src/HOL/Probability/Lebesgue_Integration.thy
author hoelzl
Tue Aug 24 14:41:37 2010 +0200 (2010-08-24)
changeset 38705 aaee86c0e237
parent 38656 d5d342611edb
child 39092 98de40859858
permissions -rw-r--r--
moved generic lemmas in Probability to HOL
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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
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begin
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section "@{text \<mu>}-null sets"
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abbreviation (in measure_space) "null_sets \<equiv> {N\<in>sets M. \<mu> N = 0}"
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lemma sums_If_finite:
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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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  "(\<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 (in sigma_algebra) "simple_function 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 g"
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  shows "finite (g ` space M)"
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  "x \<in> g ` space M \<Longrightarrow> g -` {x} \<inter> space M \<in> sets M"
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  using assms unfolding simple_function_def by auto
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lemma (in sigma_algebra) simple_function_indicator_representation:
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  fixes f ::"'a \<Rightarrow> pinfreal"
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  assumes f: "simple_function 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 (\<lambda>x. h x * indicator (- space M) x::pinfreal)" (is "simple_function ?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 f \<longleftrightarrow> simple_function 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) borel_measurable_simple_function:
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  assumes "simple_function 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 by (auto intro: finite_subset)
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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 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_const[intro, simp]:
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  "simple_function (\<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 f"
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  shows "simple_function (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 (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 f"
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  assumes "simple_function g"
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  shows "simple_function (\<lambda>x. (f x, g x))" (is "simple_function ?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 f"
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  shows "simple_function (\<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 f" and "simple_function g"
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  shows "simple_function (\<lambda>x. h (f x) (g x))"
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proof -
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  have "simple_function ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
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    using assms by auto
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  thus ?thesis by (simp_all add: comp_def)
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qed
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lemmas (in sigma_algebra) simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
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  and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
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  and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
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  and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
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  and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
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  and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
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lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
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  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
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  shows "simple_function (\<lambda>x. \<Sum>i\<in>P. f i x)"
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proof cases
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  assume "finite P" from this assms show ?thesis by induct auto
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qed auto
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lemma (in sigma_algebra) simple_function_le_measurable:
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  assumes "simple_function f" "simple_function g"
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  shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
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proof -
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  have *: "{x \<in> space M. f x \<le> g x} =
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    (\<Union>(F, G)\<in>f`space M \<times> g`space M.
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      if F \<le> G then (f -` {F} \<inter> space M) \<inter> (g -` {G} \<inter> space M) else {})"
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    apply (auto split: split_if_asm)
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    apply (rule_tac x=x in bexI)
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    apply (rule_tac x=x in bexI)
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    by simp_all
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  have **: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow>
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    (f -` {f x} \<inter> space M) \<inter> (g -` {g y} \<inter> space M) \<in> sets M"
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    using assms unfolding simple_function_def by auto
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  have "finite (f`space M \<times> g`space M)"
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    using assms unfolding simple_function_def by auto
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  thus ?thesis unfolding *
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    apply (rule finite_UN)
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    using assms unfolding simple_function_def
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    by (auto intro!: **)
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qed
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lemma setsum_indicator_disjoint_family:
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  fixes f :: "'d \<Rightarrow> 'e::semiring_1"
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  assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
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  shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
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proof -
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  have "P \<inter> {i. x \<in> A i} = {j}"
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    using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
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    by auto
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  thus ?thesis
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    unfolding indicator_def
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    by (simp add: if_distrib setsum_cases[OF `finite P`])
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qed
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lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
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  fixes u :: "'a \<Rightarrow> pinfreal"
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  assumes u: "u \<in> borel_measurable M"
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  shows "\<exists>f. (\<forall>i. simple_function (f i) \<and> (\<forall>x\<in>space M. f i x \<noteq> \<omega>)) \<and> f \<up> u"
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proof -
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  have "\<exists>f. \<forall>x j. (of_nat j \<le> u x \<longrightarrow> f x j = j*2^j) \<and>
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    (u x < of_nat j \<longrightarrow> of_nat (f x j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f x j)))"
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    (is "\<exists>f. \<forall>x j. ?P x j (f x j)")
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  proof(rule choice, rule, rule choice, rule)
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    fix x j show "\<exists>n. ?P x j n"
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    proof cases
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      assume *: "u x < of_nat j"
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      then obtain r where r: "u x = Real r" "0 \<le> r" by (cases "u x") auto
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      from reals_Archimedean6a[of "r * 2^j"]
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      obtain n :: nat where "real n \<le> r * 2 ^ j" "r * 2 ^ j < real (Suc n)"
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        using `0 \<le> r` by (auto simp: zero_le_power zero_le_mult_iff)
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      thus ?thesis using r * by (auto intro!: exI[of _ n])
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    qed auto
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  qed
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  then obtain f where top: "\<And>j x. of_nat j \<le> u x \<Longrightarrow> f x j = j*2^j" and
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    upper: "\<And>j x. u x < of_nat j \<Longrightarrow> of_nat (f x j) \<le> u x * 2^j" and
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    lower: "\<And>j x. u x < of_nat j \<Longrightarrow> u x * 2^j < of_nat (Suc (f x j))" by blast
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  { fix j x P
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    assume 1: "of_nat j \<le> u x \<Longrightarrow> P (j * 2^j)"
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    assume 2: "\<And>k. \<lbrakk> u x < of_nat j ; of_nat k \<le> u x * 2^j ; u x * 2^j < of_nat (Suc k) \<rbrakk> \<Longrightarrow> P k"
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    have "P (f x j)"
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    proof cases
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      assume "of_nat j \<le> u x" thus "P (f x j)"
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        using top[of j x] 1 by auto
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    next
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      assume "\<not> of_nat j \<le> u x"
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      hence "u x < of_nat j" "of_nat (f x j) \<le> u x * 2^j" "u x * 2^j < of_nat (Suc (f x j))"
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        using upper lower by auto
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      from 2[OF this] show "P (f x j)" .
