Real_max'[of B]
-          unfolding pinfreal_less apply- apply(subst(asm) if_P) by auto
-      qed
-      hence "\<And>n. gmeasurable (s n)" "\<And>n. gmeasure (s n) \<le> max B 0" by auto
-      note g = conjunctD2[OF has_gmeasure_nested_unions[of s, OF this assms(2)]]
-      show False using True unfolding gmeasure_lmeasure[OF g(1)] by auto
-    qed
-  next let ?B = "lmeasure (\<Union>{s n |n. n \<in> UNIV})"
-    case False note gmea_lim = glmeasurable_finite[OF mea this]
-    have ls:"\<And>n. lmeasure (s n) \<le> lmeasure (\<Union>{s n |n. n \<in> UNIV})"
-      apply(rule lmeasure_subset) using assms(1) mea by auto
-    have "\<And>n. lmeasure (s n) \<noteq> \<omega>"
-    proof(rule ccontr,safe) case goal1
-      show False using False ls[of n] unfolding goal1 by auto
-    qed
-    note gmea = glmeasurable_finite[OF assms(1) this]
-
-    have "\<And>n. gmeasure (s n) \<le> real ?B"  unfolding gmeasure_lmeasure[OF gmea_lim]
-      unfolding real_Real apply(subst if_P,rule) apply(rule measure_subset)
-      using gmea gmea_lim by auto
-    note has_gmeasure_nested_unions[of s, OF gmea this assms(2)]
-    thus ?lim unfolding gmeasure_lmeasure[OF gmea] gmeasure_lmeasure[OF gmea_lim]
-      apply-apply(subst lim_Real) by auto
-  qed
-qed
-
-lemma has_lmeasure_countable_negligible_unions: 
-  assumes "\<And>n. lmeasurable(s n)" "\<And>m n. m \<noteq> n \<Longrightarrow> negligible(s m \<inter> s n)"
-  shows "(\<lambda>m. setsum (\<lambda>n. lmeasure(s n)) {..m}) ----> (lmeasure(\<Union> { s(n) |n. n \<in> UNIV }))"
-proof- have *:"\<And>n. (\<Union> (s ` {0..n})) has_lmeasure (setsum (\<lambda>k. lmeasure(s k)) {0..n})"
-    apply(rule has_lmeasure_negligible_unions_image) using assms by auto
-  have **:"(\<Union>{\<Union>s ` {0..n} |n. n \<in> UNIV}) = (\<Union>{s n |n. n \<in> UNIV})" unfolding simple_image by fastsimp
-  have "lmeasurable (\<Union>{\<Union>s ` {0..n} |n. n \<in> UNIV}) \<and>
-    (\<lambda>n. lmeasure (\<Union>(s ` {0..n}))) ----> lmeasure (\<Union>{\<Union>s ` {0..n} |n. n \<in> UNIV})"
-    apply(rule has_lmeasure_nested_unions) apply(rule has_lmeasureD(1)[OF *])
-    apply(rule Union_mono,rule image_mono) by auto
-  note lem = conjunctD2[OF this,unfolded **] 
-  show ?thesis using lem(2) unfolding lmeasure_unique[OF *] unfolding atLeast0AtMost .
-qed
-
-lemma lmeasure_eq_0: assumes "negligible s" shows "lmeasure s = 0"
-proof- note mea=negligible_imp_lmeasurable[OF assms]
-  have *:"\<And>n. (gmeasure (s \<inter> cube n)) = 0" 
-    unfolding gmeasurable_measure_eq_0[OF mea[unfolded lmeasurable_def,rule_format]]
-    using assms by auto
-  show ?thesis
-    apply(rule lmeasure_unique) unfolding has_lmeasure_def
-    apply(rule,rule mea) unfolding * by auto
-qed
-
-lemma negligible_img_gmeasurable: fixes s::"'a::ordered_euclidean_space set"
-  assumes "negligible s" shows "gmeasurable s"
-  apply(rule glmeasurable_finite)
-  using lmeasure_eq_0[OF assms] negligible_imp_lmeasurable[OF assms] by auto
-
-
-
-
-section {* Instantiation of _::euclidean_space as measure_space *}
-
-definition lebesgue_space :: "'a::ordered_euclidean_space algebra" where
-  "lebesgue_space = \<lparr> space = UNIV, sets = lmeasurable \<rparr>"
-
-lemma lebesgue_measurable[simp]:"A \<in> sets lebesgue_space \<longleftrightarrow> lmeasurable A"
-  unfolding lebesgue_space_def by(auto simp: mem_def)
-
-lemma mem_gmeasurable[simp]: "A \<in> gmeasurable \<longleftrightarrow> gmeasurable A"
-  unfolding mem_def ..
