src/HOL/Probability/Lebesgue_Integration.thy
author hoelzl
Tue Mar 22 18:53:05 2011 +0100 (2011-03-22)
changeset 42066 6db76c88907a
parent 41981 cdf7693bbe08
child 42067 66c8281349ec
permissions -rw-r--r--
generalized Caratheodory from algebra to ring_of_sets
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(* Author: Armin Heller, Johannes Hoelzl, TU Muenchen *)
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header {*Lebesgue Integration*}
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theory Lebesgue_Integration
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imports Measure Borel_Space
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begin
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lemma extreal_indicator_pos[simp,intro]: "0 \<le> (indicator A x ::extreal)"
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  unfolding indicator_def by auto
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lemma tendsto_real_max:
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  fixes x y :: real
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  assumes "(X ---> x) net"
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  assumes "(Y ---> y) net"
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  shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
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proof -
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  have *: "\<And>x y :: real. max x y = y + ((x - y) + norm (x - y)) / 2"
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    by (auto split: split_max simp: field_simps)
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  show ?thesis
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    unfolding *
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    by (intro tendsto_add assms tendsto_divide tendsto_norm tendsto_diff) auto
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qed
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lemma (in measure_space) measure_Union:
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  assumes "finite S" "S \<subseteq> sets M" "\<And>A B. A \<in> S \<Longrightarrow> B \<in> S \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}"
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  shows "setsum \<mu> S = \<mu> (\<Union>S)"
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proof -
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  have "setsum \<mu> S = \<mu> (\<Union>i\<in>S. i)"
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    using assms by (intro measure_setsum[OF `finite S`]) (auto simp: disjoint_family_on_def)
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  also have "\<dots> = \<mu> (\<Union>S)" by (auto intro!: arg_cong[where f=\<mu>])
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  finally show ?thesis .
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qed
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lemma (in sigma_algebra) measurable_sets2[intro]:
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  assumes "f \<in> measurable M M'" "g \<in> measurable M M''"
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  and "A \<in> sets M'" "B \<in> sets M''"
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  shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
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proof -
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  have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
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    by auto
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  then show ?thesis using assms by (auto intro: measurable_sets)
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qed
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lemma incseq_extreal: "incseq f \<Longrightarrow> incseq (\<lambda>x. extreal (f x))"
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  unfolding incseq_def by auto
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lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
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proof
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  assume "\<forall>n. f n \<le> f (Suc n)" then show "incseq f" by (auto intro!: incseq_SucI)
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qed (auto simp: incseq_def)
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lemma borel_measurable_real_floor:
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  "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
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  unfolding borel.borel_measurable_iff_ge
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proof (intro allI)
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  fix a :: real
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  { fix x have "a \<le> real \<lfloor>x\<rfloor> \<longleftrightarrow> real \<lceil>a\<rceil> \<le> x"
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      using le_floor_eq[of "\<lceil>a\<rceil>" x] ceiling_le_iff[of a "\<lfloor>x\<rfloor>"]
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      unfolding real_eq_of_int by simp }
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  then have "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} = {real \<lceil>a\<rceil>..}" by auto
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  then show "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} \<in> sets borel" by auto
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qed
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lemma measure_preservingD2:
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  "f \<in> measure_preserving A B \<Longrightarrow> f \<in> measurable A B"
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  unfolding measure_preserving_def by auto
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lemma measure_preservingD3:
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  "f \<in> measure_preserving A B \<Longrightarrow> f \<in> space A \<rightarrow> space B"
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  unfolding measure_preserving_def measurable_def by auto
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lemma measure_preservingD:
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  "T \<in> measure_preserving A B \<Longrightarrow> X \<in> sets B \<Longrightarrow> measure A (T -` X \<inter> space A) = measure B X"
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  unfolding measure_preserving_def by auto
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lemma (in sigma_algebra) borel_measurable_real_natfloor[intro, simp]:
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  assumes "f \<in> borel_measurable M"
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  shows "(\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M"
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proof -
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  have "\<And>x. real (natfloor (f x)) = max 0 (real \<lfloor>f x\<rfloor>)"
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    by (auto simp: max_def natfloor_def)
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  with borel_measurable_max[OF measurable_comp[OF assms borel_measurable_real_floor] borel_measurable_const]
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  show ?thesis by (simp add: comp_def)
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qed
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lemma (in measure_space) AE_not_in:
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  assumes N: "N \<in> null_sets" shows "AE x. x \<notin> N"
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  using N by (rule AE_I') auto
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lemma sums_If_finite:
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  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
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  assumes finite: "finite {r. P r}"
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  shows "(\<lambda>r. if P r then f r else 0) sums (\<Sum>r\<in>{r. P r}. f r)" (is "?F sums _")
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proof cases
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  assume "{r. P r} = {}" hence "\<And>r. \<not> P r" by auto
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  thus ?thesis by (simp add: sums_zero)
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next
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  assume not_empty: "{r. P r} \<noteq> {}"
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  have "?F sums (\<Sum>r = 0..< Suc (Max {r. P r}). ?F r)"
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    by (rule series_zero)
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       (auto simp add: Max_less_iff[OF finite not_empty] less_eq_Suc_le[symmetric])
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  also have "(\<Sum>r = 0..< Suc (Max {r. P r}). ?F r) = (\<Sum>r\<in>{r. P r}. f r)"
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    by (subst setsum_cases)
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       (auto intro!: setsum_cong simp: Max_ge_iff[OF finite not_empty] less_Suc_eq_le)
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  finally show ?thesis .
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qed
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lemma sums_single:
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  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
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  shows "(\<lambda>r. if r = i then f r else 0) sums f i"
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  using sums_If_finite[of "\<lambda>r. r = i" f] by simp
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section "Simple function"
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text {*
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Our simple functions are not restricted to positive real numbers. Instead
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they are just functions with a finite range and are measurable when singleton
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sets are measurable.
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*}
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definition "simple_function M g \<longleftrightarrow>
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    finite (g ` space M) \<and>
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    (\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
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lemma (in sigma_algebra) simple_functionD:
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  assumes "simple_function M g"
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  shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
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proof -
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  show "finite (g ` space M)"
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    using assms unfolding simple_function_def by auto
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  have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
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  also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
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  finally show "g -` X \<inter> space M \<in> sets M" using assms
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    by (auto intro!: finite_UN simp del: UN_simps simp: simple_function_def)
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qed
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lemma (in sigma_algebra) simple_function_measurable2[intro]:
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  assumes "simple_function M f" "simple_function M g"
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  shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
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proof -
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  have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
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    by auto
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  then show ?thesis using assms[THEN simple_functionD(2)] by auto
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qed
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lemma (in sigma_algebra) simple_function_indicator_representation:
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  fixes f ::"'a \<Rightarrow> extreal"
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  assumes f: "simple_function M f" and x: "x \<in> space M"
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  shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
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  (is "?l = ?r")
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proof -
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  have "?r = (\<Sum>y \<in> f ` space M.
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    (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
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    by (auto intro!: setsum_cong2)
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  also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
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    using assms by (auto dest: simple_functionD simp: setsum_delta)
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  also have "... = f x" using x by (auto simp: indicator_def)
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  finally show ?thesis by auto
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qed
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lemma (in measure_space) simple_function_notspace:
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  "simple_function M (\<lambda>x. h x * indicator (- space M) x::extreal)" (is "simple_function M ?h")
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proof -
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  have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
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  hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
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  have "?h -` {0} \<inter> space M = space M" by auto
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  thus ?thesis unfolding simple_function_def by auto
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qed
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lemma (in sigma_algebra) simple_function_cong:
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  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
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  shows "simple_function M f \<longleftrightarrow> simple_function M g"
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proof -
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  have "f ` space M = g ` space M"
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    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
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    using assms by (auto intro!: image_eqI)
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  thus ?thesis unfolding simple_function_def using assms by simp
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qed
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lemma (in sigma_algebra) simple_function_cong_algebra:
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  assumes "sets N = sets M" "space N = space M"
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  shows "simple_function M f \<longleftrightarrow> simple_function N f"
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  unfolding simple_function_def assms ..
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lemma (in sigma_algebra) borel_measurable_simple_function:
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  assumes "simple_function M f"
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  shows "f \<in> borel_measurable M"
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proof (rule borel_measurableI)
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  fix S
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  let ?I = "f ` (f -` S \<inter> space M)"
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  have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto
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  have "finite ?I"
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    using assms unfolding simple_function_def
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    using finite_subset[of "f ` (f -` S \<inter> space M)" "f ` space M"] by auto
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  hence "?U \<in> sets M"
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    apply (rule finite_UN)
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    using assms unfolding simple_function_def by auto
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  thus "f -` S \<inter> space M \<in> sets M" unfolding * .
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qed
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lemma (in sigma_algebra) simple_function_borel_measurable:
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  fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
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  assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
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  shows "simple_function M f"
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  using assms unfolding simple_function_def
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  by (auto intro: borel_measurable_vimage)
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lemma (in sigma_algebra) simple_function_eq_borel_measurable:
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  fixes f :: "'a \<Rightarrow> extreal"
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  shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M"
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  using simple_function_borel_measurable[of f]
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    borel_measurable_simple_function[of f]
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  by (fastsimp simp: simple_function_def)
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lemma (in sigma_algebra) simple_function_const[intro, simp]:
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  "simple_function M (\<lambda>x. c)"
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  by (auto intro: finite_subset simp: simple_function_def)
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lemma (in sigma_algebra) simple_function_compose[intro, simp]:
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  assumes "simple_function M f"
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  shows "simple_function M (g \<circ> f)"
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  unfolding simple_function_def
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proof safe
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  show "finite ((g \<circ> f) ` space M)"
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    using assms unfolding simple_function_def by (auto simp: image_compose)
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next
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  fix x assume "x \<in> space M"
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  let ?G = "g -` {g (f x)} \<inter> (f`space M)"
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  have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
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    (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
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  show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
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    using assms unfolding simple_function_def *
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    by (rule_tac finite_UN) (auto intro!: finite_UN)
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qed
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lemma (in sigma_algebra) simple_function_indicator[intro, simp]:
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  assumes "A \<in> sets M"
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  shows "simple_function M (indicator A)"
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proof -
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  have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
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    by (auto simp: indicator_def)
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  hence "finite ?S" by (rule finite_subset) simp
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  moreover have "- A \<inter> space M = space M - A" by auto
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  ultimately show ?thesis unfolding simple_function_def
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    using assms by (auto simp: indicator_def_raw)
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qed
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lemma (in sigma_algebra) simple_function_Pair[intro, simp]:
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  assumes "simple_function M f"
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  assumes "simple_function M g"
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  shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
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  unfolding simple_function_def
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proof safe
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  show "finite (?p ` space M)"
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    using assms unfolding simple_function_def
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    by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
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next
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  fix x assume "x \<in> space M"
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  have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
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      (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
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    by auto
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  with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
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    using assms unfolding simple_function_def by auto
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qed
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lemma (in sigma_algebra) simple_function_compose1:
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  assumes "simple_function M f"
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  shows "simple_function M (\<lambda>x. g (f x))"
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  using simple_function_compose[OF assms, of g]
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  by (simp add: comp_def)
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lemma (in sigma_algebra) simple_function_compose2:
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  assumes "simple_function M f" and "simple_function M g"
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  shows "simple_function M (\<lambda>x. h (f x) (g x))"
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proof -
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   278
  have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
hoelzl@38656
   279
    using assms by auto
hoelzl@38656
   280
  thus ?thesis by (simp_all add: comp_def)
hoelzl@38656
   281
qed
hoelzl@35582
   282
hoelzl@38656
   283
lemmas (in sigma_algebra) simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
hoelzl@38656
   284
  and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
hoelzl@38656
   285
  and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
hoelzl@38656
   286
  and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
hoelzl@38656
   287
  and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
hoelzl@38656
   288
  and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
hoelzl@41981
   289
  and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
hoelzl@38656
   290
hoelzl@38656
   291
lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
hoelzl@41689
   292
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
hoelzl@41689
   293
  shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
hoelzl@38656
   294
proof cases
hoelzl@38656
   295
  assume "finite P" from this assms show ?thesis by induct auto
hoelzl@38656
   296
qed auto
hoelzl@35582
   297
hoelzl@41981
   298
lemma (in sigma_algebra)
hoelzl@41981
   299
  fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
hoelzl@41981
   300
  shows simple_function_extreal[intro, simp]: "simple_function M (\<lambda>x. extreal (f x))"
hoelzl@41981
   301
  by (auto intro!: simple_function_compose1[OF sf])
hoelzl@41981
   302
hoelzl@41981
   303
lemma (in sigma_algebra)
hoelzl@41981
   304
  fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
hoelzl@41981
   305
  shows simple_function_real_of_nat[intro, simp]: "simple_function M (\<lambda>x. real (f x))"
hoelzl@41981
   306
  by (auto intro!: simple_function_compose1[OF sf])
hoelzl@35582
   307
hoelzl@38656
   308
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
hoelzl@41981
   309
  fixes u :: "'a \<Rightarrow> extreal"
hoelzl@38656
   310
  assumes u: "u \<in> borel_measurable M"
hoelzl@41981
   311
  shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
hoelzl@41981
   312
             (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
hoelzl@35582
   313
proof -
hoelzl@41981
   314
  def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)"
hoelzl@41981
   315
  { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
hoelzl@41981
   316
    proof (split split_if, intro conjI impI)
hoelzl@41981
   317
      assume "\<not> real j \<le> u x"
hoelzl@41981
   318
      then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
hoelzl@41981
   319
         by (cases "u x") (auto intro!: natfloor_mono simp: mult_nonneg_nonneg)
hoelzl@41981
   320
      moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
hoelzl@41981
   321
        by (intro real_natfloor_le) (auto simp: mult_nonneg_nonneg)
hoelzl@41981
   322
      ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
hoelzl@41981
   323
        unfolding real_of_nat_le_iff by auto
hoelzl@41981
   324
    qed auto }
hoelzl@38656
   325
  note f_upper = this
hoelzl@35582
   326
hoelzl@41981
   327
  have real_f:
hoelzl@41981
   328
    "\<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)))"
hoelzl@41981
   329
    unfolding f_def by auto
hoelzl@35582
   330
hoelzl@41981
   331
  let "?g j x" = "real (f x j) / 2^j :: extreal"
hoelzl@41981
   332
  show ?thesis
hoelzl@41981
   333
  proof (intro exI[of _ ?g] conjI allI ballI)
hoelzl@41981
   334
    fix i
hoelzl@41981
   335
    have "simple_function M (\<lambda>x. real (f x i))"
hoelzl@41981
   336
    proof (intro simple_function_borel_measurable)
hoelzl@41981
   337
      show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
hoelzl@41981
   338
        using u by (auto intro!: measurable_If simp: real_f)
hoelzl@41981
   339
      have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
hoelzl@41981
   340
        using f_upper[of _ i] by auto
hoelzl@41981
   341
      then show "finite ((\<lambda>x. real (f x i))`space M)"
hoelzl@41981
   342
        by (rule finite_subset) auto
hoelzl@41981
   343
    qed
hoelzl@41981
   344
    then show "simple_function M (?g i)"
hoelzl@41981
   345
      by (auto intro: simple_function_extreal simple_function_div)
hoelzl@41981
   346
  next
hoelzl@41981
   347
    show "incseq ?g"
hoelzl@41981
   348
    proof (intro incseq_extreal incseq_SucI le_funI)
hoelzl@41981
   349
      fix x and i :: nat
hoelzl@41981
   350
      have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
hoelzl@41981
   351
      proof ((split split_if)+, intro conjI impI)
hoelzl@41981
   352
        assume "extreal (real i) \<le> u x" "\<not> extreal (real (Suc i)) \<le> u x"
hoelzl@41981
   353
        then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)"
hoelzl@41981
   354
          by (cases "u x") (auto intro!: le_natfloor)
hoelzl@38656
   355
      next
hoelzl@41981
   356
        assume "\<not> extreal (real i) \<le> u x" "extreal (real (Suc i)) \<le> u x"
hoelzl@41981
   357
        then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i"
hoelzl@41981
   358
          by (cases "u x") auto
hoelzl@41981
   359
      next
hoelzl@41981
   360
        assume "\<not> extreal (real i) \<le> u x" "\<not> extreal (real (Suc i)) \<le> u x"
hoelzl@41981
   361
        have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
hoelzl@41981
   362
          by simp
hoelzl@41981
   363
        also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
hoelzl@41981
   364
        proof cases
hoelzl@41981
   365
          assume "0 \<le> u x" then show ?thesis
hoelzl@41981
   366
            by (intro le_mult_natfloor) (cases "u x", auto intro!: mult_nonneg_nonneg)
hoelzl@41981
   367
        next
hoelzl@41981
   368
          assume "\<not> 0 \<le> u x" then show ?thesis
hoelzl@41981
   369
            by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
hoelzl@38656
   370
        qed
hoelzl@41981
   371
        also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
hoelzl@41981
   372
          by (simp add: ac_simps)
hoelzl@41981
   373
        finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" .