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    qed }
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  note fI = this
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  { fix j x
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    have "f x j = j * 2 ^ j \<longleftrightarrow> of_nat j \<le> u x"
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      by (rule fI, simp, cases "u x") (auto split: split_if_asm) }
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  note f_eq = this
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  { fix j x
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    have "f x j \<le> j * 2 ^ j"
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    proof (rule fI)
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      fix k assume *: "u x < of_nat j"
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      assume "of_nat k \<le> u x * 2 ^ j"
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      also have "\<dots> \<le> of_nat (j * 2^j)"
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        using * by (cases "u x") (auto simp: zero_le_mult_iff)
hoelzl@38656
   275
      finally show "k \<le> j*2^j" by (auto simp del: real_of_nat_mult)
hoelzl@38656
   276
    qed simp }
hoelzl@38656
   277
  note f_upper = this
hoelzl@35582
   278
hoelzl@38656
   279
  let "?g j x" = "of_nat (f x j) / 2^j :: pinfreal"
hoelzl@38656
   280
  show ?thesis unfolding simple_function_def isoton_fun_expand unfolding isoton_def
hoelzl@38656
   281
  proof (safe intro!: exI[of _ ?g])
hoelzl@38656
   282
    fix j
hoelzl@38656
   283
    have *: "?g j ` space M \<subseteq> (\<lambda>x. of_nat x / 2^j) ` {..j * 2^j}"
hoelzl@38656
   284
      using f_upper by auto
hoelzl@38656
   285
    thus "finite (?g j ` space M)" by (rule finite_subset) auto
hoelzl@38656
   286
  next
hoelzl@38656
   287
    fix j t assume "t \<in> space M"
hoelzl@38656
   288
    have **: "?g j -` {?g j t} \<inter> space M = {x \<in> space M. f t j = f x j}"
hoelzl@38656
   289
      by (auto simp: power_le_zero_eq Real_eq_Real mult_le_0_iff)
hoelzl@35582
   290
hoelzl@38656
   291
    show "?g j -` {?g j t} \<inter> space M \<in> sets M"
hoelzl@38656
   292
    proof cases
hoelzl@38656
   293
      assume "of_nat j \<le> u t"
hoelzl@38656
   294
      hence "?g j -` {?g j t} \<inter> space M = {x\<in>space M. of_nat j \<le> u x}"
hoelzl@38656
   295
        unfolding ** f_eq[symmetric] by auto
hoelzl@38656
   296
      thus "?g j -` {?g j t} \<inter> space M \<in> sets M"
hoelzl@38656
   297
        using u by auto
hoelzl@35582
   298
    next
hoelzl@38656
   299
      assume not_t: "\<not> of_nat j \<le> u t"
hoelzl@38656
   300
      hence less: "f t j < j*2^j" using f_eq[symmetric] `f t j \<le> j*2^j` by auto
hoelzl@38656
   301
      have split_vimage: "?g j -` {?g j t} \<inter> space M =
hoelzl@38656
   302
          {x\<in>space M. of_nat (f t j)/2^j \<le> u x} \<inter> {x\<in>space M. u x < of_nat (Suc (f t j))/2^j}"
hoelzl@38656
   303
        unfolding **
hoelzl@38656
   304
      proof safe
hoelzl@38656
   305
        fix x assume [simp]: "f t j = f x j"
hoelzl@38656
   306
        have *: "\<not> of_nat j \<le> u x" using not_t f_eq[symmetric] by simp
hoelzl@38656
   307
        hence "of_nat (f t j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f t j))"
hoelzl@38656
   308
          using upper lower by auto
hoelzl@38656
   309
        hence "of_nat (f t j) / 2^j \<le> u x \<and> u x < of_nat (Suc (f t j))/2^j" using *
hoelzl@38656
   310
          by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
hoelzl@38656
   311
        thus "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j" by auto
hoelzl@38656
   312
      next
hoelzl@38656
   313
        fix x
hoelzl@38656
   314
        assume "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j"
hoelzl@38656
   315
        hence "of_nat (f t j) \<le> u x * 2 ^ j \<and> u x * 2 ^ j < of_nat (Suc (f t j))"
hoelzl@38656
   316
          by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
hoelzl@38656
   317
        hence 1: "of_nat (f t j) \<le> u x * 2 ^ j" and 2: "u x * 2 ^ j < of_nat (Suc (f t j))" by auto
hoelzl@38656
   318
        note 2
hoelzl@38656
   319
        also have "\<dots> \<le> of_nat (j*2^j)"
hoelzl@38656
   320
          using less by (auto simp: zero_le_mult_iff simp del: real_of_nat_mult)
hoelzl@38656
   321
        finally have bound_ux: "u x < of_nat j"
hoelzl@38656
   322
          by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq)
hoelzl@38656
   323
        show "f t j = f x j"
hoelzl@38656
   324
        proof (rule antisym)
hoelzl@38656
   325
          from 1 lower[OF bound_ux]
hoelzl@38656
   326
          show "f t j \<le> f x j" by (cases "u x") (auto split: split_if_asm)
hoelzl@38656
   327
          from upper[OF bound_ux] 2
hoelzl@38656
   328
          show "f x j \<le> f t j" by (cases "u x") (auto split: split_if_asm)
hoelzl@38656
   329
        qed
hoelzl@38656
   330
      qed
hoelzl@38656
   331
      show ?thesis unfolding split_vimage using u by auto
hoelzl@35582
   332
    qed
hoelzl@38656
   333
  next
hoelzl@38656
   334
    fix j t assume "?g j t = \<omega>" thus False by (auto simp: power_le_zero_eq)
hoelzl@38656
   335
  next
hoelzl@38656
   336
    fix t
hoelzl@38656