-
-interpretation lebesgue: measure_space lebesgue_space lmeasure
-  apply(intro_locales) unfolding measure_space_axioms_def countably_additive_def
-  unfolding sigma_algebra_axioms_def algebra_def
-  unfolding lebesgue_measurable
-proof safe
-  fix A::"nat => _" assume as:"range A \<subseteq> sets lebesgue_space" "disjoint_family A"
-    "lmeasurable (UNION UNIV A)"
-  have *:"UNION UNIV A = \<Union>range A" by auto
-  show "(\<Sum>\<^isub>\<infinity>n. lmeasure (A n)) = lmeasure (UNION UNIV A)" 
-    unfolding psuminf_def apply(rule SUP_Lim_pinfreal)
-  proof- fix n m::nat assume mn:"m\<le>n"
-    have *:"\<And>m. (\<Sum>n<m. lmeasure (A n)) = lmeasure (\<Union>A ` {..<m})"
-      apply(subst lmeasure_unique[OF has_lmeasure_negligible_unions[where m=lmeasure]])
-      apply(rule finite_imageI) apply rule apply(subst has_lmeasure_lmeasure[THEN sym])
-    proof- fix m::nat
-      show "(\<Sum>n<m. lmeasure (A n)) = setsum lmeasure (A ` {..<m})"
-        apply(subst setsum_reindex_nonzero) unfolding o_def apply rule
-        apply(rule lmeasure_eq_0) using as(2) unfolding disjoint_family_on_def
-        apply(erule_tac x=x in ballE,safe,erule_tac x=y in ballE) by auto
-    next fix m s assume "s \<in> A ` {..<m}"
-      hence "s \<in> range A" by auto thus "lmeasurable s" using as(1) by fastsimp
-    next fix m s t assume st:"s  \<in> A ` {..<m}" "t \<in> A ` {..<m}" "s \<noteq> t"
-      from st(1-2) guess sa ta unfolding image_iff apply-by(erule bexE)+ note a=this
-      from st(3) have "sa \<noteq> ta" unfolding a by auto
-      thus "negligible (s \<inter> t)" 
-        using as(2) unfolding disjoint_family_on_def a
-        apply(erule_tac x=sa in ballE,erule_tac x=ta in ballE) by auto
-    qed
-
-    have "\<And>m. lmeasurable (\<Union>A ` {..<m})"  apply(rule lmeasurable_finite_unions)
-      apply(rule finite_imageI,rule) using as(1) by fastsimp
-    from this this show "(\<Sum>n<m. lmeasure (A n)) \<le> (\<Sum>n<n. lmeasure (A n))" unfolding *
-      apply(rule lmeasure_subset) apply(rule Union_mono) apply(rule image_mono) using mn by auto
-    
-  next have *:"UNION UNIV A = \<Union>{A n |n. n \<in> UNIV}" by auto
-    show "(\<lambda>n. \<Sum>n<n. lmeasure (A n)) ----> lmeasure (UNION UNIV A)"
-      apply(rule LIMSEQ_imp_Suc) unfolding lessThan_Suc_atMost *
-      apply(rule has_lmeasure_countable_negligible_unions)
-      using as unfolding disjoint_family_on_def subset_eq by auto
-  qed
-
-next show "lmeasure {} = 0" by auto
-next fix A::"nat => _" assume as:"range A \<subseteq> sets lebesgue_space"
-  have *:"UNION UNIV A = (\<Union>{A n |n. n \<in> UNIV})" unfolding simple_image by auto
-  show "lmeasurable (UNION UNIV A)" unfolding * using as unfolding subset_eq
-    using lmeasurable_countable_unions_strong[of A] by auto
-qed(auto simp: lebesgue_space_def mem_def)
-
-
-
-lemma lmeasurbale_closed_interval[intro]:
-  "lmeasurable {a..b::'a::ordered_euclidean_space}"
-  unfolding lmeasurable_def cube_def inter_interval by auto
-
-lemma space_lebesgue_space[simp]:"space lebesgue_space = UNIV"
-  unfolding lebesgue_space_def by auto
-
-abbreviation "gintegral \<equiv> Integration.integral"
+proof -
+  note has_gmeasure_measureI[OF assms]
+  note has_gmeasure_lmeasure[OF this]
+  thus ?thesis .
+qed
 
 lemma lebesgue_simple_function_indicator:
   fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
   assumes f:"lebesgue.simple_function f"
   shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
   apply(rule,subst lebesgue.simple_function_indicator_representation[OF f]) by auto
 
 lemma lmeasure_gmeasure:
   "gmeasurable s \<Longrightarrow> gmeasure s = real (lmeasure s)"
-  apply(subst gmeasure_lmeasure) by auto
+  by (subst gmeasure_lmeasure) auto
 
 lemma lmeasure_finite: assumes "gmeasurable s" shows "lmeasure s \<noteq> \<omega>"
   using gmeasure_lmeasure[OF assms] by auto
 
-lemma negligible_lmeasure: assumes "lmeasurable s"
-  shows "lmeasure s = 0 \<longleftrightarrow> negligible s" (is "?l = ?r")
-proof assume ?l
-  hence *:"gmeasurable s" using glmeasurable_finite[of s] assms by auto
-  show ?r unfolding gmeasurable_measure_eq_0[THEN sym,OF *]
-    unfolding lmeasure_gmeasure[OF *] using `?l` by auto
-next assume ?r
-  note g=negligible_img_gmeasurable[OF this] and measure_eq_0[OF this]
-  hence "real (lmeasure s) = 0" using lmeasure_gmeasure[of s] by auto
-  thus ?l using lmeasure_finite[OF g] apply- apply(rule real_0_imp_eq_0) by auto
+lemma negligible_iff_lebesgue_null_sets:
+  "negligible A \<longleftrightarrow> A \<in> lebesgue.null_sets"
+proof
+  assume "negligible A"
+  from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0]
+  show "A \<in> lebesgue.null_sets" by auto
+next
+  assume A: "A \<in> lebesgue.null_sets"
+  then have *:"gmeasurable A" using lmeasure_finite_gmeasurable[of A] by auto
+  show "negligible A"
+    unfolding gmeasurable_measure_eq_0[OF *, symmetric]
+    unfolding lmeasure_gmeasure[OF *] using A by auto
+qed
+
+lemma
+  fixes a b ::"'a::ordered_euclidean_space"
+  shows lmeasure_atLeastAtMost[simp]: "lmeasure {a..b} = Real (content {a..b})"
+    and lmeasure_greaterThanLessThan[simp]: "lmeasure {a <..< b} = Real (content {a..b})"
+  using has_gmeasure_interval[of a b] by (auto intro!: has_gmeasure_lmeasure)
+
+lemma lmeasure_cube:
+  "lmeasure (cube n::('a::ordered_euclidean_space) set) = (Real ((2 * real n) ^ (DIM('a))))"
+  by (intro has_gmeasure_lmeasure) auto
+
+lemma lmeasure_UNIV[intro]: "lmeasure UNIV = \<omega>"
+  unfolding lmeasure_def SUP_\<omega>
+proof (intro allI impI)
+  fix x assume "x < \<omega>"
+  then obtain r where r: "x = Real r" "0 \<le> r" by (cases x) auto
+  then obtain n where n: "r < of_nat n" using ex_less_of_nat by auto
+  show "\<exists>i\<in>UNIV. x < Real (gmeasure (UNIV \<inter> cube i))"
+  proof (intro bexI[of _ n])
+    have "x < Real (of_nat n)" using n r by auto
+    also have "Real (of_nat n) \<le> Real (gmeasure (UNIV \<inter> cube n))"
+      using gmeasure_cube_ge_n[of n] by (auto simp: real_eq_of_nat[symmetric])
+    finally show "x < Real (gmeasure (UNIV \<inter> cube n))" .