hoelzl@41981
   374
      qed simp
hoelzl@41981
   375
      then show "?g i x \<le> ?g (Suc i) x"
hoelzl@41981
   376
        by (auto simp: field_simps)
hoelzl@35582
   377
    qed
hoelzl@38656
   378
  next
hoelzl@41981
   379
    fix x show "(SUP i. ?g i x) = max 0 (u x)"
hoelzl@41981
   380
    proof (rule extreal_SUPI)
hoelzl@41981
   381
      fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
hoelzl@41981
   382
        by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
hoelzl@41981
   383
                                     mult_nonpos_nonneg mult_nonneg_nonneg)
hoelzl@41981
   384
    next
hoelzl@41981
   385
      fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
hoelzl@41981
   386
      have "\<And>i. 0 \<le> ?g i x" by (auto simp: divide_nonneg_pos)
hoelzl@41981
   387
      from order_trans[OF this *] have "0 \<le> y" by simp
hoelzl@41981
   388
      show "max 0 (u x) \<le> y"
hoelzl@41981
   389
      proof (cases y)
hoelzl@41981
   390
        case (real r)
hoelzl@41981
   391
        with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
hoelzl@41981
   392
        from real_arch_lt[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
hoelzl@41981
   393
        then have "\<exists>p. max 0 (u x) = extreal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
hoelzl@41981
   394
        then guess p .. note ux = this
hoelzl@41981
   395
        obtain m :: nat where m: "p < real m" using real_arch_lt ..
hoelzl@41981
   396
        have "p \<le> r"
hoelzl@41981
   397
        proof (rule ccontr)
hoelzl@41981
   398
          assume "\<not> p \<le> r"
hoelzl@41981
   399
          with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
hoelzl@41981
   400
          obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: inverse_eq_divide field_simps)
hoelzl@41981
   401
          then have "r * 2^max N m < p * 2^max N m - 1" by simp
hoelzl@41981
   402
          moreover
hoelzl@41981
   403
          have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m"
hoelzl@41981
   404
            using *[of "max N m"] m unfolding real_f using ux
hoelzl@41981
   405
            by (cases "0 \<le> u x") (simp_all add: max_def mult_nonneg_nonneg split: split_if_asm)
hoelzl@41981
   406
          then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
hoelzl@41981
   407
            by (metis real_natfloor_gt_diff_one less_le_trans)
hoelzl@41981
   408
          ultimately show False by auto
hoelzl@38656
   409
        qed
hoelzl@41981
   410
        then show "max 0 (u x) \<le> y" using real ux by simp
hoelzl@41981
   411
      qed (insert `0 \<le> y`, auto)
hoelzl@41981
   412
    qed
hoelzl@41981
   413
  qed (auto simp: divide_nonneg_pos)
hoelzl@41981
   414
qed
hoelzl@35582
   415
hoelzl@41981
   416
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
hoelzl@41981
   417
  fixes u :: "'a \<Rightarrow> extreal"
hoelzl@41981
   418
  assumes u: "u \<in> borel_measurable M"
hoelzl@41981
   419
  obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
hoelzl@41981
   420
    "\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
hoelzl@41981
   421
  using borel_measurable_implies_simple_function_sequence[OF u] by auto
hoelzl@41981
   422
hoelzl@41981
   423
lemma (in sigma_algebra) simple_function_If_set:
hoelzl@41981
   424
  assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
hoelzl@41981
   425
  shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
hoelzl@41981
   426
proof -
hoelzl@41981
   427
  def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
hoelzl@41981
   428
  show ?thesis unfolding simple_function_def
hoelzl@41981
   429
  proof safe
hoelzl@41981
   430
    have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
hoelzl@41981
   431
    from finite_subset[OF this] assms
hoelzl@41981
   432
    show "finite (?IF ` space M)" unfolding simple_function_def by auto
hoelzl@41981
   433
  next
hoelzl@41981
   434
    fix x assume "x \<in> space M"
hoelzl@41981
   435
    then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
hoelzl@41981
   436
      then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
hoelzl@41981
   437
      else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
hoelzl@41981
   438
      using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
hoelzl@41981
   439
    have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
hoelzl@41981
   440
      unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
hoelzl@41981
   441
    show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
hoelzl@35582
   442
  qed
hoelzl@35582
   443
qed
hoelzl@35582
   444
hoelzl@41981
   445
lemma (in sigma_algebra) simple_function_If:
hoelzl@41981
   446
  assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
hoelzl@41981
   447
  shows "simple_function M (\<lambda>x. if P x then f x else g x)"
hoelzl@35582
   448
proof -
hoelzl@41981
   449
  have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
hoelzl@41981
   450
  with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
hoelzl@38656
   451
qed
hoelzl@38656
   452
hoelzl@39092
   453
lemma (in measure_space) simple_function_restricted:
hoelzl@41981
   454
  fixes f :: "'a \<Rightarrow> extreal" assumes "A \<in> sets M"
hoelzl@41689
   455
  shows "simple_function (restricted_space A) f \<longleftrightarrow> simple_function M (\<lambda>x. f x * indicator A x)"
hoelzl@41689
   456
    (is "simple_function ?R f \<longleftrightarrow> simple_function M ?f")
hoelzl@39092
   457
proof -
hoelzl@39092
   458
  interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
hoelzl@41981
   459
  have f: "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
hoelzl@39092
   460
  proof cases
hoelzl@39092
   461
    assume "A = space M"
hoelzl@39092
   462
    then have "f`A = ?f`space M" by (fastsimp simp: image_iff)
hoelzl@39092
   463
    then show ?thesis by simp
hoelzl@39092
   464
  next
hoelzl@39092
   465
    assume "A \<noteq> space M"
hoelzl@39092
   466
    then obtain x where x: "x \<in> space M" "x \<notin> A"
hoelzl@39092
   467
      using sets_into_space `A \<in> sets M` by auto
hoelzl@39092
   468
    have *: "?f`space M = f`A \<union> {0}"
hoelzl@39092
   469
    proof (auto simp add: image_iff)
hoelzl@39092
   470
      show "\<exists>x\<in>space M. f x = 0 \<or> indicator A x = 0"
hoelzl@39092
   471
        using x by (auto intro!: bexI[of _ x])
hoelzl@39092
   472
    next
hoelzl@39092
   473
      fix x assume "x \<in> A"
hoelzl@39092
   474
      then show "\<exists>y\<in>space M. f x = f y * indicator A y"
hoelzl@39092
   475
        using `A \<in> sets M` sets_into_space by (auto intro!: bexI[of _ x])
hoelzl@39092
   476
    next
hoelzl@39092
   477
      fix x
hoelzl@41981
   478
      assume "indicator A x \<noteq> (0::extreal)"
hoelzl@39092
   479
      then have "x \<in> A" by (auto simp: indicator_def split: split_if_asm)
hoelzl@39092
   480
      moreover assume "x \<in> space M" "\<forall>y\<in>A. ?f x \<noteq> f y"
hoelzl@39092
   481
      ultimately show "f x = 0" by auto
hoelzl@39092
   482
    qed
hoelzl@39092
   483
    then show ?thesis by auto
hoelzl@39092
   484
  qed
hoelzl@39092
   485
  then show ?thesis
hoelzl@39092
   486
    unfolding simple_function_eq_borel_measurable
hoelzl@39092
   487
      R.simple_function_eq_borel_measurable
hoelzl@39092
   488
    unfolding borel_measurable_restricted[OF `A \<in> sets M`]
hoelzl@41981
   489
    using assms(1)[THEN sets_into_space]
hoelzl@41981
   490
    by (auto simp: indicator_def)
hoelzl@39092
   491
qed
hoelzl@39092
   492
hoelzl@39092
   493
lemma (in sigma_algebra) simple_function_subalgebra:
hoelzl@41689
   494
  assumes "simple_function N f"
hoelzl@41689
   495
  and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
hoelzl@41689
   496
  shows "simple_function M f"
hoelzl@41689
   497
  using assms unfolding simple_function_def by auto
hoelzl@39092
   498
hoelzl@41661
   499
lemma (in measure_space) simple_function_vimage:
hoelzl@41661
   500
  assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
hoelzl@41689
   501
    and f: "simple_function M' f"
hoelzl@41689
   502
  shows "simple_function M (\<lambda>x. f (T x))"
hoelzl@41661
   503
proof (intro simple_function_def[THEN iffD2] conjI ballI)
hoelzl@41661
   504
  interpret T: sigma_algebra M' by fact
hoelzl@41661
   505
  have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
hoelzl@41661
   506
    using T unfolding measurable_def by auto
hoelzl@41661
   507
  then show "finite ((\<lambda>x. f (T x)) ` space M)"
hoelzl@41689
   508
    using f unfolding simple_function_def by (auto intro: finite_subset)
hoelzl@41661
   509
  fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
hoelzl@41661
   510
  then have "i \<in> f ` space M'"
hoelzl@41661
   511
    using T unfolding measurable_def by auto
hoelzl@41661
   512
  then have "f -` {i} \<inter> space M' \<in> sets M'"
hoelzl@41689
   513
    using f unfolding simple_function_def by auto
hoelzl@41661
   514
  then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
hoelzl@41661
   515
    using T unfolding measurable_def by auto
hoelzl@41661
   516
  also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
hoelzl@41661
   517
    using T unfolding measurable_def by auto
hoelzl@41661
   518
  finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" .
hoelzl@40859
   519
qed
hoelzl@40859
   520
hoelzl@38656
   521
section "Simple integral"
hoelzl@38656
   522
hoelzl@41689
   523
definition simple_integral_def:
hoelzl@41689
   524
  "integral\<^isup>S M f = (\<Sum>x \<in> f ` space M. x * measure M (f -` {x} \<inter> space M))"
hoelzl@41689
   525
hoelzl@41689
   526
syntax
hoelzl@41981
   527
  "_simple_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> extreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> extreal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
hoelzl@41689
   528
hoelzl@41689
   529
translations
hoelzl@41689
   530
  "\<integral>\<^isup>S x. f \<partial>M" == "CONST integral\<^isup>S M (%x. f)"
hoelzl@35582
   531
hoelzl@38656
   532
lemma (in measure_space) simple_integral_cong:
hoelzl@38656
   533
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
hoelzl@41689
   534
  shows "integral\<^isup>S M f = integral\<^isup>S M g"
hoelzl@38656
   535
proof -
hoelzl@38656
   536
  have "f ` space M = g ` space M"
hoelzl@38656
   537
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
hoelzl@38656
   538
    using assms by (auto intro!: image_eqI)
hoelzl@38656
   539
  thus ?thesis unfolding simple_integral_def by simp
hoelzl@38656
   540
qed
hoelzl@38656
   541
hoelzl@40859
   542
lemma (in measure_space) simple_integral_cong_measure:
hoelzl@41689
   543
  assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
hoelzl@41689
   544
    and "simple_function M f"
hoelzl@41689
   545
  shows "integral\<^isup>S N f = integral\<^isup>S M f"
hoelzl@40859
   546
proof -
hoelzl@41689
   547
  interpret v: measure_space N
hoelzl@41689
   548
    by (rule measure_space_cong) fact+
hoelzl@41689
   549
  from simple_functionD[OF `simple_function M f`] assms show ?thesis
hoelzl@41689
   550
    by (auto intro!: setsum_cong simp: simple_integral_def)
hoelzl@40859
   551
qed
hoelzl@40859
   552
hoelzl@38656
   553
lemma (in measure_space) simple_integral_const[simp]:
hoelzl@41689
   554
  "(\<integral>\<^isup>Sx. c \<partial>M) = c * \<mu> (space M)"
hoelzl@38656
   555
proof (cases "space M = {}")
hoelzl@38656
   556
  case True thus ?thesis unfolding simple_integral_def by simp
hoelzl@38656
   557
next
hoelzl@38656
   558
  case False hence "(\<lambda>x. c) ` space M = {c}" by auto
hoelzl@38656
   559
  thus ?thesis unfolding simple_integral_def by simp
hoelzl@35582
   560
qed
hoelzl@35582
   561
hoelzl@38656
   562
lemma (in measure_space) simple_function_partition:
hoelzl@41981
   563
  assumes f: "simple_function M f" and g: "simple_function M g"
hoelzl@41689
   564
  shows "integral\<^isup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * \<mu> A)"
hoelzl@38656
   565
    (is "_ = setsum _ (?p ` space M)")
hoelzl@38656
   566
proof-
hoelzl@38656
   567
  let "?sub x" = "?p ` (f -` {x} \<inter> space M)"
hoelzl@38656
   568
  let ?SIGMA = "Sigma (f`space M) ?sub"
hoelzl@35582
   569
hoelzl@38656
   570
  have [intro]:
hoelzl@38656
   571
    "finite (f ` space M)"
hoelzl@38656
   572
    "finite (g ` space M)"
hoelzl@38656
   573
    using assms unfolding simple_function_def by simp_all
hoelzl@38656
   574
hoelzl@38656
   575
  { fix A
hoelzl@38656
   576
    have "?p ` (A \<inter> space M) \<subseteq>
hoelzl@38656
   577
      (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
hoelzl@38656
   578
      by auto
hoelzl@38656
   579
    hence "finite (?p ` (A \<inter> space M))"
nipkow@40786
   580
      by (rule finite_subset) auto }
hoelzl@38656
   581
  note this[intro, simp]
hoelzl@41981
   582
  note sets = simple_function_measurable2[OF f g]
hoelzl@35582
   583
hoelzl@38656
   584
  { fix x assume "x \<in> space M"
hoelzl@38656
   585
    have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
hoelzl@41981
   586
    with sets have "\<mu> (f -` {f x} \<inter> space M) = setsum \<mu> (?sub (f x))"
hoelzl@41981
   587
      by (subst measure_Union) auto }
hoelzl@41689
   588
  hence "integral\<^isup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
hoelzl@41981
   589
    unfolding simple_integral_def using f sets
hoelzl@41981
   590
    by (subst setsum_Sigma[symmetric])
hoelzl@41981
   591
       (auto intro!: setsum_cong setsum_extreal_right_distrib)
haftmann@39910
   592
  also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * \<mu> A)"
hoelzl@38656
   593
  proof -
hoelzl@38656
   594
    have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
haftmann@39910
   595
    have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
hoelzl@38656
   596
      = (\<lambda>x. (f x, ?p x)) ` space M"
hoelzl@38656
   597
    proof safe
hoelzl@38656
   598
      fix x assume "x \<in> space M"
haftmann@39910
   599
      thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M"
hoelzl@38656
   600
        by (auto intro!: image_eqI[of _ _ "?p x"])
hoelzl@38656
   601
    qed auto
hoelzl@38656
   602
    thus ?thesis
haftmann@39910
   603
      apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI)
hoelzl@38656
   604
      apply (rule_tac x="xa" in image_eqI)
hoelzl@38656
   605
      by simp_all
hoelzl@38656
   606
  qed
hoelzl@38656
   607
  finally show ?thesis .