   337
    { fix i
hoelzl@38656
   338
      have "f t i * 2 \<le> f t (Suc i)"
hoelzl@38656
   339
      proof (rule fI)
hoelzl@38656
   340
        assume "of_nat (Suc i) \<le> u t"
hoelzl@38656
   341
        hence "of_nat i \<le> u t" by (cases "u t") auto
hoelzl@38656
   342
        thus "f t i * 2 \<le> Suc i * 2 ^ Suc i" unfolding f_eq[symmetric] by simp
hoelzl@38656
   343
      next
hoelzl@38656
   344
        fix k
hoelzl@38656
   345
        assume *: "u t * 2 ^ Suc i < of_nat (Suc k)"
hoelzl@38656
   346
        show "f t i * 2 \<le> k"
hoelzl@38656
   347
        proof (rule fI)
hoelzl@38656
   348
          assume "of_nat i \<le> u t"
hoelzl@38656
   349
          hence "of_nat (i * 2^Suc i) \<le> u t * 2^Suc i"
hoelzl@38656
   350
            by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
hoelzl@38656
   351
          also have "\<dots> < of_nat (Suc k)" using * by auto
hoelzl@38656
   352
          finally show "i * 2 ^ i * 2 \<le> k"
hoelzl@38656
   353
            by (auto simp del: real_of_nat_mult)
hoelzl@38656
   354
        next
hoelzl@38656
   355
          fix j assume "of_nat j \<le> u t * 2 ^ i"
hoelzl@38656
   356
          with * show "j * 2 \<le> k" by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
hoelzl@38656
   357
        qed
hoelzl@38656
   358
      qed
hoelzl@38656
   359
      thus "?g i t \<le> ?g (Suc i) t"
hoelzl@38656
   360
        by (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps) }
hoelzl@38656
   361
    hence mono: "mono (\<lambda>i. ?g i t)" unfolding mono_iff_le_Suc by auto
hoelzl@35582
   362
hoelzl@38656
   363
    show "(SUP j. of_nat (f t j) / 2 ^ j) = u t"
hoelzl@38656
   364
    proof (rule pinfreal_SUPI)
hoelzl@38656
   365
      fix j show "of_nat (f t j) / 2 ^ j \<le> u t"
hoelzl@38656
   366
      proof (rule fI)
hoelzl@38656
   367
        assume "of_nat j \<le> u t" thus "of_nat (j * 2 ^ j) / 2 ^ j \<le> u t"
hoelzl@38656
   368
          by (cases "u t") (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps)
hoelzl@38656
   369
      next
hoelzl@38656
   370
        fix k assume "of_nat k \<le> u t * 2 ^ j"
hoelzl@38656
   371
        thus "of_nat k / 2 ^ j \<le> u t"
hoelzl@38656
   372
          by (cases "u t")
hoelzl@38656
   373
             (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps zero_le_mult_iff)
hoelzl@38656
   374
      qed
hoelzl@38656
   375
    next
hoelzl@38656
   376
      fix y :: pinfreal assume *: "\<And>j. j \<in> UNIV \<Longrightarrow> of_nat (f t j) / 2 ^ j \<le> y"
hoelzl@38656
   377
      show "u t \<le> y"
hoelzl@38656
   378
      proof (cases "u t")
hoelzl@38656
   379
        case (preal r)
hoelzl@38656
   380
        show ?thesis
hoelzl@38656
   381
        proof (rule ccontr)
hoelzl@38656
   382
          assume "\<not> u t \<le> y"
hoelzl@38656
   383
          then obtain p where p: "y = Real p" "0 \<le> p" "p < r" using preal by (cases y) auto
hoelzl@38656
   384
          with LIMSEQ_inverse_realpow_zero[of 2, unfolded LIMSEQ_iff, rule_format, of "r - p"]
hoelzl@38656
   385
          obtain n where n: "\<And>N. n \<le> N \<Longrightarrow> inverse (2^N) < r - p" by auto
hoelzl@38656
   386
          let ?N = "max n (natfloor r + 1)"
hoelzl@38656
   387
          have "u t < of_nat ?N" "n \<le> ?N"
hoelzl@38656
   388
            using ge_natfloor_plus_one_imp_gt[of r n] preal
hoelzl@38705
   389
            using real_natfloor_add_one_gt
hoelzl@38705
   390
            by (auto simp: max_def real_of_nat_Suc)
hoelzl@38656
   391
          from lower[OF this(1)]
hoelzl@38656
   392
          have "r < (real (f t ?N) + 1) / 2 ^ ?N" unfolding less_divide_eq
hoelzl@38656
   393
            using preal by (auto simp add: power_le_zero_eq zero_le_mult_iff real_of_nat_Suc split: split_if_asm)
hoelzl@38656
   394
          hence "u t < of_nat (f t ?N) / 2 ^ ?N + 1 / 2 ^ ?N"
hoelzl@38656
   395
            using preal by (auto simp: field_simps divide_real_def[symmetric])
hoelzl@38656
   396
          with n[OF `n \<le> ?N`] p preal *[of ?N]
hoelzl@38656
   397
          show False
hoelzl@38656
   398
            by (cases "f t ?N") (auto simp: power_le_zero_eq split: split_if_asm)
hoelzl@38656
   399
        qed
hoelzl@38656
   400
      next
hoelzl@38656
   401
        case infinite
hoelzl@38656
   402
        { fix j have "f t j = j*2^j" using top[of j t] infinite by simp
hoelzl@38656
   403
          hence "of_nat j \<le> y" using *[of j]
hoelzl@38656
   404
            by (cases y) (auto simp: power_le_zero_eq zero_le_mult_iff field_simps) }
hoelzl@38656
   405
        note all_less_y = this
hoelzl@38656
   406
        show ?thesis unfolding infinite
hoelzl@38656
   407
        proof (rule ccontr)
hoelzl@38656
   408
          assume "\<not> \<omega> \<le> y" then obtain r where r: "y = Real r" "0 \<le> r" by (cases y) auto
hoelzl@38656
   409
          moreover obtain n ::nat where "r < real n" using ex_less_of_nat by (auto simp: real_eq_of_nat)
hoelzl@38656
   410
          with all_less_y[of n] r show False by auto
hoelzl@38656
   411
        qed
hoelzl@38656
   412
      qed
hoelzl@38656
   413
    qed
hoelzl@35582
   414
  qed
hoelzl@35582
   415
qed
hoelzl@35582
   416
hoelzl@38656
   417
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
hoelzl@38656
   418