+  qed auto
+qed
+
+lemma atLeastAtMost_singleton_euclidean[simp]:
+  fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
+  by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a])
+
+lemma content_singleton[simp]: "content {a} = 0"
+proof -
+  have "content {a .. a} = 0"
+    by (subst content_closed_interval) auto
+  then show ?thesis by simp
+qed
+
+lemma lmeasure_singleton[simp]:
+  fixes a :: "'a::ordered_euclidean_space" shows "lmeasure {a} = 0"
+  using has_gmeasure_interval[of a a] unfolding zero_pinfreal_def
+  by (intro has_gmeasure_lmeasure)
+     (simp add: content_closed_interval DIM_positive)
+
+declare content_real[simp]
+
+lemma
+  fixes a b :: real
+  shows lmeasure_real_greaterThanAtMost[simp]:
+    "lmeasure {a <.. b} = Real (if a \<le> b then b - a else 0)"
+proof cases
+  assume "a < b"
+  then have "lmeasure {a <.. b} = lmeasure {a <..< b} + lmeasure {b}"
+    by (subst lebesgue.measure_additive)
+       (auto intro!: lebesgueI_borel arg_cong[where f=lmeasure])
+  then show ?thesis by auto
+qed auto
+
+lemma
+  fixes a b :: real
+  shows lmeasure_real_atLeastLessThan[simp]:
+    "lmeasure {a ..< b} = Real (if a \<le> b then b - a else 0)" (is ?eqlt)
+proof cases
+  assume "a < b"
+  then have "lmeasure {a ..< b} = lmeasure {a} + lmeasure {a <..< b}"
+    by (subst lebesgue.measure_additive)
+       (auto intro!: lebesgueI_borel arg_cong[where f=lmeasure])
+  then show ?thesis by auto
+qed auto
+
+interpretation borel: measure_space borel lmeasure
+proof
+  show "countably_additive borel lmeasure"
+    using lebesgue.ca unfolding countably_additive_def
+    apply safe apply (erule_tac x=A in allE) by auto
+qed auto
+
+interpretation borel: sigma_finite_measure borel lmeasure
+proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
+  show "range cube \<subseteq> sets borel" by (auto intro: borel_closed)
+  { fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
+  thus "(\<Union>i. cube i) = space borel" by auto
+  show "\<forall>i. lmeasure (cube i) \<noteq> \<omega>" unfolding lmeasure_cube by auto
+qed
+
+interpretation lebesgue: sigma_finite_measure lebesgue lmeasure
+proof
+  from borel.sigma_finite guess A ..
+  moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
+  ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lmeasure (A i) \<noteq> \<omega>)"
+    by auto
+qed
+
+lemma simple_function_has_integral:
+  fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
+  assumes f:"lebesgue.simple_function f"
+  and f':"\<forall>x. f x \<noteq> \<omega>"
+  and om:"\<forall>x\<in>range f. lmeasure (f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
+  shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
+  unfolding lebesgue.simple_integral_def
+  apply(subst lebesgue_simple_function_indicator[OF f])
+proof- case goal1
+  have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f -` {y}) x \<noteq> \<omega>"
+    "\<forall>x\<in>range f. x * lmeasure (f -` {x} \<inter> UNIV) \<noteq> \<omega>"
+    using f' om unfolding indicator_def by auto
+  show ?case unfolding space_lebesgue real_of_pinfreal_setsum'[OF *(1),THEN sym]
+    unfolding real_of_pinfreal_setsum'[OF *(2),THEN sym]
+    unfolding real_of_pinfreal_setsum space_lebesgue
+    apply(rule has_integral_setsum)
+  proof safe show "finite (range f)" using f by (auto dest: lebesgue.simple_functionD)
+    fix y::'a show "((\<lambda>x. real (f y * indicator (f -` {f y}) x)) has_integral
+      real (f y * lmeasure (f -` {f y} \<inter> UNIV))) UNIV"
+    proof(cases "f y = 0") case False
+      have mea:"gmeasurable (f -` {f y})" apply(rule lmeasure_finite_gmeasurable)
+        using assms unfolding lebesgue.simple_function_def using False by auto
+      have *:"\<And>x. real (indicator (f -` {f y}) x::pinfreal) = (if x \<in> f -` {f y} then 1 else 0)" by auto
+      show ?thesis unfolding real_of_pinfreal_mult[THEN sym]
+        apply(rule has_integral_cmul[where 'b=real, unfolded real_scaleR_def])
+        unfolding Int_UNIV_right lmeasure_gmeasure[OF mea,THEN sym]
+        unfolding measure_integral_univ[OF mea] * apply(rule integrable_integral)
+        unfolding gmeasurable_integrable[THEN sym] using mea .
+    qed auto
+  qed qed
+
+lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
+  unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
+  using assms by auto
+
+lemma simple_function_has_integral':
+  fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
+  assumes f:"lebesgue.simple_function f"
+  and i: "lebesgue.simple_integral f \<noteq> \<omega>"
+  shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
+proof- let ?f = "\<lambda>x. if f x = \<omega> then 0 else f x"
+  { fix x have "real (f x) = real (?f x)" by (cases "f x") auto } note * = this
+  have **:"{x. f x \<noteq> ?f x} = f -` {\<omega>}" by auto
+  have **:"lmeasure {x\<in>space lebesgue. f x \<noteq> ?f x} = 0"
+    using lebesgue.simple_integral_omega[OF assms] by(auto simp add:**)
+  show ?thesis apply(subst lebesgue.simple_integral_cong'[OF f _ **])
+    apply(rule lebesgue.simple_function_compose1[OF f])
+    unfolding * defer apply(rule simple_function_has_integral)
+  proof-
+    show "lebesgue.simple_function ?f"
+      using lebesgue.simple_function_compose1[OF f] .