hoelzl@35582
   608
qed
hoelzl@35582
   609
hoelzl@38656
   610
lemma (in measure_space) simple_integral_add[simp]:
hoelzl@41981
   611
  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"
hoelzl@41689
   612
  shows "(\<integral>\<^isup>Sx. f x + g x \<partial>M) = integral\<^isup>S M f + integral\<^isup>S M g"
hoelzl@35582
   613
proof -
hoelzl@38656
   614
  { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
hoelzl@38656
   615
    assume "x \<in> space M"
hoelzl@38656
   616
    hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
hoelzl@38656
   617
        "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
hoelzl@38656
   618
      by auto }
hoelzl@41981
   619
  with assms show ?thesis
hoelzl@38656
   620
    unfolding
hoelzl@41981
   621
      simple_function_partition[OF simple_function_add[OF f g] simple_function_Pair[OF f g]]
hoelzl@41981
   622
      simple_function_partition[OF f g]
hoelzl@41981
   623
      simple_function_partition[OF g f]
hoelzl@41981
   624
    by (subst (3) Int_commute)
hoelzl@41981
   625
       (auto simp add: extreal_left_distrib setsum_addf[symmetric] intro!: setsum_cong)
hoelzl@35582
   626
qed
hoelzl@35582
   627
hoelzl@38656
   628
lemma (in measure_space) simple_integral_setsum[simp]:
hoelzl@41981
   629
  assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
hoelzl@41689
   630
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
hoelzl@41689
   631
  shows "(\<integral>\<^isup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>S M (f i))"
hoelzl@38656
   632
proof cases
hoelzl@38656
   633
  assume "finite P"
hoelzl@38656
   634
  from this assms show ?thesis
hoelzl@41981
   635
    by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
hoelzl@38656
   636
qed auto
hoelzl@38656
   637
hoelzl@38656
   638
lemma (in measure_space) simple_integral_mult[simp]:
hoelzl@41981
   639
  assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
hoelzl@41689
   640
  shows "(\<integral>\<^isup>Sx. c * f x \<partial>M) = c * integral\<^isup>S M f"
hoelzl@38656
   641
proof -
hoelzl@41981
   642
  note mult = simple_function_mult[OF simple_function_const[of c] f(1)]
hoelzl@38656
   643
  { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
hoelzl@38656
   644
    assume "x \<in> space M"
hoelzl@38656
   645
    hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
hoelzl@38656
   646
      by auto }
hoelzl@41981
   647
  with assms show ?thesis
hoelzl@41981
   648
    unfolding simple_function_partition[OF mult f(1)]
hoelzl@41981
   649
              simple_function_partition[OF f(1) mult]
hoelzl@41981
   650
    by (subst setsum_extreal_right_distrib)
hoelzl@41981
   651
       (auto intro!: extreal_0_le_mult setsum_cong simp: mult_assoc)
hoelzl@40871
   652
qed
hoelzl@40871
   653
hoelzl@40859
   654
lemma (in measure_space) simple_integral_mono_AE:
hoelzl@41981
   655
  assumes f: "simple_function M f" and g: "simple_function M g"
hoelzl@40859
   656
  and mono: "AE x. f x \<le> g x"
hoelzl@41689
   657
  shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
hoelzl@40859
   658
proof -
hoelzl@40859
   659
  let "?S x" = "(g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
hoelzl@40859
   660
  have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
hoelzl@40859
   661
    "\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
hoelzl@40859
   662
  show ?thesis
hoelzl@40859
   663
    unfolding *
hoelzl@41981
   664
      simple_function_partition[OF f g]
hoelzl@41981
   665
      simple_function_partition[OF g f]
hoelzl@40859
   666
  proof (safe intro!: setsum_mono)
hoelzl@40859
   667
    fix x assume "x \<in> space M"
hoelzl@40859
   668
    then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
hoelzl@40859
   669
    show "the_elem (f`?S x) * \<mu> (?S x) \<le> the_elem (g`?S x) * \<mu> (?S x)"
hoelzl@40859
   670
    proof (cases "f x \<le> g x")
hoelzl@41981
   671
      case True then show ?thesis
hoelzl@41981
   672
        using * assms(1,2)[THEN simple_functionD(2)]
hoelzl@41981
   673
        by (auto intro!: extreal_mult_right_mono)
hoelzl@40859
   674
    next
hoelzl@40859
   675
      case False
hoelzl@40859
   676
      obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
hoelzl@40859
   677
        using mono by (auto elim!: AE_E)
hoelzl@40859
   678
      have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
hoelzl@40871
   679
      moreover have "?S x \<in> sets M" using assms
hoelzl@40871
   680
        by (rule_tac Int) (auto intro!: simple_functionD)
hoelzl@40859
   681
      ultimately have "\<mu> (?S x) \<le> \<mu> N"
hoelzl@40859
   682
        using `N \<in> sets M` by (auto intro!: measure_mono)
hoelzl@41981
   683
      moreover have "0 \<le> \<mu> (?S x)"
hoelzl@41981
   684
        using assms(1,2)[THEN simple_functionD(2)] by auto
hoelzl@41981
   685
      ultimately have "\<mu> (?S x) = 0" using `\<mu> N = 0` by auto
hoelzl@41981
   686
      then show ?thesis by simp
hoelzl@40859
   687
    qed
hoelzl@40859
   688
  qed
hoelzl@40859
   689
qed
hoelzl@40859
   690
hoelzl@38656
   691
lemma (in measure_space) simple_integral_mono:
hoelzl@41689
   692
  assumes "simple_function M f" and "simple_function M g"
hoelzl@38656
   693
  and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
hoelzl@41689
   694
  shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
hoelzl@41705
   695
  using assms by (intro simple_integral_mono_AE) auto
hoelzl@35582
   696
hoelzl@40859
   697
lemma (in measure_space) simple_integral_cong_AE:
hoelzl@41981
   698
  assumes "simple_function M f" and "simple_function M g"
hoelzl@41981
   699
  and "AE x. f x = g x"
hoelzl@41689
   700
  shows "integral\<^isup>S M f = integral\<^isup>S M g"
hoelzl@40859
   701
  using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
hoelzl@40859
   702
hoelzl@40859
   703
lemma (in measure_space) simple_integral_cong':
hoelzl@41689
   704
  assumes sf: "simple_function M f" "simple_function M g"
hoelzl@40859
   705
  and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
hoelzl@41689
   706
  shows "integral\<^isup>S M f = integral\<^isup>S M g"
hoelzl@40859
   707
proof (intro simple_integral_cong_AE sf AE_I)
hoelzl@40859
   708
  show "\<mu> {x\<in>space M. f x \<noteq> g x} = 0" by fact
hoelzl@40859
   709
  show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
hoelzl@40859
   710
    using sf[THEN borel_measurable_simple_function] by auto
hoelzl@40859
   711
qed simp
hoelzl@40859
   712
hoelzl@38656
   713
lemma (in measure_space) simple_integral_indicator:
hoelzl@38656
   714
  assumes "A \<in> sets M"
hoelzl@41689
   715
  assumes "simple_function M f"
hoelzl@41689
   716
  shows "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
hoelzl@38656
   717
    (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
hoelzl@38656
   718
proof cases
hoelzl@38656
   719
  assume "A = space M"
hoelzl@41689
   720
  moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) = integral\<^isup>S M f"
hoelzl@38656
   721
    by (auto intro!: simple_integral_cong)
hoelzl@38656
   722
  moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto
hoelzl@38656
   723
  ultimately show ?thesis by (simp add: simple_integral_def)
hoelzl@38656
   724
next
hoelzl@38656
   725
  assume "A \<noteq> space M"
hoelzl@38656
   726
  then obtain x where x: "x \<in> space M" "x \<notin> A" using sets_into_space[OF assms(1)] by auto
hoelzl@38656
   727
  have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
hoelzl@35582
   728
  proof safe
hoelzl@38656
   729
    fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
hoelzl@38656
   730
  next
hoelzl@38656
   731
    fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
hoelzl@38656
   732
      using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
hoelzl@38656
   733
  next
hoelzl@38656
   734
    show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
hoelzl@35582
   735
  qed
hoelzl@41689
   736
  have *: "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
hoelzl@38656
   737
    (\<Sum>x \<in> f ` space M \<union> {0}. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
hoelzl@38656
   738
    unfolding simple_integral_def I
hoelzl@38656
   739
  proof (rule setsum_mono_zero_cong_left)
hoelzl@38656
   740
    show "finite (f ` space M \<union> {0})"
hoelzl@38656
   741
      using assms(2) unfolding simple_function_def by auto
hoelzl@38656
   742
    show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
hoelzl@38656
   743
      using sets_into_space[OF assms(1)] by auto
hoelzl@40859
   744
    have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
hoelzl@40859
   745
      by (auto simp: image_iff)
hoelzl@38656
   746
    thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
hoelzl@38656
   747
      i * \<mu> (f -` {i} \<inter> space M \<inter> A) = 0" by auto
hoelzl@38656
   748
  next
hoelzl@38656
   749
    fix x assume "x \<in> f`A \<union> {0}"
hoelzl@38656
   750
    hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
hoelzl@38656
   751
      by (auto simp: indicator_def split: split_if_asm)
hoelzl@38656
   752
    thus "x * \<mu> (?I -` {x} \<inter> space M) =
hoelzl@38656
   753
      x * \<mu> (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
hoelzl@38656
   754
  qed
hoelzl@38656
   755
  show ?thesis unfolding *
hoelzl@38656
   756
    using assms(2) unfolding simple_function_def
hoelzl@38656
   757
    by (auto intro!: setsum_mono_zero_cong_right)
hoelzl@38656
   758
qed
hoelzl@35582
   759
hoelzl@38656
   760
lemma (in measure_space) simple_integral_indicator_only[simp]:
hoelzl@38656
   761
  assumes "A \<in> sets M"
hoelzl@41689
   762
  shows "integral\<^isup>S M (indicator A) = \<mu> A"
hoelzl@38656
   763
proof cases
hoelzl@38656
   764
  assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
hoelzl@38656
   765
  thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
hoelzl@38656
   766
next
hoelzl@41981
   767
  assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::extreal}" by auto
hoelzl@38656
   768
  thus ?thesis
hoelzl@38656
   769
    using simple_integral_indicator[OF assms simple_function_const[of 1]]
hoelzl@38656
   770
    using sets_into_space[OF assms]
hoelzl@38656
   771
    by (auto intro!: arg_cong[where f="\<mu>"])
hoelzl@38656
   772
qed
hoelzl@35582
   773
hoelzl@38656
   774
lemma (in measure_space) simple_integral_null_set:
hoelzl@41981
   775
  assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets"
hoelzl@41689
   776
  shows "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = 0"
hoelzl@38656
   777
proof -
hoelzl@41981
   778
  have "AE x. indicator N x = (0 :: extreal)"
hoelzl@40859
   779
    using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
hoelzl@41689
   780
  then have "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^isup>Sx. 0 \<partial>M)"
hoelzl@41981
   781
    using assms apply (intro simple_integral_cong_AE) by auto
hoelzl@40859
   782
  then show ?thesis by simp
hoelzl@38656
   783
qed
hoelzl@35582
   784
hoelzl@40859
   785
lemma (in measure_space) simple_integral_cong_AE_mult_indicator:
hoelzl@41689
   786
  assumes sf: "simple_function M f" and eq: "AE x. x \<in> S" and "S \<in> sets M"
hoelzl@41689
   787
  shows "integral\<^isup>S M f = (\<integral>\<^isup>Sx. f x * indicator S x \<partial>M)"
hoelzl@41705
   788
  using assms by (intro simple_integral_cong_AE) auto
hoelzl@35582
   789
hoelzl@39092
   790
lemma (in measure_space) simple_integral_restricted:
hoelzl@39092
   791
  assumes "A \<in> sets M"
hoelzl@41689
   792
  assumes sf: "simple_function M (\<lambda>x. f x * indicator A x)"
hoelzl@41689
   793
  shows "integral\<^isup>S (restricted_space A) f = (\<integral>\<^isup>Sx. f x * indicator A x \<partial>M)"
hoelzl@41689
   794
    (is "_ = integral\<^isup>S M ?f")
hoelzl@39092
   795
  unfolding simple_integral_def
hoelzl@39092
   796
proof (simp, safe intro!: setsum_mono_zero_cong_left)
hoelzl@39092
   797
  from sf show "finite (?f ` space M)"
hoelzl@39092
   798
    unfolding simple_function_def by auto
hoelzl@39092
   799
next
hoelzl@39092
   800
  fix x assume "x \<in> A"
hoelzl@39092
   801
  then show "f x \<in> ?f ` space M"
hoelzl@39092
   802
    using sets_into_space `A \<in> sets M` by (auto intro!: image_eqI[of _ _ x])
hoelzl@39092
   803
next
hoelzl@39092
   804
  fix x assume "x \<in> space M" "?f x \<notin> f`A"
hoelzl@39092
   805
  then have "x \<notin> A" by (auto simp: image_iff)
hoelzl@39092
   806
  then show "?f x * \<mu> (?f -` {?f x} \<inter> space M) = 0" by simp
hoelzl@39092
   807
next
hoelzl@39092
   808
  fix x assume "x \<in> A"
hoelzl@39092
   809
  then have "f x \<noteq> 0 \<Longrightarrow>
hoelzl@39092
   810
    f -` {f x} \<inter> A = ?f -` {f x} \<inter> space M"
hoelzl@39092
   811
    using `A \<in> sets M` sets_into_space
hoelzl@39092
   812
    by (auto simp: indicator_def split: split_if_asm)
hoelzl@39092
   813
  then show "f x * \<mu> (f -` {f x} \<inter> A) =
hoelzl@39092
   814
    f x * \<mu> (?f -` {f x} \<inter> space M)"
hoelzl@41981
   815
    unfolding extreal_mult_cancel_left by auto
hoelzl@39092
   816
qed
hoelzl@39092
   817
hoelzl@41545
   818
lemma (in measure_space) simple_integral_subalgebra:
hoelzl@41689
   819
  assumes N: "measure_space N" and [simp]: "space N = space M" "measure N = measure M"