  fixes u :: "'a \<Rightarrow> pinfreal"
hoelzl@38656
   419
  assumes "u \<in> borel_measurable M"
hoelzl@38656
   420
  obtains (x) f where "f \<up> u" "\<And>i. simple_function (f i)" "\<And>i. \<omega>\<notin>f i`space M"
hoelzl@35582
   421
proof -
hoelzl@38656
   422
  from borel_measurable_implies_simple_function_sequence[OF assms]
hoelzl@38656
   423
  obtain f where x: "\<And>i. simple_function (f i)" "f \<up> u"
hoelzl@38656
   424
    and fin: "\<And>i. \<And>x. x\<in>space M \<Longrightarrow> f i x \<noteq> \<omega>" by auto
hoelzl@38656
   425
  { fix i from fin[of _ i] have "\<omega> \<notin> f i`space M" by fastsimp }
hoelzl@38656
   426
  with x show thesis by (auto intro!: that[of f])
hoelzl@38656
   427
qed
hoelzl@38656
   428
hoelzl@38656
   429
section "Simple integral"
hoelzl@38656
   430
hoelzl@38656
   431
definition (in measure_space)
hoelzl@38656
   432
  "simple_integral f = (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M))"
hoelzl@35582
   433
hoelzl@38656
   434
lemma (in measure_space) simple_integral_cong:
hoelzl@38656
   435
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
hoelzl@38656
   436
  shows "simple_integral f = simple_integral g"
hoelzl@38656
   437
proof -
hoelzl@38656
   438
  have "f ` space M = g ` space M"
hoelzl@38656
   439
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
hoelzl@38656
   440
    using assms by (auto intro!: image_eqI)
hoelzl@38656
   441
  thus ?thesis unfolding simple_integral_def by simp
hoelzl@38656
   442
qed
hoelzl@38656
   443
hoelzl@38656
   444
lemma (in measure_space) simple_integral_const[simp]:
hoelzl@38656
   445
  "simple_integral (\<lambda>x. c) = c * \<mu> (space M)"
hoelzl@38656
   446
proof (cases "space M = {}")
hoelzl@38656
   447
  case True thus ?thesis unfolding simple_integral_def by simp
hoelzl@38656
   448
next
hoelzl@38656
   449
  case False hence "(\<lambda>x. c) ` space M = {c}" by auto
hoelzl@38656
   450
  thus ?thesis unfolding simple_integral_def by simp
hoelzl@35582
   451
qed
hoelzl@35582
   452
hoelzl@38656
   453
lemma (in measure_space) simple_function_partition:
hoelzl@38656
   454
  assumes "simple_function f" and "simple_function g"
hoelzl@38656
   455
  shows "simple_integral f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. contents (f`A) * \<mu> A)"
hoelzl@38656
   456
    (is "_ = setsum _ (?p ` space M)")
hoelzl@38656
   457
proof-
hoelzl@38656
   458
  let "?sub x" = "?p ` (f -` {x} \<inter> space M)"
hoelzl@38656
   459
  let ?SIGMA = "Sigma (f`space M) ?sub"
hoelzl@35582
   460
hoelzl@38656
   461
  have [intro]:
hoelzl@38656
   462
    "finite (f ` space M)"
hoelzl@38656
   463
    "finite (g ` space M)"
hoelzl@38656
   464
    using assms unfolding simple_function_def by simp_all
hoelzl@38656
   465
hoelzl@38656
   466
  { fix A
hoelzl@38656
   467
    have "?p ` (A \<inter> space M) \<subseteq>
hoelzl@38656
   468
      (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
hoelzl@38656
   469
      by auto
hoelzl@38656
   470
    hence "finite (?p ` (A \<inter> space M))"
hoelzl@38656
   471
      by (rule finite_subset) (auto intro: finite_SigmaI finite_imageI) }
hoelzl@38656
   472
  note this[intro, simp]
hoelzl@35582
   473
hoelzl@38656
   474
  { fix x assume "x \<in> space M"
hoelzl@38656
   475
    have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
hoelzl@38656
   476
    moreover {
hoelzl@38656
   477
      fix x y
hoelzl@38656
   478
      have *: "f -` {f x} \<inter> g -` {g x} \<inter> space M
hoelzl@38656
   479
          = (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)" by auto
hoelzl@38656
   480
      assume "x \<in> space M" "y \<in> space M"
hoelzl@38656
   481
      hence "f -` {f x} \<inter> g -` {g x} \<inter> space M \<in> sets M"
hoelzl@38656
   482
        using assms unfolding simple_function_def * by auto }
hoelzl@38656
   483
    ultimately
hoelzl@38656
   484
    have "\<mu> (f -` {f x} \<inter> space M) = setsum (\<mu>) (?sub (f x))"
hoelzl@38656
   485
      by (subst measure_finitely_additive) auto }
hoelzl@38656
   486
  hence "simple_integral f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
hoelzl@38656
   487
    unfolding simple_integral_def
hoelzl@38656
   488
    by (subst setsum_Sigma[symmetric],
hoelzl@38656
   489
       auto intro!: setsum_cong simp: setsum_right_distrib[symmetric])
hoelzl@38656
   490
  also have "\<dots> = (\<Sum>A\<in>?p ` space M. contents (f`A) * \<mu> A)"
hoelzl@38656
   491
  proof -
hoelzl@38656
   492
    have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
hoelzl@38656
   493
    have "(\<lambda>A. (contents (f ` A), A)) ` ?p ` space M
hoelzl@38656
   494
      = (\<lambda>x. (f x, ?p x)) ` space M"
hoelzl@38656
   495
    proof safe
hoelzl@38656
   496
      fix x assume "x \<in> space M"
hoelzl@38656
   497
      thus "(f x, ?p x) \<in> (\<lambda>A. (contents (f`A), A)) ` ?p ` space M"
hoelzl@38656
   498
        by (auto intro!: image_eqI[of _ _ "?p x"])
hoelzl@38656
   499
    qed auto
hoelzl@38656
   500
    thus ?thesis
hoelzl@38656
   501
      apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (contents (f`A), A)"] inj_onI)
hoelzl@38656
   502
      apply (rule_tac x="xa" in image_eqI)
hoelzl@38656
   503
      by simp_all
hoelzl@38656
   504
  qed
hoelzl@38656
   505
  finally show ?thesis .