+    show "\<forall>x. ?f x \<noteq> \<omega>" by auto
+    show "\<forall>x\<in>range ?f. lmeasure (?f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
+    proof (safe, simp, safe, rule ccontr)
+      fix y assume "f y \<noteq> \<omega>" "f y \<noteq> 0"
+      hence "(\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y} = f -` {f y}"
+        by (auto split: split_if_asm)
+      moreover assume "lmeasure ((\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y}) = \<omega>"
+      ultimately have "lmeasure (f -` {f y}) = \<omega>" by simp
+      moreover
+      have "f y * lmeasure (f -` {f y}) \<noteq> \<omega>" using i f
+        unfolding lebesgue.simple_integral_def setsum_\<omega> lebesgue.simple_function_def
+        by auto
+      ultimately have "f y = 0" by (auto split: split_if_asm)
+      then show False using `f y \<noteq> 0` by simp
+    qed
+  qed
+qed
+
+lemma (in measure_space) positive_integral_monotone_convergence:
+  fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pinfreal"
+  assumes i: "\<And>i. f i \<in> borel_measurable M" and mono: "\<And>x. mono (\<lambda>n. f n x)"
+  and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
+  shows "u \<in> borel_measurable M"
+  and "(\<lambda>i. positive_integral (f i)) ----> positive_integral u" (is ?ilim)
+proof -
+  from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u]
+  show ?ilim using mono lim i by auto
+  have "(SUP i. f i) = u" using mono lim SUP_Lim_pinfreal
+    unfolding fun_eq_iff SUPR_fun_expand mono_def by auto
+  moreover have "(SUP i. f i) \<in> borel_measurable M"
+    using i by (rule borel_measurable_SUP)
+  ultimately show "u \<in> borel_measurable M" by simp
+qed
+
+lemma positive_integral_has_integral:
+  fixes f::"'a::ordered_euclidean_space => pinfreal"
+  assumes f:"f \<in> borel_measurable lebesgue"
+  and int_om:"lebesgue.positive_integral f \<noteq> \<omega>"
+  and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *)
+  shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.positive_integral f))) UNIV"
+proof- let ?i = "lebesgue.positive_integral f"
+  from lebesgue.borel_measurable_implies_simple_function_sequence[OF f]
+  guess u .. note conjunctD2[OF this,rule_format] note u = conjunctD2[OF this(1)] this(2)
+  let ?u = "\<lambda>i x. real (u i x)" and ?f = "\<lambda>x. real (f x)"
+  have u_simple:"\<And>k. lebesgue.simple_integral (u k) = lebesgue.positive_integral (u k)"
+    apply(subst lebesgue.positive_integral_eq_simple_integral[THEN sym,OF u(1)]) ..
+  have int_u_le:"\<And>k. lebesgue.simple_integral (u k) \<le> lebesgue.positive_integral f"
+    unfolding u_simple apply(rule lebesgue.positive_integral_mono)
+    using isoton_Sup[OF u(3)] unfolding le_fun_def by auto
+  have u_int_om:"\<And>i. lebesgue.simple_integral (u i) \<noteq> \<omega>"
+  proof- case goal1 thus ?case using int_u_le[of i] int_om by auto qed
+
+  note u_int = simple_function_has_integral'[OF u(1) this]
+  have "(\<lambda>x. real (f x)) integrable_on UNIV \<and>
+    (\<lambda>k. Integration.integral UNIV (\<lambda>x. real (u k x))) ----> Integration.integral UNIV (\<lambda>x. real (f x))"
+    apply(rule monotone_convergence_increasing) apply(rule,rule,rule u_int)
+  proof safe case goal1 show ?case apply(rule real_of_pinfreal_mono) using u(2,3) by auto
+  next case goal2 show ?case using u(3) apply(subst lim_Real[THEN sym])
+      prefer 3 apply(subst Real_real') defer apply(subst Real_real')
+      using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] using f_om u by auto
+  next case goal3
+    show ?case apply(rule bounded_realI[where B="real (lebesgue.positive_integral f)"])
+      apply safe apply(subst abs_of_nonneg) apply(rule integral_nonneg,rule) apply(rule u_int)
+      unfolding integral_unique[OF u_int] defer apply(rule real_of_pinfreal_mono[OF _ int_u_le])
+      using u int_om by auto
+  qed note int = conjunctD2[OF this]
+
+  have "(\<lambda>i. lebesgue.simple_integral (u i)) ----> ?i" unfolding u_simple
+    apply(rule lebesgue.positive_integral_monotone_convergence(2))
+    apply(rule lebesgue.borel_measurable_simple_function[OF u(1)])
+    using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] by auto
+  hence "(\<lambda>i. real (lebesgue.simple_integral (u i))) ----> real ?i" apply-
+    apply(subst lim_Real[THEN sym]) prefer 3
+    apply(subst Real_real') defer apply(subst Real_real')
+    using u f_om int_om u_int_om by auto
+  note * = LIMSEQ_unique[OF this int(2)[unfolded integral_unique[OF u_int]]]
+  show ?thesis unfolding * by(rule integrable_integral[OF int(1)])
+qed
+
+lemma lebesgue_integral_has_integral:
+  fixes f::"'a::ordered_euclidean_space => real"
+  assumes f:"lebesgue.integrable f"
+  shows "(f has_integral (lebesgue.integral f)) UNIV"
+proof- let ?n = "\<lambda>x. - min (f x) 0" and ?p = "\<lambda>x. max (f x) 0"
+  have *:"f = (\<lambda>x. ?p x - ?n x)" apply rule by auto
+  note f = lebesgue.integrableD[OF f]
+  show ?thesis unfolding lebesgue.integral_def apply(subst *)
+  proof(rule has_integral_sub) case goal1
+    have *:"\<forall>x. Real (f x) \<noteq> \<omega>" by auto
+    note lebesgue.borel_measurable_Real[OF f(1)]
+    from positive_integral_has_integral[OF this f(2) *]
+    show ?case unfolding real_Real_max .