hoelzl@41689
   820
  shows "integral\<^isup>S N = integral\<^isup>S M"
hoelzl@41689
   821
  unfolding simple_integral_def_raw by simp
hoelzl@39092
   822
hoelzl@40859
   823
lemma (in measure_space) simple_integral_vimage:
hoelzl@41831
   824
  assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
hoelzl@41689
   825
    and f: "simple_function M' f"
hoelzl@41689
   826
  shows "integral\<^isup>S M' f = (\<integral>\<^isup>S x. f (T x) \<partial>M)"
hoelzl@40859
   827
proof -
hoelzl@41831
   828
  interpret T: measure_space M' by (rule measure_space_vimage[OF T])
hoelzl@41689
   829
  show "integral\<^isup>S M' f = (\<integral>\<^isup>S x. f (T x) \<partial>M)"
hoelzl@41689
   830
    unfolding simple_integral_def
hoelzl@41661
   831
  proof (intro setsum_mono_zero_cong_right ballI)
hoelzl@41661
   832
    show "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
hoelzl@41831
   833
      using T unfolding measurable_def measure_preserving_def by auto
hoelzl@41661
   834
    show "finite (f ` space M')"
hoelzl@41689
   835
      using f unfolding simple_function_def by auto
hoelzl@41661
   836
  next
hoelzl@41661
   837
    fix i assume "i \<in> f ` space M' - (\<lambda>x. f (T x)) ` space M"
hoelzl@41661
   838
    then have "T -` (f -` {i} \<inter> space M') \<inter> space M = {}" by (auto simp: image_iff)
hoelzl@41831
   839
    with f[THEN T.simple_functionD(2), THEN measure_preservingD[OF T(2)], of "{i}"]
hoelzl@41689
   840
    show "i * T.\<mu> (f -` {i} \<inter> space M') = 0" by simp
hoelzl@41661
   841
  next
hoelzl@41661
   842
    fix i assume "i \<in> (\<lambda>x. f (T x)) ` space M"
hoelzl@41661
   843
    then have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
hoelzl@41831
   844
      using T unfolding measurable_def measure_preserving_def by auto
hoelzl@41831
   845
    with f[THEN T.simple_functionD(2), THEN measure_preservingD[OF T(2)], of "{i}"]
hoelzl@41689
   846
    show "i * T.\<mu> (f -` {i} \<inter> space M') = i * \<mu> ((\<lambda>x. f (T x)) -` {i} \<inter> space M)"
hoelzl@41661
   847
      by auto
hoelzl@41661
   848
  qed
hoelzl@40859
   849
qed
hoelzl@40859
   850
hoelzl@41981
   851
lemma (in measure_space) simple_integral_cmult_indicator:
hoelzl@41981
   852
  assumes A: "A \<in> sets M"
hoelzl@41981
   853
  shows "(\<integral>\<^isup>Sx. c * indicator A x \<partial>M) = c * \<mu> A"
hoelzl@41981
   854
  using simple_integral_mult[OF simple_function_indicator[OF A]]
hoelzl@41981
   855
  unfolding simple_integral_indicator_only[OF A] by simp
hoelzl@41981
   856
hoelzl@41981
   857
lemma (in measure_space) simple_integral_positive:
hoelzl@41981
   858
  assumes f: "simple_function M f" and ae: "AE x. 0 \<le> f x"
hoelzl@41981
   859
  shows "0 \<le> integral\<^isup>S M f"
hoelzl@41981
   860
proof -
hoelzl@41981
   861
  have "integral\<^isup>S M (\<lambda>x. 0) \<le> integral\<^isup>S M f"
hoelzl@41981
   862
    using simple_integral_mono_AE[OF _ f ae] by auto
hoelzl@41981
   863
  then show ?thesis by simp
hoelzl@41981
   864
qed
hoelzl@41981
   865
hoelzl@41689
   866
section "Continuous positive integration"
hoelzl@41689
   867
hoelzl@41689
   868
definition positive_integral_def:
hoelzl@41981
   869
  "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^isup>S M g)"
hoelzl@35692
   870
hoelzl@41689
   871
syntax
hoelzl@41981
   872
  "_positive_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> extreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> extreal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
hoelzl@41689
   873
hoelzl@41689
   874
translations
hoelzl@41689
   875
  "\<integral>\<^isup>+ x. f \<partial>M" == "CONST integral\<^isup>P M (%x. f)"
hoelzl@40872
   876
hoelzl@40873
   877
lemma (in measure_space) positive_integral_cong_measure:
hoelzl@41689
   878
  assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
hoelzl@41689
   879
  shows "integral\<^isup>P N f = integral\<^isup>P M f"
hoelzl@41981
   880
  unfolding positive_integral_def
hoelzl@41981
   881
  unfolding simple_function_cong_algebra[OF assms(2,3), symmetric]
hoelzl@41981
   882
  using AE_cong_measure[OF assms]
hoelzl@41981
   883
  using simple_integral_cong_measure[OF assms]
hoelzl@41981
   884
  by (auto intro!: SUP_cong)
hoelzl@41981
   885
hoelzl@41981
   886
lemma (in measure_space) positive_integral_positive:
hoelzl@41981
   887
  "0 \<le> integral\<^isup>P M f"
hoelzl@41981
   888
  by (auto intro!: le_SUPI2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
hoelzl@40873
   889
hoelzl@41981
   890
lemma (in measure_space) positive_integral_def_finite:
hoelzl@41981
   891
  "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^isup>S M g)"
hoelzl@41981
   892
    (is "_ = SUPR ?A ?f")
hoelzl@41981
   893
  unfolding positive_integral_def
hoelzl@41981
   894
proof (safe intro!: antisym SUP_leI)
hoelzl@41981
   895
  fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
hoelzl@41981
   896
  let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
hoelzl@41981
   897
  note gM = g(1)[THEN borel_measurable_simple_function]
hoelzl@41981
   898
  have \<mu>G_pos: "0 \<le> \<mu> ?G" using gM by auto
hoelzl@41981
   899
  let "?g y x" = "if g x = \<infinity> then y else max 0 (g x)"
hoelzl@41981
   900
  from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
hoelzl@41981
   901
    apply (safe intro!: simple_function_max simple_function_If)
hoelzl@41981
   902
    apply (force simp: max_def le_fun_def split: split_if_asm)+
hoelzl@41981
   903
    done
hoelzl@41981
   904
  show "integral\<^isup>S M g \<le> SUPR ?A ?f"
hoelzl@41981
   905
  proof cases
hoelzl@41981
   906
    have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
hoelzl@41981
   907
    assume "\<mu> ?G = 0"
hoelzl@41981
   908
    with gM have "AE x. x \<notin> ?G" by (simp add: AE_iff_null_set)
hoelzl@41981
   909
    with gM g show ?thesis
hoelzl@41981
   910
      by (intro le_SUPI2[OF g0] simple_integral_mono_AE)
hoelzl@41981
   911
         (auto simp: max_def intro!: simple_function_If)
hoelzl@41981
   912
  next
hoelzl@41981
   913
    assume \<mu>G: "\<mu> ?G \<noteq> 0"
hoelzl@41981
   914
    have "SUPR ?A (integral\<^isup>S M) = \<infinity>"
hoelzl@41981
   915
    proof (intro SUP_PInfty)
hoelzl@41981
   916
      fix n :: nat
hoelzl@41981
   917
      let ?y = "extreal (real n) / (if \<mu> ?G = \<infinity> then 1 else \<mu> ?G)"
hoelzl@41981
   918
      have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>G \<mu>G_pos by (auto simp: extreal_divide_eq)
hoelzl@41981
   919
      then have "?g ?y \<in> ?A" by (rule g_in_A)
hoelzl@41981
   920
      have "real n \<le> ?y * \<mu> ?G"
hoelzl@41981
   921
        using \<mu>G \<mu>G_pos by (cases "\<mu> ?G") (auto simp: field_simps)
hoelzl@41981
   922
      also have "\<dots> = (\<integral>\<^isup>Sx. ?y * indicator ?G x \<partial>M)"
hoelzl@41981
   923
        using `0 \<le> ?y` `?g ?y \<in> ?A` gM
hoelzl@41981
   924
        by (subst simple_integral_cmult_indicator) auto
hoelzl@41981
   925
      also have "\<dots> \<le> integral\<^isup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
hoelzl@41981
   926
        by (intro simple_integral_mono) auto
hoelzl@41981
   927
      finally show "\<exists>i\<in>?A. real n \<le> integral\<^isup>S M i"
hoelzl@41981
   928
        using `?g ?y \<in> ?A` by blast
hoelzl@41981
   929
    qed
hoelzl@41981
   930
    then show ?thesis by simp
hoelzl@41981
   931
  qed
hoelzl@41981
   932
qed (auto intro: le_SUPI)
hoelzl@40873
   933
hoelzl@41981
   934
lemma (in measure_space) positive_integral_mono_AE:
hoelzl@41981
   935
  assumes ae: "AE x. u x \<le> v x" shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
hoelzl@41981
   936
  unfolding positive_integral_def
hoelzl@41981
   937
proof (safe intro!: SUP_mono)
hoelzl@41981
   938
  fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
hoelzl@41981
   939
  from ae[THEN AE_E] guess N . note N = this
hoelzl@41981
   940
  then have ae_N: "AE x. x \<notin> N" by (auto intro: AE_not_in)
hoelzl@41981
   941
  let "?n x" = "n x * indicator (space M - N) x"
hoelzl@41981
   942
  have "AE x. n x \<le> ?n x" "simple_function M ?n"
hoelzl@41981
   943
    using n N ae_N by auto
hoelzl@41981
   944
  moreover
hoelzl@41981
   945
  { fix x have "?n x \<le> max 0 (v x)"
hoelzl@41981
   946
    proof cases
hoelzl@41981
   947
      assume x: "x \<in> space M - N"
hoelzl@41981
   948
      with N have "u x \<le> v x" by auto
hoelzl@41981
   949
      with n(2)[THEN le_funD, of x] x show ?thesis
hoelzl@41981
   950
        by (auto simp: max_def split: split_if_asm)
hoelzl@41981
   951
    qed simp }
hoelzl@41981
   952
  then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
hoelzl@41981
   953
  moreover have "integral\<^isup>S M n \<le> integral\<^isup>S M ?n"
hoelzl@41981
   954
    using ae_N N n by (auto intro!: simple_integral_mono_AE)
hoelzl@41981
   955
  ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^isup>S M n \<le> integral\<^isup>S M m"
hoelzl@41981
   956
    by force
hoelzl@38656
   957
qed
hoelzl@38656
   958
hoelzl@41981
   959
lemma (in measure_space) positive_integral_mono:
hoelzl@41981
   960
  "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^isup>P M u \<le> integral\<^isup>P M v"
hoelzl@41981
   961
  by (auto intro: positive_integral_mono_AE)
hoelzl@40859
   962
hoelzl@40859
   963
lemma (in measure_space) positive_integral_cong_AE:
hoelzl@41689
   964
  "AE x. u x = v x \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
hoelzl@40859
   965
  by (auto simp: eq_iff intro!: positive_integral_mono_AE)
hoelzl@40859
   966
hoelzl@41981
   967
lemma (in measure_space) positive_integral_cong:
hoelzl@41981
   968
  "(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
hoelzl@41981
   969
  by (auto intro: positive_integral_cong_AE)
hoelzl@40859
   970
hoelzl@41981
   971
lemma (in measure_space) positive_integral_eq_simple_integral:
hoelzl@41981
   972
  assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
hoelzl@41981
   973
proof -
hoelzl@41981
   974
  let "?f x" = "f x * indicator (space M) x"
hoelzl@41981
   975
  have f': "simple_function M ?f" using f by auto
hoelzl@41981
   976
  with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
hoelzl@41981
   977
    by (auto simp: fun_eq_iff max_def split: split_indicator)
hoelzl@41981
   978
  have "integral\<^isup>P M ?f \<le> integral\<^isup>S M ?f" using f'
hoelzl@41981
   979
    by (force intro!: SUP_leI simple_integral_mono simp: le_fun_def positive_integral_def)
hoelzl@41981
   980
  moreover have "integral\<^isup>S M ?f \<le> integral\<^isup>P M ?f"
hoelzl@41981
   981
    unfolding positive_integral_def
hoelzl@41981
   982
    using f' by (auto intro!: le_SUPI)
hoelzl@41981
   983
  ultimately show ?thesis
hoelzl@41981
   984
    by (simp cong: positive_integral_cong simple_integral_cong)
hoelzl@41981
   985
qed
hoelzl@41981
   986
hoelzl@41981
   987
lemma (in measure_space) positive_integral_eq_simple_integral_AE:
hoelzl@41981
   988
  assumes f: "simple_function M f" "AE x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
hoelzl@41981
   989
proof -
hoelzl@41981
   990
  have "AE x. f x = max 0 (f x)" using f by (auto split: split_max)
hoelzl@41981
   991
  with f have "integral\<^isup>P M f = integral\<^isup>S M (\<lambda>x. max 0 (f x))"
hoelzl@41981
   992
    by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
hoelzl@41981
   993
             add: positive_integral_eq_simple_integral)
hoelzl@41981
   994
  with assms show ?thesis
hoelzl@41981
   995
    by (auto intro!: simple_integral_cong_AE split: split_max)
hoelzl@41981
   996
qed
hoelzl@40873
   997
hoelzl@38656
   998
lemma (in measure_space) positive_integral_SUP_approx:
hoelzl@41981
   999
  assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
hoelzl@41981
  1000
  and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
hoelzl@41689
  1001
  shows "integral\<^isup>S M u \<le> (SUP i. integral\<^isup>P M (f i))" (is "_ \<le> ?S")
hoelzl@41981
  1002
proof (rule extreal_le_mult_one_interval)
hoelzl@41981
  1003
  have "0 \<le> (SUP i. integral\<^isup>P M (f i))"
hoelzl@41981
  1004
    using f(3) by (auto intro!: le_SUPI2 positive_integral_positive)
hoelzl@41981
  1005
  then show "(SUP i. integral\<^isup>P M (f i)) \<noteq> -\<infinity>" by auto
hoelzl@41981
  1006
  have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
hoelzl@41981
  1007
    using u(3) by auto
hoelzl@41981
  1008
  fix a :: extreal assume "0 < a" "a < 1"
hoelzl@38656
  1009
  hence "a \<noteq> 0" by auto
hoelzl@38656
  1010
  let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
hoelzl@38656
  1011
  have B: "\<And>i. ?B i \<in> sets M"
hoelzl@41689
  1012
    using f `simple_function M u` by (auto simp: borel_measurable_simple_function)
hoelzl@38656
  1013
hoelzl@38656
  1014
  let "?uB i x" = "u x * indicator (?B i) x"
hoelzl@38656
  1015
hoelzl@38656
  1016
  { fix i have "?B i \<subseteq> ?B (Suc i)"
hoelzl@38656
  1017
    proof safe
hoelzl@38656
  1018
      fix i x assume "a * u x \<le> f i x"
hoelzl@38656
  1019
      also have "\<dots> \<le> f (Suc i) x"
hoelzl@41981
  1020
        using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
hoelzl@38656
  1021
      finally show "a * u x \<le> f (Suc i) x" .