hoelzl@35582
   506
qed
hoelzl@35582
   507
hoelzl@38656
   508
lemma (in measure_space) simple_integral_add[simp]:
hoelzl@38656
   509
  assumes "simple_function f" and "simple_function g"
hoelzl@38656
   510
  shows "simple_integral (\<lambda>x. f x + g x) = simple_integral f + simple_integral g"
hoelzl@35582
   511
proof -
hoelzl@38656
   512
  { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
hoelzl@38656
   513
    assume "x \<in> space M"
hoelzl@38656
   514
    hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
hoelzl@38656
   515
        "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
hoelzl@38656
   516
      by auto }
hoelzl@38656
   517
  thus ?thesis
hoelzl@38656
   518
    unfolding
hoelzl@38656
   519
      simple_function_partition[OF simple_function_add[OF assms] simple_function_Pair[OF assms]]
hoelzl@38656
   520
      simple_function_partition[OF `simple_function f` `simple_function g`]
hoelzl@38656
   521
      simple_function_partition[OF `simple_function g` `simple_function f`]
hoelzl@38656
   522
    apply (subst (3) Int_commute)
hoelzl@38656
   523
    by (auto simp add: field_simps setsum_addf[symmetric] intro!: setsum_cong)
hoelzl@35582
   524
qed
hoelzl@35582
   525
hoelzl@38656
   526
lemma (in measure_space) simple_integral_setsum[simp]:
hoelzl@38656
   527
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
hoelzl@38656
   528
  shows "simple_integral (\<lambda>x. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. simple_integral (f i))"
hoelzl@38656
   529
proof cases
hoelzl@38656
   530
  assume "finite P"
hoelzl@38656
   531
  from this assms show ?thesis
hoelzl@38656
   532
    by induct (auto simp: simple_function_setsum simple_integral_add)
hoelzl@38656
   533
qed auto
hoelzl@38656
   534
hoelzl@38656
   535
lemma (in measure_space) simple_integral_mult[simp]:
hoelzl@38656
   536
  assumes "simple_function f"
hoelzl@38656
   537
  shows "simple_integral (\<lambda>x. c * f x) = c * simple_integral f"
hoelzl@38656
   538
proof -
hoelzl@38656
   539
  note mult = simple_function_mult[OF simple_function_const[of c] assms]
hoelzl@38656
   540
  { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
hoelzl@38656
   541
    assume "x \<in> space M"
hoelzl@38656
   542
    hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
hoelzl@38656
   543
      by auto }
hoelzl@38656
   544
  thus ?thesis
hoelzl@38656
   545
    unfolding simple_function_partition[OF mult assms]
hoelzl@38656
   546
      simple_function_partition[OF assms mult]
hoelzl@38656
   547
    by (auto simp: setsum_right_distrib field_simps intro!: setsum_cong)
hoelzl@35582
   548
qed
hoelzl@35582
   549
hoelzl@38656
   550
lemma (in measure_space) simple_integral_mono:
hoelzl@38656
   551
  assumes "simple_function f" and "simple_function g"
hoelzl@38656
   552
  and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
hoelzl@38656
   553
  shows "simple_integral f \<le> simple_integral g"
hoelzl@38656
   554
  unfolding
hoelzl@38656
   555
    simple_function_partition[OF `simple_function f` `simple_function g`]
hoelzl@38656
   556
    simple_function_partition[OF `simple_function g` `simple_function f`]
hoelzl@38656
   557
  apply (subst Int_commute)
hoelzl@38656
   558
proof (safe intro!: setsum_mono)
hoelzl@38656
   559
  fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
hoelzl@38656
   560
  assume "x \<in> space M"
hoelzl@38656
   561
  hence "f ` ?S = {f x}" "g ` ?S = {g x}" by auto
hoelzl@38656
   562
  thus "contents (f`?S) * \<mu> ?S \<le> contents (g`?S) * \<mu> ?S"
hoelzl@38656
   563
    using mono[OF `x \<in> space M`] by (auto intro!: mult_right_mono)
hoelzl@35582
   564
qed
hoelzl@35582
   565
hoelzl@38656
   566
lemma (in measure_space) simple_integral_indicator:
hoelzl@38656
   567
  assumes "A \<in> sets M"
hoelzl@38656
   568
  assumes "simple_function f"
hoelzl@38656
   569
  shows "simple_integral (\<lambda>x. f x * indicator A x) =
hoelzl@38656
   570
    (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
hoelzl@38656
   571
proof cases
hoelzl@38656
   572
  assume "A = space M"
hoelzl@38656
   573
  moreover hence "simple_integral (\<lambda>x. f x * indicator A x) = simple_integral f"
hoelzl@38656
   574
    by (auto intro!: simple_integral_cong)
hoelzl@38656
   575
  moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto
hoelzl@38656
   576
  ultimately show ?thesis by (simp add: simple_integral_def)
hoelzl@38656
   577
next
hoelzl@38656
   578
  assume "A \<noteq> space M"
hoelzl@38656
   579
  then obtain x where x: "x \<in> space M" "x \<notin> A" using sets_into_space[OF assms(1)] by auto
hoelzl@38656
   580
  have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
hoelzl@35582
   581
  proof safe
hoelzl@38656
   582
    fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
hoelzl@38656
   583
  next
hoelzl@38656
   584
    fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
hoelzl@38656
   585
      using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
hoelzl@38656
   586
  next
hoelzl@38656
   587
    show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
hoelzl@35582
   588
  qed
hoelzl@38656
   589
  have *: "simple_integral (\<lambda>x. f x * indicator A x) =
hoelzl@38656
   590
    (\<Sum>x \<in> f ` space M \<union> {0}. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
hoelzl@38656
   591
    unfolding simple_integral_def I
hoelzl@38656
   592
  proof (rule setsum_mono_zero_cong_left)
hoelzl@38656
   593
    show "finite (f ` space M \<union> {0})"
hoelzl@38656
   594
      using assms(2) unfolding simple_function_def by auto
hoelzl@38656
   595
    show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
hoelzl@38656
   596
      using sets_into_space[OF assms(1)] by auto
hoelzl@38656
   597
    have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}" by (auto simp: image_iff)
hoelzl@38656
   598
    thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
hoelzl@38656
   599
      i * \<mu> (f -` {i} \<inter> space M \<inter> A) = 0" by auto
hoelzl@38656
   600
  next
hoelzl@38656
   601
    fix x assume "x \<in> f`A \<union> {0}"
hoelzl@38656
   602
    hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
hoelzl@38656
   603
      by (auto simp: indicator_def split: split_if_asm)
hoelzl@38656
   604
    thus "x * \<mu> (?I -` {x} \<inter> space M) =
hoelzl@38656
   605
      x * \<mu> (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
hoelzl@38656
   606
  qed
hoelzl@38656
   607
  show ?thesis unfolding *
hoelzl@38656
   608
    using assms(2) unfolding simple_function_def
hoelzl@38656
   609