+  next case goal2
+    have *:"\<forall>x. Real (- f x) \<noteq> \<omega>" by auto
+    note lebesgue.borel_measurable_uminus[OF f(1)]
+    note lebesgue.borel_measurable_Real[OF this]
+    from positive_integral_has_integral[OF this f(3) *]
+    show ?case unfolding real_Real_max minus_min_eq_max by auto
+  qed
+qed
+
+lemma continuous_on_imp_borel_measurable:
+  fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
+  assumes "continuous_on UNIV f"
+  shows "f \<in> borel_measurable lebesgue"
+  apply(rule lebesgue.borel_measurableI)
+  using continuous_open_preimage[OF assms] unfolding vimage_def by auto
+
+lemma (in measure_space) integral_monotone_convergence_pos':
+  assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
+  and pos: "\<And>x i. 0 \<le> f i x"
+  and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
+  and ilim: "(\<lambda>i. integral (f i)) ----> x"
+  shows "integrable u \<and> integral u = x"
+  using integral_monotone_convergence_pos[OF assms] by auto
+
+definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where
+  "e2p x = (\<lambda>i\<in>{..<DIM('a)}. x$$i)"
+
+definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where
+  "p2e x = (\<chi>\<chi> i. x i)"
+
+lemma bij_euclidean_component:
+  "bij_betw (e2p::'a::ordered_euclidean_space \<Rightarrow> _) (UNIV :: 'a set)
+  ({..<DIM('a)} \<rightarrow>\<^isub>E (UNIV :: real set))"
+  unfolding bij_betw_def e2p_def_raw
+proof let ?e = "\<lambda>x.\<lambda>i\<in>{..<DIM('a::ordered_euclidean_space)}. (x::'a)$$i"
+  show "inj ?e" unfolding inj_on_def restrict_def apply(subst euclidean_eq) apply safe
+    apply(drule_tac x=i in fun_cong) by auto
+  { fix x::"nat \<Rightarrow> real" assume x:"\<forall>i. i \<notin> {..<DIM('a)} \<longrightarrow> x i = undefined"
+    hence "x = ?e (\<chi>\<chi> i. x i)" apply-apply(rule,case_tac "xa<DIM('a)") by auto
+    hence "x \<in> range ?e" by fastsimp
+  } thus "range ?e = ({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)}"
+    unfolding extensional_def using DIM_positive by auto
+qed
+
+lemma bij_p2e:
+  "bij_betw (p2e::_ \<Rightarrow> 'a::ordered_euclidean_space) ({..<DIM('a)} \<rightarrow>\<^isub>E (UNIV :: real set))
+  (UNIV :: 'a set)" (is "bij_betw ?p ?U _")
+  unfolding bij_betw_def
+proof show "inj_on ?p ?U" unfolding inj_on_def p2e_def
+    apply(subst euclidean_eq) apply(safe,rule) unfolding extensional_def
+    apply(case_tac "xa<DIM('a)") by auto
+  { fix x::'a have "x \<in> ?p ` extensional {..<DIM('a)}"
+      unfolding image_iff apply(rule_tac x="\<lambda>i. if i<DIM('a) then x$$i else undefined" in bexI)
+      apply(subst euclidean_eq,safe) unfolding p2e_def extensional_def by auto
+  } thus "?p ` ?U = UNIV" by auto
+qed
+
+lemma e2p_p2e[simp]: fixes z::"'a::ordered_euclidean_space"
+  assumes "x \<in> extensional {..<DIM('a)}"
+  shows "e2p (p2e x::'a) = x"
+proof fix i::nat
+  show "e2p (p2e x::'a) i = x i" unfolding e2p_def p2e_def restrict_def
+    using assms unfolding extensional_def by auto
+qed
+
+lemma p2e_e2p[simp]: fixes x::"'a::ordered_euclidean_space"
+  shows "p2e (e2p x) = x"
+  apply(subst euclidean_eq) unfolding e2p_def p2e_def restrict_def by auto
+
+interpretation borel_product: product_sigma_finite "\<lambda>x. borel::real algebra" "\<lambda>x. lmeasure"
+  by default
+
+lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)"
+  unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto
+
+lemma borel_vimage_algebra_eq:
+  "sigma_algebra.vimage_algebra
+         (borel :: ('a::ordered_euclidean_space) algebra) ({..<DIM('a)} \<rightarrow>\<^isub>E UNIV) p2e =
+  sigma (product_algebra (\<lambda>x. \<lparr> space = UNIV::real set, sets = range (\<lambda>a. {..<a}) \<rparr>) {..<DIM('a)} )"
+proof- note bor = borel_eq_lessThan
+  def F \<equiv> "product_algebra (\<lambda>x. \<lparr> space = UNIV::real set, sets = range (\<lambda>a. {..<a}) \<rparr>) {..<DIM('a)}"
+  def E \<equiv> "\<lparr>space = (UNIV::'a set), sets = range lessThan\<rparr>"
+  have *:"(({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)}) = space F" unfolding F_def by auto
+  show ?thesis unfolding F_def[symmetric] * bor
+  proof(rule vimage_algebra_sigma,unfold E_def[symmetric])
+    show "sets E \<subseteq> Pow (space E)" "p2e \<in> space F \<rightarrow> space E" unfolding E_def by auto
+  next fix A assume "A \<in> sets F"
+    hence A:"A \<in> (Pi\<^isub>E {..<DIM('a)}) ` ({..<DIM('a)} \<rightarrow> range lessThan)"
+      unfolding F_def product_algebra_def algebra.simps .