hoelzl@38656
  1022
    qed }
hoelzl@38656
  1023
  note B_mono = this
hoelzl@35582
  1024
hoelzl@41981
  1025
  note B_u = Int[OF u(1)[THEN simple_functionD(2)] B]
hoelzl@38656
  1026
hoelzl@41981
  1027
  let "?B' i n" = "(u -` {i} \<inter> space M) \<inter> ?B n"
hoelzl@41981
  1028
  have measure_conv: "\<And>i. \<mu> (u -` {i} \<inter> space M) = (SUP n. \<mu> (?B' i n))"
hoelzl@41981
  1029
  proof -
hoelzl@41981
  1030
    fix i
hoelzl@41981
  1031
    have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
hoelzl@41981
  1032
    have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
hoelzl@41981
  1033
    have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
hoelzl@41981
  1034
    proof safe
hoelzl@41981
  1035
      fix x i assume x: "x \<in> space M"
hoelzl@41981
  1036
      show "x \<in> (\<Union>i. ?B' (u x) i)"
hoelzl@41981
  1037
      proof cases
hoelzl@41981
  1038
        assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
hoelzl@41981
  1039
      next
hoelzl@41981
  1040
        assume "u x \<noteq> 0"
hoelzl@41981
  1041
        with `a < 1` u_range[OF `x \<in> space M`]
hoelzl@41981
  1042
        have "a * u x < 1 * u x"
hoelzl@41981
  1043
          by (intro extreal_mult_strict_right_mono) (auto simp: image_iff)
hoelzl@41981
  1044
        also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def SUPR_apply)
hoelzl@41981
  1045
        finally obtain i where "a * u x < f i x" unfolding SUPR_def
hoelzl@41981
  1046
          by (auto simp add: less_Sup_iff)
hoelzl@41981
  1047
        hence "a * u x \<le> f i x" by auto
hoelzl@41981
  1048
        thus ?thesis using `x \<in> space M` by auto
hoelzl@41981
  1049
      qed
hoelzl@40859
  1050
    qed
hoelzl@41981
  1051
    then show "?thesis i" using continuity_from_below[OF 1 2] by simp
hoelzl@41981
  1052
  qed
hoelzl@38656
  1053
hoelzl@41689
  1054
  have "integral\<^isup>S M u = (SUP i. integral\<^isup>S M (?uB i))"
hoelzl@41689
  1055
    unfolding simple_integral_indicator[OF B `simple_function M u`]
hoelzl@41981
  1056
  proof (subst SUPR_extreal_setsum, safe)
hoelzl@38656
  1057
    fix x n assume "x \<in> space M"
hoelzl@41981
  1058
    with u_range show "incseq (\<lambda>i. u x * \<mu> (?B' (u x) i))" "\<And>i. 0 \<le> u x * \<mu> (?B' (u x) i)"
hoelzl@41981
  1059
      using B_mono B_u by (auto intro!: measure_mono extreal_mult_left_mono incseq_SucI simp: extreal_zero_le_0_iff)
hoelzl@38656
  1060
  next
hoelzl@41981
  1061
    show "integral\<^isup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (?B' i n))"
hoelzl@41981
  1062
      using measure_conv u_range B_u unfolding simple_integral_def
hoelzl@41981
  1063
      by (auto intro!: setsum_cong SUPR_extreal_cmult[symmetric])
hoelzl@38656
  1064
  qed
hoelzl@38656
  1065
  moreover
hoelzl@41689
  1066
  have "a * (SUP i. integral\<^isup>S M (?uB i)) \<le> ?S"
hoelzl@41981
  1067
    apply (subst SUPR_extreal_cmult[symmetric])
hoelzl@38705
  1068
  proof (safe intro!: SUP_mono bexI)
hoelzl@38656
  1069
    fix i
hoelzl@41689
  1070
    have "a * integral\<^isup>S M (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x \<partial>M)"
hoelzl@41981
  1071
      using B `simple_function M u` u_range
hoelzl@41981
  1072
      by (subst simple_integral_mult) (auto split: split_indicator)
hoelzl@41689
  1073
    also have "\<dots> \<le> integral\<^isup>P M (f i)"
hoelzl@38656
  1074
    proof -
hoelzl@41981
  1075
      have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
hoelzl@41981
  1076
      show ?thesis using f(3) * u_range `0 < a`
hoelzl@41981
  1077
        by (subst positive_integral_eq_simple_integral[symmetric])
hoelzl@41981
  1078
           (auto intro!: positive_integral_mono split: split_indicator)
hoelzl@38656
  1079
    qed
hoelzl@41689
  1080
    finally show "a * integral\<^isup>S M (?uB i) \<le> integral\<^isup>P M (f i)"
hoelzl@38656
  1081
      by auto
hoelzl@41981
  1082
  next
hoelzl@41981
  1083
    fix i show "0 \<le> \<integral>\<^isup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
hoelzl@41981
  1084
      by (intro simple_integral_positive) (auto split: split_indicator)
hoelzl@41981
  1085
  qed (insert `0 < a`, auto)
hoelzl@41689
  1086
  ultimately show "a * integral\<^isup>S M u \<le> ?S" by simp
hoelzl@35582
  1087
qed
hoelzl@35582
  1088
hoelzl@41981
  1089
lemma (in measure_space) incseq_positive_integral:
hoelzl@41981
  1090
  assumes "incseq f" shows "incseq (\<lambda>i. integral\<^isup>P M (f i))"
hoelzl@41981
  1091
proof -
hoelzl@41981
  1092
  have "\<And>i x. f i x \<le> f (Suc i) x"
hoelzl@41981
  1093
    using assms by (auto dest!: incseq_SucD simp: le_fun_def)
hoelzl@41981
  1094
  then show ?thesis
hoelzl@41981
  1095
    by (auto intro!: incseq_SucI positive_integral_mono)
hoelzl@41981
  1096
qed
hoelzl@41981
  1097
hoelzl@35582
  1098
text {* Beppo-Levi monotone convergence theorem *}
hoelzl@41981
  1099
lemma (in measure_space) positive_integral_monotone_convergence_SUP:
hoelzl@41981
  1100
  assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
hoelzl@41981
  1101
  shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
hoelzl@41981
  1102
proof (rule antisym)
hoelzl@41981
  1103
  show "(SUP j. integral\<^isup>P M (f j)) \<le> (\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M)"
hoelzl@41981
  1104
    by (auto intro!: SUP_leI le_SUPI positive_integral_mono)
hoelzl@38656
  1105
next
hoelzl@41981
  1106
  show "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^isup>P M (f j))"
hoelzl@41981
  1107
    unfolding positive_integral_def_finite[of "\<lambda>x. SUP i. f i x"]
hoelzl@41981
  1108
  proof (safe intro!: SUP_leI)
hoelzl@41981
  1109
    fix g assume g: "simple_function M g"
hoelzl@41981
  1110
      and "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
hoelzl@41981
  1111
    moreover then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}"
hoelzl@41981
  1112
      using f by (auto intro!: le_SUPI2)
hoelzl@41981
  1113
    ultimately show "integral\<^isup>S M g \<le> (SUP j. integral\<^isup>P M (f j))"
hoelzl@41981
  1114
      by (intro  positive_integral_SUP_approx[OF f g _ g'])
hoelzl@41981
  1115
         (auto simp: le_fun_def max_def SUPR_apply)
hoelzl@35582
  1116
  qed
hoelzl@35582
  1117
qed
hoelzl@35582
  1118
hoelzl@41981
  1119
lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE:
hoelzl@41981
  1120
  assumes f: "\<And>i. AE x. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
hoelzl@41981
  1121
  shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
hoelzl@40859
  1122
proof -
hoelzl@41981
  1123
  from f have "AE x. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
hoelzl@41981
  1124
    by (simp add: AE_all_countable)
hoelzl@41981
  1125
  from this[THEN AE_E] guess N . note N = this
hoelzl@41981
  1126
  let "?f i x" = "if x \<in> space M - N then f i x else 0"
hoelzl@41981
  1127
  have f_eq: "AE x. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ N])
hoelzl@41981
  1128
  then have "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^isup>+ x. (SUP i. ?f i x) \<partial>M)"
hoelzl@41981
  1129
    by (auto intro!: positive_integral_cong_AE)
hoelzl@41981
  1130
  also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. ?f i x \<partial>M))"
hoelzl@41981
  1131
  proof (rule positive_integral_monotone_convergence_SUP)
hoelzl@41981
  1132
    show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
hoelzl@41981
  1133
    { fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
hoelzl@41981
  1134
        using f N(3) by (intro measurable_If_set) auto
hoelzl@41981
  1135
      fix x show "0 \<le> ?f i x"
hoelzl@41981
  1136
        using N(1) by auto }
hoelzl@40859
  1137
  qed
hoelzl@41981
  1138
  also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. f i x \<partial>M))"
hoelzl@41981
  1139
    using f_eq by (force intro!: arg_cong[where f="SUPR UNIV"] positive_integral_cong_AE ext)
hoelzl@41981
  1140
  finally show ?thesis .
hoelzl@41981
  1141
qed
hoelzl@41981
  1142
hoelzl@41981
  1143
lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE_incseq:
hoelzl@41981
  1144
  assumes f: "incseq f" "\<And>i. AE x. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
hoelzl@41981
  1145
  shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
hoelzl@41981
  1146
  using f[unfolded incseq_Suc_iff le_fun_def]
hoelzl@41981
  1147
  by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
hoelzl@41981
  1148
     auto
hoelzl@41981
  1149
hoelzl@41981
  1150
lemma (in measure_space) positive_integral_monotone_convergence_simple:
hoelzl@41981
  1151
  assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
hoelzl@41981
  1152
  shows "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
hoelzl@41981
  1153
  using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
hoelzl@41981
  1154
    f(3)[THEN borel_measurable_simple_function] f(2)]
hoelzl@41981
  1155
  by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPR UNIV"] ext)
hoelzl@41981
  1156
hoelzl@41981
  1157
lemma positive_integral_max_0:
hoelzl@41981
  1158
  "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M) = integral\<^isup>P M f"
hoelzl@41981
  1159
  by (simp add: le_fun_def positive_integral_def)
hoelzl@41981
  1160
hoelzl@41981
  1161
lemma (in measure_space) positive_integral_cong_pos:
hoelzl@41981
  1162
  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
hoelzl@41981
  1163
  shows "integral\<^isup>P M f = integral\<^isup>P M g"
hoelzl@41981
  1164
proof -
hoelzl@41981
  1165
  have "integral\<^isup>P M (\<lambda>x. max 0 (f x)) = integral\<^isup>P M (\<lambda>x. max 0 (g x))"
hoelzl@41981
  1166
  proof (intro positive_integral_cong)
hoelzl@41981
  1167
    fix x assume "x \<in> space M"
hoelzl@41981
  1168
    from assms[OF this] show "max 0 (f x) = max 0 (g x)"
hoelzl@41981
  1169
      by (auto split: split_max)
hoelzl@41981
  1170
  qed
hoelzl@41981
  1171
  then show ?thesis by (simp add: positive_integral_max_0)
hoelzl@40859
  1172
qed
hoelzl@40859
  1173
hoelzl@38656
  1174
lemma (in measure_space) SUP_simple_integral_sequences:
hoelzl@41981
  1175
  assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
hoelzl@41981
  1176
  and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
hoelzl@41981
  1177
  and eq: "AE x. (SUP i. f i x) = (SUP i. g i x)"
hoelzl@41689
  1178
  shows "(SUP i. integral\<^isup>S M (f i)) = (SUP i. integral\<^isup>S M (g i))"
hoelzl@38656
  1179
    (is "SUPR _ ?F = SUPR _ ?G")
hoelzl@38656
  1180
proof -
hoelzl@41981
  1181
  have "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
hoelzl@41981
  1182
    using f by (rule positive_integral_monotone_convergence_simple)
hoelzl@41981
  1183
  also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. g i x) \<partial>M)"
hoelzl@41981
  1184
    unfolding eq[THEN positive_integral_cong_AE] ..
hoelzl@38656
  1185
  also have "\<dots> = (SUP i. ?G i)"
hoelzl@41981
  1186
    using g by (rule positive_integral_monotone_convergence_simple[symmetric])
hoelzl@41981
  1187
  finally show ?thesis by simp
hoelzl@38656
  1188
qed
hoelzl@38656
  1189
hoelzl@38656
  1190
lemma (in measure_space) positive_integral_const[simp]:
hoelzl@41981
  1191
  "0 \<le> c \<Longrightarrow> (\<integral>\<^isup>+ x. c \<partial>M) = c * \<mu> (space M)"
hoelzl@38656
  1192
  by (subst positive_integral_eq_simple_integral) auto
hoelzl@38656
  1193
hoelzl@41661
  1194
lemma (in measure_space) positive_integral_vimage:
hoelzl@41981
  1195
  assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
hoelzl@41981
  1196
  and f: "f \<in> borel_measurable M'"
hoelzl@41689
  1197
  shows "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
hoelzl@41661
  1198
proof -
hoelzl@41831
  1199
  interpret T: measure_space M' by (rule measure_space_vimage[OF T])
hoelzl@41981
  1200
  from T.borel_measurable_implies_simple_function_sequence'[OF f]
hoelzl@41981
  1201
  guess f' . note f' = this
hoelzl@41981
  1202
  let "?f i x" = "f' i (T x)"
hoelzl@41981
  1203
  have inc: "incseq ?f" using f' by (force simp: le_fun_def incseq_def)
hoelzl@41981
  1204
  have sup: "\<And>x. (SUP i. ?f i x) = max 0 (f (T x))"
hoelzl@41981
  1205
    using f'(4) .
hoelzl@41981
  1206
  have sf: "\<And>i. simple_function M (\<lambda>x. f' i (T x))"
hoelzl@41981
  1207
    using simple_function_vimage[OF T(1) measure_preservingD2[OF T(2)] f'(1)] .
hoelzl@41689
  1208
  show "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
hoelzl@41981
  1209
    using
hoelzl@41981
  1210
      T.positive_integral_monotone_convergence_simple[OF f'(2,5,1)]
hoelzl@41981
  1211
      positive_integral_monotone_convergence_simple[OF inc f'(5) sf]
hoelzl@41981
  1212
    by (simp add: positive_integral_max_0 simple_integral_vimage[OF T f'(1)] f')
hoelzl@41661
  1213
qed
hoelzl@41661
  1214
hoelzl@38656
  1215
lemma (in measure_space) positive_integral_linear:
hoelzl@41981
  1216
  assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
hoelzl@41981
  1217
  and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
hoelzl@41689
  1218
  shows "(\<integral>\<^isup>+ x. a * f x + g x \<partial>M) = a * integral\<^isup>P M f + integral\<^isup>P M g"
hoelzl@41689
  1219
    (is "integral\<^isup>P M ?L = _")
hoelzl@35582
  1220
proof -
hoelzl@41981
  1221
  from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
hoelzl@41981
  1222
  note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
hoelzl@41981
  1223
  from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
hoelzl@41981
  1224
  note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
hoelzl@38656
  1225
  let "?L' i x" = "a * u i x + v i x"
hoelzl@38656
  1226
hoelzl@41981
  1227
  have "?L \<in> borel_measurable M" using assms by auto
hoelzl@38656
  1228
  from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
hoelzl@41981
  1229
  note l = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
hoelzl@41981
  1230
hoelzl@41981
  1231
  have inc: "incseq (\<lambda>i. a * integral\<^isup>S M (u i))" "incseq (\<lambda>i. integral\<^isup>S M (v i))"
hoelzl@41981
  1232
    using u v `0 \<le> a`
hoelzl@41981
  1233
    by (auto simp: incseq_Suc_iff le_fun_def
hoelzl@41981
  1234
             intro!: add_mono extreal_mult_left_mono simple_integral_mono)
hoelzl@41981
  1235
  have pos: "\<And>i. 0 \<le> integral\<^isup>S M (u i)" "\<And>i. 0 \<le> integral\<^isup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^isup>S M (u i)"
hoelzl@41981
  1236
    using u v `0 \<le> a` by (auto simp: simple_integral_positive)
hoelzl@41981
  1237
  { fix i from pos[of i] have "a * integral\<^isup>S M (u i) \<noteq> -\<infinity>" "integral\<^isup>S M (v i) \<noteq> -\<infinity>"
hoelzl@41981
  1238
      by (auto split: split_if_asm) }
hoelzl@41981
  1239
  note not_MInf = this
hoelzl@41981
  1240
hoelzl@41981
  1241
  have l': "(SUP i. integral\<^isup>S M (l i)) = (SUP i. integral\<^isup>S M (?L' i))"
hoelzl@41981
  1242
  proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
hoelzl@41981
  1243
    show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
hoelzl@41981
  1244
      using u v  `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
hoelzl@41981
  1245
      by (auto intro!: add_mono extreal_mult_left_mono extreal_add_nonneg_nonneg)
hoelzl@41981
  1246
    { fix x
hoelzl@41981
  1247
      { 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]
hoelzl@41981
  1248
          by auto }
hoelzl@41981
  1249
      then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
hoelzl@41981
  1250
        using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
hoelzl@41981
  1251
        by (subst SUPR_extreal_cmult[symmetric, OF u(6) `0 \<le> a`])
hoelzl@41981
  1252
           (auto intro!: SUPR_extreal_add
hoelzl@41981
  1253
                 simp: incseq_Suc_iff le_fun_def add_mono extreal_mult_left_mono extreal_add_nonneg_nonneg) }
hoelzl@41981
  1254
    then show "AE x. (SUP i. l i x) = (SUP i. ?L' i x)"
hoelzl@41981
  1255
      unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
hoelzl@41981
  1256
      by (intro AE_I2) (auto split: split_max simp add: extreal_add_nonneg_nonneg)
hoelzl@38656
  1257
  qed
hoelzl@41981
  1258
  also have "\<dots> = (SUP i. a * integral\<^isup>S M (u i) + integral\<^isup>S M (v i))"
hoelzl@41981
  1259
    using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPR UNIV"] ext)
hoelzl@41981
  1260
  finally have "(\<integral>\<^isup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^isup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+x. max 0 (g x) \<partial>M)"
hoelzl@41981
  1261
    unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
hoelzl@41981
  1262
    unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
hoelzl@41981
  1263
    apply (subst SUPR_extreal_cmult[symmetric, OF pos(1) `0 \<le> a`])
hoelzl@41981
  1264
    apply (subst SUPR_extreal_add[symmetric, OF inc not_MInf]) .