    by (auto intro!: setsum_mono_zero_cong_right)
hoelzl@38656
   610
qed
hoelzl@35582
   611
hoelzl@38656
   612
lemma (in measure_space) simple_integral_indicator_only[simp]:
hoelzl@38656
   613
  assumes "A \<in> sets M"
hoelzl@38656
   614
  shows "simple_integral (indicator A) = \<mu> A"
hoelzl@38656
   615
proof cases
hoelzl@38656
   616
  assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
hoelzl@38656
   617
  thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
hoelzl@38656
   618
next
hoelzl@38656
   619
  assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::pinfreal}" by auto
hoelzl@38656
   620
  thus ?thesis
hoelzl@38656
   621
    using simple_integral_indicator[OF assms simple_function_const[of 1]]
hoelzl@38656
   622
    using sets_into_space[OF assms]
hoelzl@38656
   623
    by (auto intro!: arg_cong[where f="\<mu>"])
hoelzl@38656
   624
qed
hoelzl@35582
   625
hoelzl@38656
   626
lemma (in measure_space) simple_integral_null_set:
hoelzl@38656
   627
  assumes "simple_function u" "N \<in> null_sets"
hoelzl@38656
   628
  shows "simple_integral (\<lambda>x. u x * indicator N x) = 0"
hoelzl@38656
   629
proof -
hoelzl@38656
   630
  have "simple_integral (\<lambda>x. u x * indicator N x) \<le>
hoelzl@38656
   631
    simple_integral (\<lambda>x. \<omega> * indicator N x)"
hoelzl@38656
   632
    using assms
hoelzl@38656
   633
    by (safe intro!: simple_integral_mono simple_function_mult simple_function_indicator simple_function_const) simp
hoelzl@38656
   634
  also have "... = 0" apply(subst simple_integral_mult)
hoelzl@38656
   635
    using assms(2) by auto
hoelzl@38656
   636
  finally show ?thesis by auto
hoelzl@38656
   637
qed
hoelzl@35582
   638
hoelzl@38656
   639
lemma (in measure_space) simple_integral_cong':
hoelzl@38656
   640
  assumes f: "simple_function f" and g: "simple_function g"
hoelzl@38656
   641
  and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
hoelzl@38656
   642
  shows "simple_integral f = simple_integral g"
hoelzl@38656
   643
proof -
hoelzl@38656
   644
  let ?h = "\<lambda>h. \<lambda>x. (h x * indicator {x\<in>space M. f x = g x} x
hoelzl@38656
   645
    + h x * indicator {x\<in>space M. f x \<noteq> g x} x
hoelzl@38656
   646
    + h x * indicator (-space M) x::pinfreal)"
hoelzl@38656
   647
  have *:"\<And>h. h = ?h h" unfolding indicator_def apply rule by auto
hoelzl@38656
   648
  have mea_neq:"{x \<in> space M. f x \<noteq> g x} \<in> sets M" using f g by (auto simp: borel_measurable_simple_function)
hoelzl@38656
   649
  then have mea_nullset: "{x \<in> space M. f x \<noteq> g x} \<in> null_sets" using mea by auto
hoelzl@38656
   650
  have h1:"\<And>h::_=>pinfreal. simple_function h \<Longrightarrow>
hoelzl@38656
   651
    simple_function (\<lambda>x. h x * indicator {x\<in>space M. f x = g x} x)"
hoelzl@38656
   652
    apply(safe intro!: simple_function_add simple_function_mult simple_function_indicator)
hoelzl@38656
   653
    using f g by (auto simp: borel_measurable_simple_function)
hoelzl@38656
   654
  have h2:"\<And>h::_\<Rightarrow>pinfreal. simple_function h \<Longrightarrow>
hoelzl@38656
   655
    simple_function (\<lambda>x. h x * indicator {x\<in>space M. f x \<noteq> g x} x)"
hoelzl@38656
   656
    apply(safe intro!: simple_function_add simple_function_mult simple_function_indicator)
hoelzl@38656
   657
    by(rule mea_neq)
hoelzl@38656
   658
  have **:"\<And>a b c d e f. a = b \<Longrightarrow> c = d \<Longrightarrow> e = f \<Longrightarrow> a+c+e = b+d+f" by auto
hoelzl@38656
   659
  note *** = simple_integral_add[OF simple_function_add[OF h1 h2] simple_function_notspace]
hoelzl@38656
   660
    simple_integral_add[OF h1 h2]
hoelzl@38656
   661
  show ?thesis apply(subst *[of g]) apply(subst *[of f])
hoelzl@38656
   662
    unfolding ***[OF f f] ***[OF g g]
hoelzl@38656
   663
  proof(rule **) case goal1 show ?case apply(rule arg_cong[where f=simple_integral]) apply rule 
hoelzl@38656
   664
      unfolding indicator_def by auto
hoelzl@38656
   665
  next note * = simple_integral_null_set[OF _ mea_nullset]
hoelzl@38656
   666
    case goal2 show ?case unfolding *[OF f] *[OF g] ..
hoelzl@38656
   667
  next case goal3 show ?case apply(rule simple_integral_cong) by auto
hoelzl@35582
   668
  qed
hoelzl@35582
   669
qed
hoelzl@35582
   670
hoelzl@35692
   671
section "Continuous posititve integration"
hoelzl@35692
   672
hoelzl@38656
   673
definition (in measure_space)
hoelzl@38656
   674
  "positive_integral f =
hoelzl@38656
   675
    (SUP g : {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}. simple_integral g)"
hoelzl@35582
   676
hoelzl@38656
   677
lemma (in measure_space) positive_integral_alt1:
hoelzl@38656
   678
  "positive_integral f =
hoelzl@38656
   679
    (SUP g : {g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}. simple_integral g)"
hoelzl@38656
   680
  unfolding positive_integral_def SUPR_def
hoelzl@38656
   681
proof (safe intro!: arg_cong[where f=Sup])
hoelzl@38656
   682
  fix g let ?g = "\<lambda>x. if x \<in> space M then g x else f x"
hoelzl@38656
   683
  assume "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
hoelzl@38656
   684
  hence "?g \<le> f" "simple_function ?g" "simple_integral g = simple_integral ?g"
hoelzl@38656
   685
    "\<omega> \<notin> g`space M"
hoelzl@38656
   686
    unfolding le_fun_def by (auto cong: simple_function_cong simple_integral_cong)
hoelzl@38656
   687
  thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}"
hoelzl@38656
   688
    by auto
hoelzl@38656
   689
next
hoelzl@38656
   690
  fix g assume "simple_function g" "g \<le> f" "\<omega> \<notin> g`space M"
hoelzl@38656
   691
  hence "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
hoelzl@38656
   692
    by (auto simp add: le_fun_def image_iff)
hoelzl@38656
   693
  thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}"
hoelzl@38656
   694
    by auto
hoelzl@35582
   695
qed
hoelzl@35582
   696
hoelzl@38656
   697
lemma (in measure_space) positive_integral_alt:
hoelzl@38656
   698
  "positive_integral f =
hoelzl@38656
   699
    (SUP g : {g. simple_function g \<and> g \<le> f}. simple_integral g)"
hoelzl@38656
   700
  apply(rule order_class.antisym) unfolding positive_integral_def 
hoelzl@38656
   701
  apply(rule SUPR_subset) apply blast apply(rule SUPR_mono_lim)
hoelzl@38656
   702
proof safe fix u assume u:"simple_function u" and uf:"u \<le> f"
hoelzl@38656
   703
  let ?u = "\<lambda>n x. if u x = \<omega> then Real (real (n::nat)) else u x"
hoelzl@38656
   704
  have su:"\<And>n. simple_function (?u n)" using simple_function_compose1[OF u] .