+    then guess B unfolding image_iff .. note B=this
+    hence "\<forall>x<DIM('a). B x \<in> range lessThan" by auto
+    hence "\<forall>x. \<exists>xa. x<DIM('a) \<longrightarrow> B x = {..<xa}" unfolding image_iff by auto
+    from choice[OF this] guess b .. note b=this
+    hence b':"\<forall>i<DIM('a). Sup (B i) = b i" using Sup_lessThan by auto
+
+    show "A \<in> (\<lambda>X. p2e -` X \<inter> space F) ` sets E" unfolding image_iff B
+    proof(rule_tac x="{..< \<chi>\<chi> i. Sup (B i)}" in bexI)
+      show "Pi\<^isub>E {..<DIM('a)} B = p2e -` {..<(\<chi>\<chi> i. Sup (B i))::'a} \<inter> space F"
+        unfolding F_def E_def product_algebra_def algebra.simps
+      proof(rule,unfold subset_eq,rule_tac[!] ballI)
+        fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} B"
+        hence *:"\<forall>i<DIM('a). x i < b i" "\<forall>i\<ge>DIM('a). x i = undefined"
+          unfolding Pi_def extensional_def using b by auto
+        have "(p2e x::'a) < (\<chi>\<chi> i. Sup (B i))" unfolding less_prod_def eucl_less[of "p2e x"]
+          apply safe unfolding euclidean_lambda_beta b'[rule_format] p2e_def using * by auto
+        moreover have "x \<in> extensional {..<DIM('a)}"
+          using *(2) unfolding extensional_def by auto
+        ultimately show "x \<in> p2e -` {..<(\<chi>\<chi> i. Sup (B i)) ::'a} \<inter>
+          (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})" by auto
+      next fix x assume as:"x \<in> p2e -` {..<(\<chi>\<chi> i. Sup (B i))::'a} \<inter>
+          (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
+        hence "p2e x < ((\<chi>\<chi> i. Sup (B i))::'a)" by auto
+        hence "\<forall>i<DIM('a). x i \<in> B i" apply-apply(subst(asm) eucl_less)
+          unfolding p2e_def using b b' by auto
+        thus "x \<in> Pi\<^isub>E {..<DIM('a)} B" using as unfolding Pi_def extensional_def by auto
+      qed
+      show "{..<(\<chi>\<chi> i. Sup (B i))::'a} \<in> sets E" unfolding E_def algebra.simps by auto
+    qed
+  next fix A assume "A \<in> sets E"
+    then guess a unfolding E_def algebra.simps image_iff .. note a = this(2)
+    def B \<equiv> "\<lambda>i. {..<a $$ i}"
+    show "p2e -` A \<inter> space F \<in> sets F" unfolding F_def
+      unfolding product_algebra_def algebra.simps image_iff
+      apply(rule_tac x=B in bexI) apply rule unfolding subset_eq apply(rule_tac[1-2] ballI)
+    proof- show "B \<in> {..<DIM('a)} \<rightarrow> range lessThan" unfolding B_def by auto
+      fix x assume as:"x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
+      hence "p2e x \<in> A" by auto
+      hence "\<forall>i<DIM('a). x i \<in> B i" unfolding B_def a lessThan_iff
+        apply-apply(subst (asm) eucl_less) unfolding p2e_def by auto
+      thus "x \<in> Pi\<^isub>E {..<DIM('a)} B" using as unfolding Pi_def extensional_def by auto
+    next fix x assume x:"x \<in> Pi\<^isub>E {..<DIM('a)} B"
+      moreover have "p2e x \<in> A" unfolding a lessThan_iff p2e_def apply(subst eucl_less)
+        using x unfolding Pi_def extensional_def B_def by auto
+      ultimately show "x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})" by auto
+    qed
+  qed
+qed
+
+lemma Real_mult_nonneg: assumes "x \<ge> 0" "y \<ge> 0"
+  shows "Real (x * y) = Real x * Real y" using assms by auto
+
+lemma Real_setprod: assumes "\<forall>x\<in>A. f x \<ge> 0" shows "Real (setprod f A) = setprod (\<lambda>x. Real (f x)) A"
+proof(cases "finite A")
+  case True thus ?thesis using assms
+  proof(induct A) case (insert x A)
+    have "0 \<le> setprod f A" apply(rule setprod_nonneg) using insert by auto
+    thus ?case unfolding setprod_insert[OF insert(1-2)] apply-
+      apply(subst Real_mult_nonneg) prefer 3 apply(subst insert(3)[THEN sym])
+      using insert by auto
+  qed auto
+qed auto
+
+lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
+  apply(rule image_Int[THEN sym]) using bij_euclidean_component
+  unfolding bij_betw_def by auto
+
+lemma Int_stable_cuboids: fixes x::"'a::ordered_euclidean_space"
+  shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). e2p ` {a..b})\<rparr>"
+  unfolding Int_stable_def algebra.select_convs
+proof safe fix a b x y::'a
+  have *:"e2p ` {a..b} \<inter> e2p ` {x..y} =
+    (\<lambda>(a, b). e2p ` {a..b}) (\<chi>\<chi> i. max (a $$ i) (x $$ i), \<chi>\<chi> i. min (b $$ i) (y $$ i)::'a)"
+    unfolding e2p_Int inter_interval by auto
+  show "e2p ` {a..b} \<inter> e2p ` {x..y} \<in> range (\<lambda>(a, b). e2p ` {a..b::'a})" unfolding *
+    apply(rule range_eqI) ..