hoelzl@41981
  1265
  then show ?thesis by (simp add: positive_integral_max_0)
hoelzl@38656
  1266
qed
hoelzl@38656
  1267
hoelzl@38656
  1268
lemma (in measure_space) positive_integral_cmult:
hoelzl@41981
  1269
  assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
hoelzl@41689
  1270
  shows "(\<integral>\<^isup>+ x. c * f x \<partial>M) = c * integral\<^isup>P M f"
hoelzl@41981
  1271
proof -
hoelzl@41981
  1272
  have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
hoelzl@41981
  1273
    by (auto split: split_max simp: extreal_zero_le_0_iff)
hoelzl@41981
  1274
  have "(\<integral>\<^isup>+ x. c * f x \<partial>M) = (\<integral>\<^isup>+ x. c * max 0 (f x) \<partial>M)"
hoelzl@41981
  1275
    by (simp add: positive_integral_max_0)
hoelzl@41981
  1276
  then show ?thesis
hoelzl@41981
  1277
    using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" "\<lambda>x. 0"] f
hoelzl@41981
  1278
    by (auto simp: positive_integral_max_0)
hoelzl@41981
  1279
qed
hoelzl@38656
  1280
hoelzl@41096
  1281
lemma (in measure_space) positive_integral_multc:
hoelzl@41981
  1282
  assumes "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
hoelzl@41689
  1283
  shows "(\<integral>\<^isup>+ x. f x * c \<partial>M) = integral\<^isup>P M f * c"
hoelzl@41096
  1284
  unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
hoelzl@41096
  1285
hoelzl@38656
  1286
lemma (in measure_space) positive_integral_indicator[simp]:
hoelzl@41689
  1287
  "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x\<partial>M) = \<mu> A"
hoelzl@41544
  1288
  by (subst positive_integral_eq_simple_integral)
hoelzl@41544
  1289
     (auto simp: simple_function_indicator simple_integral_indicator)
hoelzl@38656
  1290
hoelzl@38656
  1291
lemma (in measure_space) positive_integral_cmult_indicator:
hoelzl@41981
  1292
  "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * \<mu> A"
hoelzl@41544
  1293
  by (subst positive_integral_eq_simple_integral)
hoelzl@41544
  1294
     (auto simp: simple_function_indicator simple_integral_indicator)
hoelzl@38656
  1295
hoelzl@38656
  1296
lemma (in measure_space) positive_integral_add:
hoelzl@41981
  1297
  assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
hoelzl@41981
  1298
  and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
hoelzl@41689
  1299
  shows "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = integral\<^isup>P M f + integral\<^isup>P M g"
hoelzl@41981
  1300
proof -
hoelzl@41981
  1301
  have ae: "AE x. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
hoelzl@41981
  1302
    using assms by (auto split: split_max simp: extreal_add_nonneg_nonneg)
hoelzl@41981
  1303
  have "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = (\<integral>\<^isup>+ x. max 0 (f x + g x) \<partial>M)"
hoelzl@41981
  1304
    by (simp add: positive_integral_max_0)
hoelzl@41981
  1305
  also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
hoelzl@41981
  1306
    unfolding ae[THEN positive_integral_cong_AE] ..
hoelzl@41981
  1307
  also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+ x. max 0 (g x) \<partial>M)"
hoelzl@41981
  1308
    using positive_integral_linear[of "\<lambda>x. max 0 (f x)" 1 "\<lambda>x. max 0 (g x)"] f g
hoelzl@41981
  1309
    by auto
hoelzl@41981
  1310
  finally show ?thesis
hoelzl@41981
  1311
    by (simp add: positive_integral_max_0)
hoelzl@41981
  1312
qed
hoelzl@38656
  1313
hoelzl@38656
  1314
lemma (in measure_space) positive_integral_setsum:
hoelzl@41981
  1315
  assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x. 0 \<le> f i x"
hoelzl@41689
  1316
  shows "(\<integral>\<^isup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>P M (f i))"
hoelzl@38656
  1317
proof cases
hoelzl@41981
  1318
  assume f: "finite P"
hoelzl@41981
  1319
  from assms have "AE x. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
hoelzl@41981
  1320
  from f this assms(1) show ?thesis
hoelzl@38656
  1321
  proof induct
hoelzl@38656
  1322
    case (insert i P)
hoelzl@41981
  1323
    then have "f i \<in> borel_measurable M" "AE x. 0 \<le> f i x"
hoelzl@41981
  1324
      "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x. 0 \<le> (\<Sum>i\<in>P. f i x)"
hoelzl@41981
  1325
      by (auto intro!: borel_measurable_extreal_setsum setsum_nonneg)
hoelzl@38656
  1326
    from positive_integral_add[OF this]
hoelzl@38656
  1327
    show ?case using insert by auto
hoelzl@38656
  1328
  qed simp
hoelzl@38656
  1329
qed simp
hoelzl@38656
  1330
hoelzl@41981
  1331
lemma (in measure_space) positive_integral_Markov_inequality:
hoelzl@41981
  1332
  assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c" "c \<noteq> \<infinity>"
hoelzl@41981
  1333
  shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
hoelzl@41981
  1334
    (is "\<mu> ?A \<le> _ * ?PI")
hoelzl@41981
  1335
proof -
hoelzl@41981
  1336
  have "?A \<in> sets M"
hoelzl@41981
  1337
    using `A \<in> sets M` u by auto
hoelzl@41981
  1338
  hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
hoelzl@41981
  1339
    using positive_integral_indicator by simp
hoelzl@41981
  1340
  also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
hoelzl@41981
  1341
    by (auto intro!: positive_integral_mono_AE
hoelzl@41981
  1342
      simp: indicator_def extreal_zero_le_0_iff)
hoelzl@41981
  1343
  also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
hoelzl@41981
  1344
    using assms
hoelzl@41981
  1345
    by (auto intro!: positive_integral_cmult borel_measurable_indicator simp: extreal_zero_le_0_iff)
hoelzl@41981
  1346
  finally show ?thesis .
hoelzl@41981
  1347
qed
hoelzl@41981
  1348
hoelzl@41981
  1349
lemma (in measure_space) positive_integral_noteq_infinite:
hoelzl@41981
  1350
  assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
hoelzl@41981
  1351
  and "integral\<^isup>P M g \<noteq> \<infinity>"
hoelzl@41981
  1352
  shows "AE x. g x \<noteq> \<infinity>"
hoelzl@41981
  1353
proof (rule ccontr)
hoelzl@41981
  1354
  assume c: "\<not> (AE x. g x \<noteq> \<infinity>)"
hoelzl@41981
  1355
  have "\<mu> {x\<in>space M. g x = \<infinity>} \<noteq> 0"
hoelzl@41981
  1356
    using c g by (simp add: AE_iff_null_set)
hoelzl@41981
  1357
  moreover have "0 \<le> \<mu> {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
hoelzl@41981
  1358
  ultimately have "0 < \<mu> {x\<in>space M. g x = \<infinity>}" by auto
hoelzl@41981
  1359
  then have "\<infinity> = \<infinity> * \<mu> {x\<in>space M. g x = \<infinity>}" by auto
hoelzl@41981
  1360
  also have "\<dots> \<le> (\<integral>\<^isup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
hoelzl@41981
  1361
    using g by (subst positive_integral_cmult_indicator) auto
hoelzl@41981
  1362
  also have "\<dots> \<le> integral\<^isup>P M g"
hoelzl@41981
  1363
    using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
hoelzl@41981
  1364
  finally show False using `integral\<^isup>P M g \<noteq> \<infinity>` by auto
hoelzl@41981
  1365
qed
hoelzl@41981
  1366
hoelzl@38656
  1367
lemma (in measure_space) positive_integral_diff:
hoelzl@41981
  1368
  assumes f: "f \<in> borel_measurable M"
hoelzl@41981
  1369
  and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
hoelzl@41981
  1370
  and fin: "integral\<^isup>P M g \<noteq> \<infinity>"
hoelzl@41981
  1371
  and mono: "AE x. g x \<le> f x"
hoelzl@41689
  1372
  shows "(\<integral>\<^isup>+ x. f x - g x \<partial>M) = integral\<^isup>P M f - integral\<^isup>P M g"
hoelzl@38656
  1373
proof -
hoelzl@41981
  1374
  have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x. 0 \<le> f x - g x"
hoelzl@41981
  1375
    using assms by (auto intro: extreal_diff_positive)
hoelzl@41981
  1376
  have pos_f: "AE x. 0 \<le> f x" using mono g by auto
hoelzl@41981
  1377
  { fix a b :: extreal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
hoelzl@41981
  1378
      by (cases rule: extreal2_cases[of a b]) auto }
hoelzl@41981
  1379
  note * = this
hoelzl@41981
  1380
  then have "AE x. f x = f x - g x + g x"
hoelzl@41981
  1381
    using mono positive_integral_noteq_infinite[OF g fin] assms by auto
hoelzl@41981
  1382
  then have **: "integral\<^isup>P M f = (\<integral>\<^isup>+x. f x - g x \<partial>M) + integral\<^isup>P M g"
hoelzl@41981
  1383
    unfolding positive_integral_add[OF diff g, symmetric]
hoelzl@41981
  1384
    by (rule positive_integral_cong_AE)
hoelzl@41981
  1385
  show ?thesis unfolding **
hoelzl@41981
  1386
    using fin positive_integral_positive[of g]
hoelzl@41981
  1387
    by (cases rule: extreal2_cases[of "\<integral>\<^isup>+ x. f x - g x \<partial>M" "integral\<^isup>P M g"]) auto
hoelzl@38656
  1388
qed
hoelzl@38656
  1389
hoelzl@41981
  1390
lemma (in measure_space) positive_integral_suminf:
hoelzl@41981
  1391
  assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> f i x"
hoelzl@41981
  1392
  shows "(\<integral>\<^isup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^isup>P M (f i))"
hoelzl@38656
  1393
proof -
hoelzl@41981
  1394
  have all_pos: "AE x. \<forall>i. 0 \<le> f i x"
hoelzl@41981
  1395
    using assms by (auto simp: AE_all_countable)
hoelzl@41981
  1396
  have "(\<Sum>i. integral\<^isup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^isup>P M (f i))"
hoelzl@41981
  1397
    using positive_integral_positive by (rule suminf_extreal_eq_SUPR)
hoelzl@41981
  1398
  also have "\<dots> = (SUP n. \<integral>\<^isup>+x. (\<Sum>i<n. f i x) \<partial>M)"
hoelzl@41981
  1399
    unfolding positive_integral_setsum[OF f] ..
hoelzl@41981
  1400
  also have "\<dots> = \<integral>\<^isup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
hoelzl@41981
  1401
    by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
hoelzl@41981
  1402
       (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
hoelzl@41981
  1403
  also have "\<dots> = \<integral>\<^isup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
hoelzl@41981
  1404
    by (intro positive_integral_cong_AE) (auto simp: suminf_extreal_eq_SUPR)
hoelzl@41981
  1405
  finally show ?thesis by simp
hoelzl@38656
  1406
qed
hoelzl@38656
  1407
hoelzl@38656
  1408
text {* Fatou's lemma: convergence theorem on limes inferior *}
hoelzl@38656
  1409
lemma (in measure_space) positive_integral_lim_INF:
hoelzl@41981
  1410
  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
hoelzl@41981
  1411
  assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> u i x"
hoelzl@41981
  1412
  shows "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
hoelzl@38656
  1413
proof -
hoelzl@41981
  1414
  have pos: "AE x. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
hoelzl@41981
  1415
  have "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
hoelzl@41981
  1416
    (SUP n. \<integral>\<^isup>+ x. (INF i:{n..}. u i x) \<partial>M)"
hoelzl@41981
  1417
    unfolding liminf_SUPR_INFI using pos u
hoelzl@41981
  1418
    by (intro positive_integral_monotone_convergence_SUP_AE)
hoelzl@41981
  1419
       (elim AE_mp, auto intro!: AE_I2 intro: le_INFI INF_subset)
hoelzl@41981
  1420
  also have "\<dots> \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
hoelzl@41981
  1421
    unfolding liminf_SUPR_INFI
hoelzl@41981
  1422
    by (auto intro!: SUP_mono exI le_INFI positive_integral_mono INF_leI)
hoelzl@38656
  1423
  finally show ?thesis .
hoelzl@35582
  1424
qed
hoelzl@35582
  1425
hoelzl@38656
  1426
lemma (in measure_space) measure_space_density:
hoelzl@41981
  1427
  assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x"
hoelzl@41689
  1428
    and M'[simp]: "M' = (M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)\<rparr>)"
hoelzl@41689
  1429
  shows "measure_space M'"
hoelzl@41689
  1430
proof -
hoelzl@41689
  1431
  interpret M': sigma_algebra M' by (intro sigma_algebra_cong) auto
hoelzl@41689
  1432
  show ?thesis
hoelzl@41689
  1433
  proof
hoelzl@41981
  1434
    have pos: "\<And>A. AE x. 0 \<le> u x * indicator A x"
hoelzl@41981
  1435
      using u by (auto simp: extreal_zero_le_0_iff)
hoelzl@41981
  1436
    then show "positive M' (measure M')" unfolding M'
hoelzl@41981
  1437
      using u(1) by (auto simp: positive_def intro!: positive_integral_positive)
hoelzl@41689
  1438
    show "countably_additive M' (measure M')"
hoelzl@41689
  1439
    proof (intro countably_additiveI)
hoelzl@41689
  1440
      fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M'"
hoelzl@41981
  1441
      then have *: "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
hoelzl@41981
  1442
        using u by (auto intro: borel_measurable_indicator)
hoelzl@41981
  1443
      assume disj: "disjoint_family A"
hoelzl@41981
  1444
      have "(\<Sum>n. measure M' (A n)) = (\<integral>\<^isup>+ x. (\<Sum>n. u x * indicator (A n) x) \<partial>M)"
hoelzl@41981
  1445
        unfolding M' using u(1) *
hoelzl@41981
  1446
        by (simp add: positive_integral_suminf[OF _ pos, symmetric])
hoelzl@41981
  1447
      also have "\<dots> = (\<integral>\<^isup>+ x. u x * (\<Sum>n. indicator (A n) x) \<partial>M)" using u
hoelzl@41981
  1448
        by (intro positive_integral_cong_AE)
hoelzl@41981
  1449
           (elim AE_mp, auto intro!: AE_I2 suminf_cmult_extreal)
hoelzl@41981
  1450
      also have "\<dots> = (\<integral>\<^isup>+ x. u x * indicator (\<Union>n. A n) x \<partial>M)"
hoelzl@41981
  1451
        unfolding suminf_indicator[OF disj] ..