hoelzl@38656
   705
  show "\<exists>b. \<forall>n. b n \<in> {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g ` space M} \<and>
hoelzl@38656
   706
    (\<lambda>n. simple_integral (b n)) ----> simple_integral u"
hoelzl@38656
   707
    apply(rule_tac x="?u" in exI, safe) apply(rule su)
hoelzl@38656
   708
  proof- fix n::nat have "?u n \<le> u" unfolding le_fun_def by auto
hoelzl@38656
   709
    also note uf finally show "?u n \<le> f" .
hoelzl@38656
   710
    let ?s = "{x \<in> space M. u x = \<omega>}"
hoelzl@38656
   711
    show "(\<lambda>n. simple_integral (?u n)) ----> simple_integral u"
hoelzl@38656
   712
    proof(cases "\<mu> ?s = 0")
hoelzl@38656
   713
      case True have *:"\<And>n. {x\<in>space M. ?u n x \<noteq> u x} = {x\<in>space M. u x = \<omega>}" by auto 
hoelzl@38656
   714
      have *:"\<And>n. simple_integral (?u n) = simple_integral u"
hoelzl@38656
   715
        apply(rule simple_integral_cong'[OF su u]) unfolding * True ..
hoelzl@38656
   716
      show ?thesis unfolding * by auto 
hoelzl@38656
   717
    next case False note m0=this
hoelzl@38656
   718
      have s:"{x \<in> space M. u x = \<omega>} \<in> sets M" using u  by (auto simp: borel_measurable_simple_function)
hoelzl@38656
   719
      have "\<omega> = simple_integral (\<lambda>x. \<omega> * indicator {x\<in>space M. u x = \<omega>} x)"
hoelzl@38656
   720
        apply(subst simple_integral_mult) using s
hoelzl@38656
   721
        unfolding simple_integral_indicator_only[OF s] using False by auto
hoelzl@38656
   722
      also have "... \<le> simple_integral u"
hoelzl@38656
   723
        apply (rule simple_integral_mono)
hoelzl@38656
   724
        apply (rule simple_function_mult)
hoelzl@38656
   725
        apply (rule simple_function_const)
hoelzl@38656
   726
        apply(rule ) prefer 3 apply(subst indicator_def)
hoelzl@38656
   727
        using s u by auto
hoelzl@38656
   728
      finally have *:"simple_integral u = \<omega>" by auto
hoelzl@38656
   729
      show ?thesis unfolding * Lim_omega_pos
hoelzl@38656
   730
      proof safe case goal1
hoelzl@38656
   731
        from real_arch_simple[of "B / real (\<mu> ?s)"] guess N0 .. note N=this
hoelzl@38656
   732
        def N \<equiv> "Suc N0" have N:"real N \<ge> B / real (\<mu> ?s)" "N > 0"
hoelzl@38656
   733
          unfolding N_def using N by auto
hoelzl@38656
   734
        show ?case apply-apply(rule_tac x=N in exI,safe) 
hoelzl@38656
   735
        proof- case goal1
hoelzl@38656
   736
          have "Real B \<le> Real (real N) * \<mu> ?s"
hoelzl@38656
   737
          proof(cases "\<mu> ?s = \<omega>")
hoelzl@38656
   738
            case True thus ?thesis using `B>0` N by auto
hoelzl@38656
   739
          next case False
hoelzl@38656
   740
            have *:"B \<le> real N * real (\<mu> ?s)" 
hoelzl@38656
   741
              using N(1) apply-apply(subst (asm) pos_divide_le_eq)
hoelzl@38656
   742
              apply rule using m0 False by auto
hoelzl@38656
   743
            show ?thesis apply(subst Real_real'[THEN sym,OF False])
hoelzl@38656
   744
              apply(subst pinfreal_times,subst if_P) defer
hoelzl@38656
   745
              apply(subst pinfreal_less_eq,subst if_P) defer
hoelzl@38656
   746
              using * N `B>0` by(auto intro: mult_nonneg_nonneg)
hoelzl@38656
   747
          qed
hoelzl@38656
   748
          also have "... \<le> Real (real n) * \<mu> ?s"
hoelzl@38656
   749
            apply(rule mult_right_mono) using goal1 by auto
hoelzl@38656
   750
          also have "... = simple_integral (\<lambda>x. Real (real n) * indicator ?s x)" 
hoelzl@38656
   751
            apply(subst simple_integral_mult) apply(rule simple_function_indicator[OF s])
hoelzl@38656
   752
            unfolding simple_integral_indicator_only[OF s] ..
hoelzl@38656
   753
          also have "... \<le> simple_integral (\<lambda>x. if u x = \<omega> then Real (real n) else u x)"
hoelzl@38656
   754
            apply(rule simple_integral_mono) apply(rule simple_function_mult)
hoelzl@38656
   755
            apply(rule simple_function_const)
hoelzl@38656
   756
            apply(rule simple_function_indicator) apply(rule s su)+ by auto
hoelzl@38656
   757
          finally show ?case .