+qed
+
+lemma Int_stable_cuboids': fixes x::"'a::ordered_euclidean_space"
+  shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>"
+  unfolding Int_stable_def algebra.select_convs
+  apply safe unfolding inter_interval by auto
+
+lemma product_borel_eq_vimage:
+  "sigma (product_algebra (\<lambda>x. borel) {..<DIM('a::ordered_euclidean_space)}) =
+  sigma_algebra.vimage_algebra borel (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})
+  (p2e:: _ \<Rightarrow> 'a::ordered_euclidean_space)"
+  unfolding borel_vimage_algebra_eq unfolding borel_eq_lessThan
+  apply(subst sigma_product_algebra_sigma_eq[where S="\<lambda>i. \<lambda>n. lessThan (real n)"])
+  unfolding lessThan_iff
+proof- fix i assume i:"i<DIM('a)"
+  show "(\<lambda>n. {..<real n}) \<up> space \<lparr>space = UNIV, sets = range lessThan\<rparr>"
+    by(auto intro!:real_arch_lt isotoneI)
+qed auto
+
+lemma inj_on_disjoint_family_on: assumes "disjoint_family_on A S" "inj f"
+  shows "disjoint_family_on (\<lambda>x. f ` A x) S"
+  unfolding disjoint_family_on_def
+proof(rule,rule,rule)
+  fix x1 x2 assume x:"x1 \<in> S" "x2 \<in> S" "x1 \<noteq> x2"
+  show "f ` A x1 \<inter> f ` A x2 = {}"
+  proof(rule ccontr) case goal1
+    then obtain z where z:"z \<in> f ` A x1 \<inter> f ` A x2" by auto
+    then obtain z1 z2 where z12:"z1 \<in> A x1" "z2 \<in> A x2" "f z1 = z" "f z2 = z" by auto
+    hence "z1 = z2" using assms(2) unfolding inj_on_def by blast
+    hence "x1 = x2" using z12(1-2) using assms[unfolded disjoint_family_on_def] using x by auto
+    thus False using x(3) by auto
+  qed
+qed
+
+declare restrict_extensional[intro]
+
+lemma e2p_extensional[intro]:"e2p (y::'a::ordered_euclidean_space) \<in> extensional {..<DIM('a)}"
+  unfolding e2p_def by auto
+
+lemma e2p_image_vimage: fixes A::"'a::ordered_euclidean_space set"
+  shows "e2p ` A = p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
+proof(rule set_eqI,rule)
+  fix x assume "x \<in> e2p ` A" then guess y unfolding image_iff .. note y=this
+  show "x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
+    apply safe apply(rule vimageI[OF _ y(1)]) unfolding y p2e_e2p by auto
+next fix x assume "x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
+  thus "x \<in> e2p ` A" unfolding image_iff apply(rule_tac x="p2e x" in bexI) apply(subst e2p_p2e) by auto
+qed
+
+lemma lmeasure_measure_eq_borel_prod:
+  fixes A :: "('a::ordered_euclidean_space) set"
+  assumes "A \<in> sets borel"
+  shows "lmeasure A = borel_product.product_measure {..<DIM('a)} (e2p ` A :: (nat \<Rightarrow> real) set)"
+proof (rule measure_unique_Int_stable[where X=A and A=cube])
+  interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
+  show "Int_stable \<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
+    (is "Int_stable ?E" ) using Int_stable_cuboids' .
+  show "borel = sigma ?E" using borel_eq_atLeastAtMost .
+  show "\<And>i. lmeasure (cube i) \<noteq> \<omega>" unfolding lmeasure_cube by auto
+  show "\<And>X. X \<in> sets ?E \<Longrightarrow>
+    lmeasure X = borel_product.product_measure {..<DIM('a)} (e2p ` X :: (nat \<Rightarrow> real) set)"
+  proof- case goal1 then obtain a b where X:"X = {a..b}" by auto
+    { presume *:"X \<noteq> {} \<Longrightarrow> ?case"
+      show ?case apply(cases,rule *,assumption) by auto }
+    def XX \<equiv> "\<lambda>i. {a $$ i .. b $$ i}" assume  "X \<noteq> {}"  note X' = this[unfolded X interval_ne_empty]
+    have *:"Pi\<^isub>E {..<DIM('a)} XX = e2p ` X" apply(rule set_eqI)
+    proof fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} XX"
+      thus "x \<in> e2p ` X" unfolding image_iff apply(rule_tac x="\<chi>\<chi> i. x i" in bexI)
+        unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by rule auto
+    next fix x assume "x \<in> e2p ` X" then guess y unfolding image_iff .. note y = this
+      show "x \<in> Pi\<^isub>E {..<DIM('a)} XX" unfolding y using y(1)
+        unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by auto
+    qed
+    have "lmeasure X = (\<Prod>x<DIM('a). Real (b $$ x - a $$ x))"  using X' apply- unfolding X
+      unfolding lmeasure_atLeastAtMost content_closed_interval apply(subst Real_setprod) by auto
+    also have "... = (\<Prod>i<DIM('a). lmeasure (XX i))" apply(rule setprod_cong2)
+      unfolding XX_def lmeasure_atLeastAtMost apply(subst content_real) using X' by auto
+    also have "... = borel_product.product_measure {..<DIM('a)} (e2p ` X)" unfolding *[THEN sym]
+      apply(rule fprod.measure_times[THEN sym]) unfolding XX_def by auto
+    finally show ?case .