hoelzl@41981
  1452
      finally show "(\<Sum>n. measure M' (A n)) = measure M' (\<Union>x. A x)"
hoelzl@41981
  1453
        unfolding M' by simp
hoelzl@41689
  1454
    qed
hoelzl@38656
  1455
  qed
hoelzl@38656
  1456
qed
hoelzl@35582
  1457
hoelzl@38656
  1458
lemma (in measure_space) positive_integral_null_set:
hoelzl@41689
  1459
  assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = 0"
hoelzl@38656
  1460
proof -
hoelzl@41689
  1461
  have "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
hoelzl@40859
  1462
  proof (intro positive_integral_cong_AE AE_I)
hoelzl@40859
  1463
    show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
hoelzl@40859
  1464
      by (auto simp: indicator_def)
hoelzl@40859
  1465
    show "\<mu> N = 0" "N \<in> sets M"
hoelzl@40859
  1466
      using assms by auto
hoelzl@35582
  1467
  qed
hoelzl@40859
  1468
  then show ?thesis by simp
hoelzl@38656
  1469
qed
hoelzl@35582
  1470
hoelzl@41981
  1471
lemma (in measure_space) positive_integral_translated_density:
hoelzl@41981
  1472
  assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
hoelzl@41981
  1473
  assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
hoelzl@41981
  1474
    and M': "M' = (M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)\<rparr>)"
hoelzl@41981
  1475
  shows "integral\<^isup>P M' g = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
hoelzl@38656
  1476
proof -
hoelzl@41981
  1477
  from measure_space_density[OF f M']
hoelzl@41981
  1478
  interpret T: measure_space M' .
hoelzl@41981
  1479
  have borel[simp]:
hoelzl@41981
  1480
    "borel_measurable M' = borel_measurable M"
hoelzl@41981
  1481
    "simple_function M' = simple_function M"
hoelzl@41981
  1482
    unfolding measurable_def simple_function_def_raw by (auto simp: M')
hoelzl@41981
  1483
  from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess G . note G = this
hoelzl@41981
  1484
  note G' = borel_measurable_simple_function[OF this(1)] simple_functionD[OF G(1)]
hoelzl@41981
  1485
  note G'(2)[simp]
hoelzl@41981
  1486
  { fix P have "AE x. P x \<Longrightarrow> AE x in M'. P x"
hoelzl@41981
  1487
      using positive_integral_null_set[of _ f]
hoelzl@41981
  1488
      unfolding T.almost_everywhere_def almost_everywhere_def
hoelzl@41981
  1489
      by (auto simp: M') }
hoelzl@41981
  1490
  note ac = this
hoelzl@41981
  1491
  from G(4) g(2) have G_M': "AE x in M'. (SUP i. G i x) = g x"
hoelzl@41981
  1492
    by (auto intro!: ac split: split_max)
hoelzl@41981
  1493
  { fix i
hoelzl@41981
  1494
    let "?I y x" = "indicator (G i -` {y} \<inter> space M) x"
hoelzl@41981
  1495
    { fix x assume *: "x \<in> space M" "0 \<le> f x" "0 \<le> g x"
hoelzl@41981
  1496
      then have [simp]: "G i ` space M \<inter> {y. G i x = y \<and> x \<in> space M} = {G i x}" by auto
hoelzl@41981
  1497
      from * G' G have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * (\<Sum>y\<in>G i`space M. (y * ?I y x))"
hoelzl@41981
  1498
        by (subst setsum_extreal_right_distrib) (auto simp: ac_simps)
hoelzl@41981
  1499
      also have "\<dots> = f x * G i x"
hoelzl@41981
  1500
        by (simp add: indicator_def if_distrib setsum_cases)
hoelzl@41981
  1501
      finally have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * G i x" . }
hoelzl@41981
  1502
    note to_singleton = this
hoelzl@41981
  1503
    have "integral\<^isup>P M' (G i) = integral\<^isup>S M' (G i)"
hoelzl@41981
  1504
      using G T.positive_integral_eq_simple_integral by simp
hoelzl@41981
  1505
    also have "\<dots> = (\<Sum>y\<in>G i`space M. y * (\<integral>\<^isup>+x. f x * ?I y x \<partial>M))"
hoelzl@41981
  1506
      unfolding simple_integral_def M' by simp
hoelzl@41981
  1507
    also have "\<dots> = (\<Sum>y\<in>G i`space M. (\<integral>\<^isup>+x. y * (f x * ?I y x) \<partial>M))"
hoelzl@41981
  1508
      using f G' G by (auto intro!: setsum_cong positive_integral_cmult[symmetric])
hoelzl@41981
  1509
    also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) \<partial>M)"
hoelzl@41981
  1510
      using f G' G by (auto intro!: positive_integral_setsum[symmetric])
hoelzl@41981
  1511
    finally have "integral\<^isup>P M' (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)"
hoelzl@41981
  1512
      using f g G' to_singleton by (auto intro!: positive_integral_cong_AE) }
hoelzl@41981
  1513
  note [simp] = this
hoelzl@41981
  1514
  have "integral\<^isup>P M' g = (SUP i. integral\<^isup>P M' (G i))" using G'(1) G_M'(1) G
hoelzl@41981
  1515
    using T.positive_integral_monotone_convergence_SUP[symmetric, OF `incseq G`]
hoelzl@41981
  1516
    by (simp cong: T.positive_integral_cong_AE)
hoelzl@41981
  1517
  also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f x * G i x \<partial>M))" by simp
hoelzl@41981
  1518
  also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. f x * G i x) \<partial>M)"
hoelzl@41981
  1519
    using f G' G(2)[THEN incseq_SucD] G
hoelzl@41981
  1520
    by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
hoelzl@41981
  1521
       (auto simp: extreal_mult_left_mono le_fun_def extreal_zero_le_0_iff)
hoelzl@41981
  1522
  also have "\<dots> = (\<integral>\<^isup>+x. f x * g x \<partial>M)" using f G' G g
hoelzl@41981
  1523
    by (intro positive_integral_cong_AE)
hoelzl@41981
  1524
       (auto simp add: SUPR_extreal_cmult split: split_max)
hoelzl@41981
  1525
  finally show "integral\<^isup>P M' g = (\<integral>\<^isup>+x. f x * g x \<partial>M)" .
hoelzl@35582
  1526
qed
hoelzl@35582
  1527
hoelzl@38656
  1528
lemma (in measure_space) positive_integral_0_iff:
hoelzl@41981
  1529
  assumes u: "u \<in> borel_measurable M" and pos: "AE x. 0 \<le> u x"
hoelzl@41689
  1530
  shows "integral\<^isup>P M u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
hoelzl@38656
  1531
    (is "_ \<longleftrightarrow> \<mu> ?A = 0")
hoelzl@35582
  1532
proof -
hoelzl@41981
  1533
  have u_eq: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u"
hoelzl@38656
  1534
    by (auto intro!: positive_integral_cong simp: indicator_def)
hoelzl@38656
  1535
  show ?thesis
hoelzl@38656
  1536
  proof
hoelzl@38656
  1537
    assume "\<mu> ?A = 0"
hoelzl@41981
  1538
    with positive_integral_null_set[of ?A u] u
hoelzl@41981
  1539
    show "integral\<^isup>P M u = 0" by (simp add: u_eq)
hoelzl@38656
  1540
  next
hoelzl@41981
  1541
    { fix r :: extreal and n :: nat assume gt_1: "1 \<le> real n * r"
hoelzl@41981
  1542
      then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_extreal_def)
hoelzl@41981
  1543
      then have "0 \<le> r" by (auto simp add: extreal_zero_less_0_iff) }
hoelzl@41981
  1544
    note gt_1 = this
hoelzl@41689
  1545
    assume *: "integral\<^isup>P M u = 0"
hoelzl@41981
  1546
    let "?M n" = "{x \<in> space M. 1 \<le> real (n::nat) * u x}"
hoelzl@38656
  1547
    have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
hoelzl@38656
  1548
    proof -
hoelzl@41981
  1549
      { fix n :: nat
hoelzl@41981
  1550
        from positive_integral_Markov_inequality[OF u pos, of ?A "extreal (real n)"]
hoelzl@41981
  1551
        have "\<mu> (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
hoelzl@41981
  1552
        moreover have "0 \<le> \<mu> (?M n \<inter> ?A)" using u by auto
hoelzl@41981
  1553
        ultimately have "\<mu> (?M n \<inter> ?A) = 0" by auto }
hoelzl@38656
  1554
      thus ?thesis by simp
hoelzl@35582
  1555
    qed
hoelzl@38656
  1556
    also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
hoelzl@38656
  1557
    proof (safe intro!: continuity_from_below)
hoelzl@38656
  1558
      fix n show "?M n \<inter> ?A \<in> sets M"
hoelzl@41981
  1559
        using u by (auto intro!: Int)
hoelzl@38656
  1560
    next
hoelzl@41981
  1561
      show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
hoelzl@41981
  1562
      proof (safe intro!: incseq_SucI)
hoelzl@41981
  1563
        fix n :: nat and x
hoelzl@41981
  1564
        assume *: "1 \<le> real n * u x"
hoelzl@41981
  1565
        also from gt_1[OF this] have "real n * u x \<le> real (Suc n) * u x"
hoelzl@41981
  1566
          using `0 \<le> u x` by (auto intro!: extreal_mult_right_mono)
hoelzl@41981
  1567
        finally show "1 \<le> real (Suc n) * u x" by auto
hoelzl@41981
  1568
      qed
hoelzl@38656
  1569
    qed
hoelzl@41981
  1570
    also have "\<dots> = \<mu> {x\<in>space M. 0 < u x}"
hoelzl@41981
  1571
    proof (safe intro!: arg_cong[where f="\<mu>"] dest!: gt_1)
hoelzl@41981
  1572
      fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
hoelzl@38656
  1573
      show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
hoelzl@38656
  1574
      proof (cases "u x")
hoelzl@41981
  1575
        case (real r) with `0 < u x` have "0 < r" by auto
hoelzl@41981
  1576
        obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
hoelzl@41981
  1577
        hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
hoelzl@41981
  1578
        hence "1 \<le> real j * r" using real `0 < r` by auto
hoelzl@41981
  1579
        thus ?thesis using `0 < r` real by (auto simp: one_extreal_def)
hoelzl@41981
  1580
      qed (insert `0 < u x`, auto)
hoelzl@41981
  1581
    qed auto
hoelzl@41981
  1582
    finally have "\<mu> {x\<in>space M. 0 < u x} = 0" by simp
hoelzl@41981
  1583
    moreover
hoelzl@41981
  1584
    from pos have "AE x. \<not> (u x < 0)" by auto
hoelzl@41981
  1585
    then have "\<mu> {x\<in>space M. u x < 0} = 0"
hoelzl@41981
  1586
      using AE_iff_null_set u by auto
hoelzl@41981
  1587
    moreover have "\<mu> {x\<in>space M. u x \<noteq> 0} = \<mu> {x\<in>space M. u x < 0} + \<mu> {x\<in>space M. 0 < u x}"
hoelzl@41981
  1588
      using u by (subst measure_additive) (auto intro!: arg_cong[where f=\<mu>])
hoelzl@41981
  1589
    ultimately show "\<mu> ?A = 0" by simp
hoelzl@35582
  1590
  qed
hoelzl@35582
  1591
qed
hoelzl@35582
  1592
hoelzl@41705
  1593
lemma (in measure_space) positive_integral_0_iff_AE:
hoelzl@41705
  1594
  assumes u: "u \<in> borel_measurable M"
hoelzl@41981
  1595
  shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. u x \<le> 0)"
hoelzl@41705
  1596
proof -
hoelzl@41981
  1597
  have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
hoelzl@41705
  1598
    using u by auto
hoelzl@41981
  1599
  from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
hoelzl@41981
  1600
  have "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. max 0 (u x) = 0)"
hoelzl@41981
  1601
    unfolding positive_integral_max_0
hoelzl@41981
  1602
    using AE_iff_null_set[OF sets] u by auto
hoelzl@41981
  1603
  also have "\<dots> \<longleftrightarrow> (AE x. u x \<le> 0)" by (auto split: split_max)
hoelzl@41981
  1604
  finally show ?thesis .
hoelzl@41705
  1605
qed
hoelzl@41705
  1606
hoelzl@39092
  1607
lemma (in measure_space) positive_integral_restricted:
hoelzl@41981
  1608
  assumes A: "A \<in> sets M"
hoelzl@41689
  1609
  shows "integral\<^isup>P (restricted_space A) f = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
hoelzl@41689
  1610
    (is "integral\<^isup>P ?R f = integral\<^isup>P M ?f")
hoelzl@39092
  1611
proof -
hoelzl@41981
  1612
  interpret R: measure_space ?R
hoelzl@41981
  1613
    by (rule restricted_measure_space) fact
hoelzl@41981
  1614
  let "?I g x" = "g x * indicator A x :: extreal"
hoelzl@39092
  1615
  show ?thesis
hoelzl@41981
  1616
    unfolding positive_integral_def
hoelzl@41981
  1617
    unfolding simple_function_restricted[OF A]
hoelzl@41981
  1618
    unfolding AE_restricted[OF A]
hoelzl@41981
  1619
  proof (safe intro!: SUPR_eq)
hoelzl@41981
  1620
    fix g assume g: "simple_function M (?I g)" and le: "g \<le> max 0 \<circ> f"
hoelzl@41981
  1621
    show "\<exists>j\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> ?I f}.
hoelzl@41981
  1622
      integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M j"
hoelzl@41981
  1623
    proof (safe intro!: bexI[of _ "?I g"])
hoelzl@41981
  1624
      show "integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M (?I g)"
hoelzl@41981
  1625
        using g A by (simp add: simple_integral_restricted)
hoelzl@41981
  1626
      show "?I g \<le> max 0 \<circ> ?I f"
hoelzl@41981
  1627
        using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
hoelzl@41981
  1628
    qed fact
hoelzl@39092
  1629
  next
hoelzl@41981
  1630
    fix g assume g: "simple_function M g" and le: "g \<le> max 0 \<circ> ?I f"
hoelzl@41981
  1631
    show "\<exists>i\<in>{g. simple_function M (?I g) \<and> g \<le> max 0 \<circ> f}.
hoelzl@41981
  1632
      integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) i"
hoelzl@41981
  1633
    proof (safe intro!: bexI[of _ "?I g"])
hoelzl@41981
  1634
      show "?I g \<le> max 0 \<circ> f"
hoelzl@41981
  1635
        using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
hoelzl@41981
  1636
      from le have "\<And>x. g x \<le> ?I (?I g) x"
hoelzl@41981
  1637
        by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
hoelzl@41981
  1638
      then show "integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) (?I g)"
hoelzl@41981
  1639
        using A g by (auto intro!: simple_integral_mono simp: simple_integral_restricted)
hoelzl@41981
  1640
      show "simple_function M (?I (?I g))" using g A by auto
hoelzl@41981
  1641
    qed
hoelzl@39092
  1642
  qed
hoelzl@39092
  1643
qed
hoelzl@39092
  1644
hoelzl@41545
  1645
lemma (in measure_space) positive_integral_subalgebra:
hoelzl@41981
  1646
  assumes f: "f \<in> borel_measurable N" "AE x in N. 0 \<le> f x"
hoelzl@41689
  1647
  and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
hoelzl@41689
  1648
  and sa: "sigma_algebra N"
hoelzl@41689
  1649
  shows "integral\<^isup>P N f = integral\<^isup>P M f"
hoelzl@39092
  1650
proof -
hoelzl@41689
  1651
  interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
hoelzl@41981
  1652
  from N.borel_measurable_implies_simple_function_sequence'[OF f(1)] guess fs . note fs = this
hoelzl@41981
  1653
  note sf = simple_function_subalgebra[OF fs(1) N(1,2)]
hoelzl@41981
  1654
  from N.positive_integral_monotone_convergence_simple[OF fs(2,5,1), symmetric]
hoelzl@41689
  1655
  have "integral\<^isup>P N f = (SUP i. \<Sum>x\<in>fs i ` space M. x * N.\<mu> (fs i -` {x} \<inter> space M))"
hoelzl@41981
  1656
    unfolding fs(4) positive_integral_max_0
hoelzl@41981
  1657
    unfolding simple_integral_def `space N = space M` by simp
hoelzl@41689
  1658
  also have "\<dots> = (SUP i. \<Sum>x\<in>fs i ` space M. x * \<mu> (fs i -` {x} \<inter> space M))"
hoelzl@41981
  1659
    using N N.simple_functionD(2)[OF fs(1)] unfolding `space N = space M` by auto
hoelzl@41689
  1660
  also have "\<dots> = integral\<^isup>P M f"
hoelzl@41981
  1661
    using positive_integral_monotone_convergence_simple[OF fs(2,5) sf, symmetric]
hoelzl@41981
  1662
    unfolding fs(4) positive_integral_max_0
hoelzl@41981
  1663
    unfolding simple_integral_def `space N = space M` by simp
hoelzl@41689
  1664
  finally show ?thesis .