+  qed
+
+  show "range cube \<subseteq> sets \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
+    unfolding cube_def_raw by auto
+  have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastsimp
+  thus "cube \<up> space \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
+    apply-apply(rule isotoneI) apply(rule cube_subset_Suc) by auto
+  show "A \<in> sets borel " by fact
+  show "measure_space borel lmeasure" by default
+  show "measure_space borel
+     (\<lambda>a::'a set. finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` a))"
+    apply default unfolding countably_additive_def
+  proof safe fix A::"nat \<Rightarrow> 'a set" assume A:"range A \<subseteq> sets borel" "disjoint_family A"
+      "(\<Union>i. A i) \<in> sets borel"
+    note fprod.ca[unfolded countably_additive_def,rule_format]
+    note ca = this[of "\<lambda> n. e2p ` (A n)"]
+    show "(\<Sum>\<^isub>\<infinity>n. finite_product_sigma_finite.measure
+        (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` A n)) =
+           finite_product_sigma_finite.measure (\<lambda>x. borel)
+            (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` (\<Union>i. A i))" unfolding image_UN
+    proof(rule ca) show "range (\<lambda>n. e2p ` A n) \<subseteq> sets
+       (sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))"
+        unfolding product_borel_eq_vimage
+      proof case goal1
+        then guess y unfolding image_iff .. note y=this(2)
+        show ?case unfolding borel.in_vimage_algebra y apply-
+          apply(rule_tac x="A y" in bexI,rule e2p_image_vimage)
+          using A(1) by auto
+      qed
+
+      show "disjoint_family (\<lambda>n. e2p ` A n)" apply(rule inj_on_disjoint_family_on)
+        using bij_euclidean_component using A(2) unfolding bij_betw_def by auto
+      show "(\<Union>n. e2p ` A n) \<in> sets (sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))"
+        unfolding product_borel_eq_vimage borel.in_vimage_algebra
+      proof(rule bexI[OF _ A(3)],rule set_eqI,rule)
+        fix x assume x:"x \<in> (\<Union>n. e2p ` A n)" hence "p2e x \<in> (\<Union>i. A i)" by auto
+        moreover have "x \<in> extensional {..<DIM('a)}"
+          using x unfolding extensional_def e2p_def_raw by auto
+        ultimately show "x \<in> p2e -` (\<Union>i. A i) \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter>
+          extensional {..<DIM('a)})" by auto
+      next fix x assume x:"x \<in> p2e -` (\<Union>i. A i) \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter>
+          extensional {..<DIM('a)})"
+        hence "p2e x \<in> (\<Union>i. A i)" by auto
+        hence "\<exists>n. x \<in> e2p ` A n" apply safe apply(rule_tac x=i in exI)
+          unfolding image_iff apply(rule_tac x="p2e x" in bexI)
+          apply(subst e2p_p2e) using x by auto
+        thus "x \<in> (\<Union>n. e2p ` A n)" by auto
+      qed
+    qed
+  qed auto
+qed
+
+lemma e2p_p2e'[simp]: fixes x::"'a::ordered_euclidean_space"
+  assumes "A \<subseteq> extensional {..<DIM('a)}"
+  shows "e2p ` (p2e ` A ::'a set) = A"
+  apply(rule set_eqI) unfolding image_iff Bex_def apply safe defer
+  apply(rule_tac x="p2e x" in exI,safe) using assms by auto
+
+lemma range_p2e:"range (p2e::_\<Rightarrow>'a::ordered_euclidean_space) = UNIV"
+  apply safe defer unfolding image_iff apply(rule_tac x="\<lambda>i. x $$ i" in bexI)
+  unfolding p2e_def by auto
+
+lemma p2e_inv_extensional:"(A::'a::ordered_euclidean_space set)
+  = p2e ` (p2e -` A \<inter> extensional {..<DIM('a)})"
+  unfolding p2e_def_raw apply safe unfolding image_iff
+proof- fix x assume "x\<in>A"
+  let ?y = "\<lambda>i. if i<DIM('a) then x$$i else undefined"
+  have *:"Chi ?y = x" apply(subst euclidean_eq) by auto
+  show "\<exists>xa\<in>Chi -` A \<inter> extensional {..<DIM('a)}. x = Chi xa" apply(rule_tac x="?y" in bexI)
+    apply(subst euclidean_eq) unfolding extensional_def using `x\<in>A` by(auto simp: *)
+qed
+
+lemma borel_fubini_positiv_integral:
+  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pinfreal"
+  assumes f: "f \<in> borel_measurable borel"
+  shows "borel.positive_integral f =
+          borel_product.product_positive_integral {..<DIM('a)} (f \<circ> p2e)"
+proof- def U \<equiv> "(({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)}):: (nat \<Rightarrow> real) set"
+  interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
+  have "\<And>x. \<exists>i::nat. x < real i" by (metis real_arch_lt)
+  hence "(\<lambda>n::nat. {..<real n}) \<up> UNIV" apply-apply(rule isotoneI) by auto
+  hence *:"sigma_algebra.vimage_algebra borel U (p2e:: _ \<Rightarrow> 'a)
+    = sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)})"
+    unfolding U_def apply-apply(subst borel_vimage_algebra_eq)
+    apply(subst sigma_product_algebra_sigma_eq[where S="\<lambda>x. \<lambda>n. {..<(\<chi>\<chi> i. real n)}", THEN sym])
+    unfolding borel_eq_lessThan[THEN sym] by auto
+  show ?thesis unfolding borel.positive_integral_vimage[unfolded space_borel,OF bij_p2e]
+    apply(subst fprod.positive_integral_cong_measure[THEN sym, of "\<lambda>A. lmeasure (p2e ` A)"])
+    unfolding U_def[symmetric] *[THEN sym] o_def
+  proof- fix A assume A:"A \<in> sets (sigma_algebra.vimage_algebra borel U (p2e ::_ \<Rightarrow> 'a))"
+    hence *:"A \<subseteq> extensional {..<DIM('a)}" unfolding U_def by auto
+    from A guess B unfolding borel.in_vimage_algebra U_def .. note B=this
+    have "(p2e ` A::'a set) \<in> sets borel" unfolding B apply(subst Int_left_commute)
+      apply(subst Int_absorb1) unfolding p2e_inv_extensional[of B,THEN sym] using B(1) by auto
+    from lmeasure_measure_eq_borel_prod[OF this] show "lmeasure (p2e ` A::'a set) =
+      finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} A"
+      unfolding e2p_p2e'[OF *] .
+  qed auto
+qed
+
+lemma borel_fubini:
+  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
+  assumes f: "f \<in> borel_measurable borel"
+  shows "borel.integral f = borel_product.product_integral {..<DIM('a)} (f \<circ> p2e)"
+proof- interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
+  have 1:"(\<lambda>x. Real (f x)) \<in> borel_measurable borel" using f by auto
+  have 2:"(\<lambda>x. Real (- f x)) \<in> borel_measurable borel" using f by auto
+  show ?thesis unfolding fprod.integral_def borel.integral_def
+    unfolding borel_fubini_positiv_integral[OF 1] borel_fubini_positiv_integral[OF 2]
+    unfolding o_def ..
 qed
 
 end