hoelzl@39092
  1665
qed
hoelzl@39092
  1666
hoelzl@35692
  1667
section "Lebesgue Integral"
hoelzl@35692
  1668
hoelzl@41689
  1669
definition integrable where
hoelzl@41689
  1670
  "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
hoelzl@41981
  1671
    (\<integral>\<^isup>+ x. extreal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^isup>+ x. extreal (- f x) \<partial>M) \<noteq> \<infinity>"
hoelzl@35692
  1672
hoelzl@41689
  1673
lemma integrableD[dest]:
hoelzl@41689
  1674
  assumes "integrable M f"
hoelzl@41981
  1675
  shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. extreal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^isup>+ x. extreal (- f x) \<partial>M) \<noteq> \<infinity>"
hoelzl@38656
  1676
  using assms unfolding integrable_def by auto
hoelzl@35692
  1677
hoelzl@41689
  1678
definition lebesgue_integral_def:
hoelzl@41981
  1679
  "integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. extreal (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. extreal (- f x) \<partial>M))"
hoelzl@41689
  1680
hoelzl@41689
  1681
syntax
hoelzl@41689
  1682
  "_lebesgue_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
hoelzl@41689
  1683
hoelzl@41689
  1684
translations
hoelzl@41689
  1685
  "\<integral> x. f \<partial>M" == "CONST integral\<^isup>L M (%x. f)"
hoelzl@38656
  1686
hoelzl@41981
  1687
lemma (in measure_space) integrableE:
hoelzl@41981
  1688
  assumes "integrable M f"
hoelzl@41981
  1689
  obtains r q where
hoelzl@41981
  1690
    "(\<integral>\<^isup>+x. extreal (f x)\<partial>M) = extreal r"
hoelzl@41981
  1691
    "(\<integral>\<^isup>+x. extreal (-f x)\<partial>M) = extreal q"
hoelzl@41981
  1692
    "f \<in> borel_measurable M" "integral\<^isup>L M f = r - q"
hoelzl@41981
  1693
  using assms unfolding integrable_def lebesgue_integral_def
hoelzl@41981
  1694
  using positive_integral_positive[of "\<lambda>x. extreal (f x)"]
hoelzl@41981
  1695
  using positive_integral_positive[of "\<lambda>x. extreal (-f x)"]
hoelzl@41981
  1696
  by (cases rule: extreal2_cases[of "(\<integral>\<^isup>+x. extreal (-f x)\<partial>M)" "(\<integral>\<^isup>+x. extreal (f x)\<partial>M)"]) auto
hoelzl@41981
  1697
hoelzl@38656
  1698
lemma (in measure_space) integral_cong:
hoelzl@41689
  1699
  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
hoelzl@41689
  1700
  shows "integral\<^isup>L M f = integral\<^isup>L M g"
hoelzl@41689
  1701
  using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def)
hoelzl@35582
  1702
hoelzl@40859
  1703
lemma (in measure_space) integral_cong_measure:
hoelzl@41689
  1704
  assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
hoelzl@41689
  1705
  shows "integral\<^isup>L N f = integral\<^isup>L M f"
hoelzl@41981
  1706
  by (simp add: positive_integral_cong_measure[OF assms] lebesgue_integral_def)
hoelzl@40859
  1707
hoelzl@40859
  1708
lemma (in measure_space) integral_cong_AE:
hoelzl@40859
  1709
  assumes cong: "AE x. f x = g x"
hoelzl@41689
  1710
  shows "integral\<^isup>L M f = integral\<^isup>L M g"
hoelzl@40859
  1711
proof -
hoelzl@41981
  1712
  have *: "AE x. extreal (f x) = extreal (g x)"
hoelzl@41981
  1713
    "AE x. extreal (- f x) = extreal (- g x)" using cong by auto
hoelzl@41981
  1714
  show ?thesis
hoelzl@41981
  1715
    unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
hoelzl@40859
  1716
qed
hoelzl@40859
  1717
hoelzl@38656
  1718
lemma (in measure_space) integrable_cong:
hoelzl@41689
  1719
  "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
hoelzl@38656
  1720
  by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
hoelzl@38656
  1721
hoelzl@38656
  1722
lemma (in measure_space) integral_eq_positive_integral:
hoelzl@41981
  1723
  assumes f: "\<And>x. 0 \<le> f x"
hoelzl@41981
  1724
  shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. extreal (f x) \<partial>M)"
hoelzl@35582
  1725
proof -
hoelzl@41981
  1726
  { fix x have "max 0 (extreal (- f x)) = 0" using f[of x] by (simp split: split_max) }
hoelzl@41981
  1727
  then have "0 = (\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>M)" by simp
hoelzl@41981
  1728
  also have "\<dots> = (\<integral>\<^isup>+ x. extreal (- f x) \<partial>M)" unfolding positive_integral_max_0 ..
hoelzl@41981
  1729
  finally show ?thesis
hoelzl@41981
  1730
    unfolding lebesgue_integral_def by simp
hoelzl@35582
  1731
qed
hoelzl@35582
  1732
hoelzl@41661
  1733
lemma (in measure_space) integral_vimage:
hoelzl@41831
  1734
  assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
hoelzl@41831
  1735
  assumes f: "f \<in> borel_measurable M'"
hoelzl@41831
  1736
  shows "integral\<^isup>L M' f = (\<integral>x. f (T x) \<partial>M)"
hoelzl@40859
  1737
proof -
hoelzl@41831
  1738
  interpret T: measure_space M' by (rule measure_space_vimage[OF T])
hoelzl@41831
  1739
  from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
hoelzl@41981
  1740
  have borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable M'" "(\<lambda>x. extreal (- f x)) \<in> borel_measurable M'"
hoelzl@41661
  1741
    and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
hoelzl@41831
  1742
    using f by (auto simp: comp_def)
hoelzl@41831
  1743
  then show ?thesis
hoelzl@41689
  1744
    using f unfolding lebesgue_integral_def integrable_def
hoelzl@41831
  1745
    by (auto simp: borel[THEN positive_integral_vimage[OF T]])
hoelzl@41831
  1746
qed
hoelzl@41831
  1747
hoelzl@41831
  1748
lemma (in measure_space) integrable_vimage:
hoelzl@41831
  1749
  assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
hoelzl@41831
  1750
  assumes f: "integrable M' f"
hoelzl@41831
  1751
  shows "integrable M (\<lambda>x. f (T x))"
hoelzl@41831
  1752
proof -
hoelzl@41831
  1753
  interpret T: measure_space M' by (rule measure_space_vimage[OF T])
hoelzl@41831
  1754
  from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
hoelzl@41981
  1755
  have borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable M'" "(\<lambda>x. extreal (- f x)) \<in> borel_measurable M'"
hoelzl@41831
  1756
    and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
hoelzl@41831
  1757
    using f by (auto simp: comp_def)
hoelzl@41831
  1758
  then show ?thesis
hoelzl@41831
  1759
    using f unfolding lebesgue_integral_def integrable_def
hoelzl@41831
  1760
    by (auto simp: borel[THEN positive_integral_vimage[OF T]])
hoelzl@40859
  1761
qed
hoelzl@40859
  1762
hoelzl@38656
  1763
lemma (in measure_space) integral_minus[intro, simp]:
hoelzl@41689
  1764
  assumes "integrable M f"
hoelzl@41689
  1765
  shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^isup>L M f"
hoelzl@41689
  1766
  using assms by (auto simp: integrable_def lebesgue_integral_def)
hoelzl@38656
  1767
hoelzl@38656
  1768
lemma (in measure_space) integral_of_positive_diff:
hoelzl@41689
  1769
  assumes integrable: "integrable M u" "integrable M v"
hoelzl@38656
  1770
  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"
hoelzl@41689
  1771
  shows "integrable M f" and "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
hoelzl@35582
  1772
proof -
hoelzl@41981
  1773
  let "?f x" = "max 0 (extreal (f x))"
hoelzl@41981
  1774
  let "?mf x" = "max 0 (extreal (- f x))"
hoelzl@41981
  1775
  let "?u x" = "max 0 (extreal (u x))"
hoelzl@41981
  1776
  let "?v x" = "max 0 (extreal (v x))"
hoelzl@38656
  1777
hoelzl@38656
  1778
  from borel_measurable_diff[of u v] integrable
hoelzl@38656
  1779
  have f_borel: "?f \<in> borel_measurable M" and
hoelzl@38656
  1780
    mf_borel: "?mf \<in> borel_measurable M" and
hoelzl@38656
  1781
    v_borel: "?v \<in> borel_measurable M" and
hoelzl@38656
  1782
    u_borel: "?u \<in> borel_measurable M" and
hoelzl@38656
  1783
    "f \<in> borel_measurable M"
hoelzl@38656
  1784
    by (auto simp: f_def[symmetric] integrable_def)
hoelzl@35582
  1785
hoelzl@41981
  1786
  have "(\<integral>\<^isup>+ x. extreal (u x - v x) \<partial>M) \<le> integral\<^isup>P M ?u"
hoelzl@41981
  1787
    using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
hoelzl@41981
  1788
  moreover have "(\<integral>\<^isup>+ x. extreal (v x - u x) \<partial>M) \<le> integral\<^isup>P M ?v"
hoelzl@41981
  1789
    using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
hoelzl@41689
  1790
  ultimately show f: "integrable M f"
hoelzl@41689
  1791
    using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
hoelzl@41981
  1792
    by (auto simp: integrable_def f_def positive_integral_max_0)
hoelzl@35582
  1793
hoelzl@38656
  1794
  have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
hoelzl@41981
  1795
    unfolding f_def using pos by (simp split: split_max)
hoelzl@41981
  1796
  then have "(\<integral>\<^isup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^isup>+ x. ?v x + ?f x \<partial>M)" by simp
hoelzl@41981
  1797
  then have "real (integral\<^isup>P M ?u + integral\<^isup>P M ?mf) =
hoelzl@41689
  1798
      real (integral\<^isup>P M ?v + integral\<^isup>P M ?f)"
hoelzl@41981
  1799
    using positive_integral_add[OF u_borel _ mf_borel]
hoelzl@41981
  1800
    using positive_integral_add[OF v_borel _ f_borel]
hoelzl@38656
  1801
    by auto
hoelzl@41689
  1802
  then show "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
hoelzl@41981
  1803
    unfolding positive_integral_max_0
hoelzl@41981
  1804
    unfolding pos[THEN integral_eq_positive_integral]
hoelzl@41981
  1805
    using integrable f by (auto elim!: integrableE)
hoelzl@35582
  1806
qed
hoelzl@35582
  1807
hoelzl@38656
  1808
lemma (in measure_space) integral_linear:
hoelzl@41689
  1809
  assumes "integrable M f" "integrable M g" and "0 \<le> a"
hoelzl@41689
  1810
  shows "integrable M (\<lambda>t. a * f t + g t)"
hoelzl@41981
  1811
  and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g" (is ?EQ)
hoelzl@38656
  1812
proof -
hoelzl@41981
  1813
  let "?f x" = "max 0 (extreal (f x))"
hoelzl@41981
  1814
  let "?g x" = "max 0 (extreal (g x))"
hoelzl@41981
  1815
  let "?mf x" = "max 0 (extreal (- f x))"
hoelzl@41981
  1816
  let "?mg x" = "max 0 (extreal (- g x))"
hoelzl@38656
  1817
  let "?p t" = "max 0 (a * f t) + max 0 (g t)"
hoelzl@38656
  1818
  let "?n t" = "max 0 (- (a * f t)) + max 0 (- g t)"
hoelzl@38656
  1819
hoelzl@41981
  1820
  from assms have linear:
hoelzl@41981
  1821
    "(\<integral>\<^isup>+ x. extreal a * ?f x + ?g x \<partial>M) = extreal a * integral\<^isup>P M ?f + integral\<^isup>P M ?g"
hoelzl@41981
  1822
    "(\<integral>\<^isup>+ x. extreal a * ?mf x + ?mg x \<partial>M) = extreal a * integral\<^isup>P M ?mf + integral\<^isup>P M ?mg"
hoelzl@41981
  1823
    by (auto intro!: positive_integral_linear simp: integrable_def)
hoelzl@35582
  1824
hoelzl@41981
  1825
  have *: "(\<integral>\<^isup>+x. extreal (- ?p x) \<partial>M) = 0" "(\<integral>\<^isup>+x. extreal (- ?n x) \<partial>M) = 0"
hoelzl@41981
  1826
    using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def)
hoelzl@41981
  1827
  have **: "\<And>x. extreal a * ?f x + ?g x = max 0 (extreal (?p x))"
hoelzl@41981
  1828
           "\<And>x. extreal a * ?mf x + ?mg x = max 0 (extreal (?n x))"
hoelzl@41981
  1829
    using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff)
hoelzl@35582
  1830
hoelzl@41689
  1831
  have "integrable M ?p" "integrable M ?n"
hoelzl@38656
  1832
      "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
hoelzl@41981
  1833
    using linear assms unfolding integrable_def ** *
hoelzl@41981
  1834
    by (auto simp: positive_integral_max_0)
hoelzl@38656
  1835
  note diff = integral_of_positive_diff[OF this]
hoelzl@38656
  1836
hoelzl@41689
  1837
  show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
hoelzl@41981
  1838
  from assms linear show ?EQ
hoelzl@41981
  1839
    unfolding diff(2) ** positive_integral_max_0
hoelzl@41981
  1840
    unfolding lebesgue_integral_def *
hoelzl@41981
  1841
    by (auto elim!: integrableE simp: field_simps)
hoelzl@38656
  1842
qed
hoelzl@38656
  1843
hoelzl@38656
  1844
lemma (in measure_space) integral_add[simp, intro]:
hoelzl@41689
  1845
  assumes "integrable M f" "integrable M g"
hoelzl@41689
  1846
  shows "integrable M (\<lambda>t. f t + g t)"
hoelzl@41689
  1847
  and "(\<integral> t. f t + g t \<partial>M) = integral\<^isup>L M f + integral\<^isup>L M g"
hoelzl@38656
  1848
  using assms integral_linear[where a=1] by auto
hoelzl@38656
  1849
hoelzl@38656
  1850
lemma (in measure_space) integral_zero[simp, intro]:
hoelzl@41689
  1851
  shows "integrable M (\<lambda>x. 0)" "(\<integral> x.0 \<partial>M) = 0"
hoelzl@41689
  1852
  unfolding integrable_def lebesgue_integral_def
hoelzl@38656
  1853
  by (auto simp add: borel_measurable_const)
hoelzl@35582
  1854
hoelzl@38656
  1855
lemma (in measure_space) integral_cmult[simp, intro]:
hoelzl@41689
  1856
  assumes "integrable M f"