hoelzl@38656
   758
        qed qed qed
hoelzl@38656
   759
    fix x assume x:"\<omega> = (if u x = \<omega> then Real (real n) else u x)" "x \<in> space M"
hoelzl@38656
   760
    hence "u x = \<omega>" apply-apply(rule ccontr) by auto
hoelzl@38656
   761
    hence "\<omega> = Real (real n)" using x by auto
hoelzl@38656
   762
    thus False by auto
hoelzl@35582
   763
  qed
hoelzl@35582
   764
qed
hoelzl@35582
   765
hoelzl@38656
   766
lemma (in measure_space) positive_integral_cong:
hoelzl@38656
   767
  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
hoelzl@38656
   768
  shows "positive_integral f = positive_integral g"
hoelzl@38656
   769
proof -
hoelzl@38656
   770
  have "\<And>h. (\<forall>x\<in>space M. h x \<le> f x \<and> h x \<noteq> \<omega>) = (\<forall>x\<in>space M. h x \<le> g x \<and> h x \<noteq> \<omega>)"
hoelzl@38656
   771
    using assms by auto
hoelzl@38656
   772
  thus ?thesis unfolding positive_integral_alt1 by auto
hoelzl@38656
   773
qed
hoelzl@38656
   774
hoelzl@38656
   775
lemma (in measure_space) positive_integral_eq_simple_integral:
hoelzl@38656
   776
  assumes "simple_function f"
hoelzl@38656
   777
  shows "positive_integral f = simple_integral f"
hoelzl@38656
   778
  unfolding positive_integral_alt
hoelzl@38656
   779
proof (safe intro!: pinfreal_SUPI)
hoelzl@38656
   780
  fix g assume "simple_function g" "g \<le> f"
hoelzl@38656
   781
  with assms show "simple_integral g \<le> simple_integral f"
hoelzl@38656
   782
    by (auto intro!: simple_integral_mono simp: le_fun_def)
hoelzl@38656
   783
next
hoelzl@38656
   784
  fix y assume "\<forall>x. x\<in>{g. simple_function g \<and> g \<le> f} \<longrightarrow> simple_integral x \<le> y"
hoelzl@38656
   785
  with assms show "simple_integral f \<le> y" by auto
hoelzl@38656
   786
qed
hoelzl@35582
   787
hoelzl@38656
   788
lemma (in measure_space) positive_integral_mono:
hoelzl@38656
   789
  assumes mono: "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x"
hoelzl@38656
   790
  shows "positive_integral u \<le> positive_integral v"
hoelzl@38656
   791
  unfolding positive_integral_alt1
hoelzl@38656
   792
proof (safe intro!: SUPR_mono)
hoelzl@38656
   793
  fix a assume a: "simple_function a" and "\<forall>x\<in>space M. a x \<le> u x \<and> a x \<noteq> \<omega>"
hoelzl@38656
   794
  with mono have "\<forall>x\<in>space M. a x \<le> v x \<and> a x \<noteq> \<omega>" by fastsimp
hoelzl@38656
   795
  with a show "\<exists>b\<in>{g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> v x \<and> g x \<noteq> \<omega>)}. simple_integral a \<le> simple_integral b"
hoelzl@38656
   796
    by (auto intro!: bexI[of _ a])
hoelzl@38656
   797
qed
hoelzl@38656
   798
hoelzl@38656
   799
lemma (in measure_space) positive_integral_SUP_approx:
hoelzl@38656
   800
  assumes "f \<up> s"
hoelzl@38656
   801
  and f: "\<And>i. f i \<in> borel_measurable M"
hoelzl@38656
   802
  and "simple_function u"
hoelzl@38656
   803
  and le: "u \<le> s" and real: "\<omega> \<notin> u`space M"
hoelzl@38656
   804
  shows "simple_integral u \<le> (SUP i. positive_integral (f i))" (is "_ \<le> ?S")
hoelzl@38656
   805
proof (rule pinfreal_le_mult_one_interval)
hoelzl@38656
   806
  fix a :: pinfreal assume "0 < a" "a < 1"
hoelzl@38656
   807
  hence "a \<noteq> 0" by auto
hoelzl@38656
   808
  let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
hoelzl@38656
   809
  have B: "\<And>i. ?B i \<in> sets M"
hoelzl@38656
   810
    using f `simple_function u` by (auto simp: borel_measurable_simple_function)
hoelzl@38656
   811
hoelzl@38656
   812
  let "?uB i x" = "u x * indicator (?B i) x"
hoelzl@38656
   813
hoelzl@38656
   814
  { fix i have "?B i \<subseteq> ?B (Suc i)"
hoelzl@38656
   815
    proof safe
hoelzl@38656
   816
      fix i x assume "a * u x \<le> f i x"
hoelzl@38656
   817
      also have "\<dots> \<le> f (Suc i) x"
hoelzl@38656
   818
        using `f \<up> s` unfolding isoton_def le_fun_def by auto
hoelzl@38656
   819
      finally show "a * u x \<le> f (Suc i) x" .
hoelzl@38656
   820
    qed }
hoelzl@38656
   821
  note B_mono = this
hoelzl@35582
   822
hoelzl@38656
   823
  have u: "\<And>x. x \<in> space M \<Longrightarrow> u -` {u x} \<inter> space M \<in> sets M"
hoelzl@38656
   824
    using `simple_function u` by (auto simp add: simple_function_def)
hoelzl@38656
   825
hoelzl@38656
   826
  { fix i
hoelzl@38656
   827
    have "(\<lambda>n. (u -` {i} \<inter> space M) \<inter> ?B n) \<up> (u -` {i} \<inter> space M)" using B_mono unfolding isoton_def
hoelzl@38656
   828
    proof safe
hoelzl@38656
   829
      fix x assume "x \<in> space M"
hoelzl@38656
   830
      show "x \<in> (\<Union>i. (u -` {u x} \<inter> space M) \<inter> ?B i)"
hoelzl@38656
   831
      proof cases
hoelzl@38656
   832
        assume "u x = 0" thus ?thesis using `x \<in> space M` by simp
hoelzl@38656
   833
      next
hoelzl@38656
   834
        assume "u x \<noteq> 0"
hoelzl@38656
   835
        with `a < 1` real `x \<in> space M`
hoelzl@38656
   836
        have "a * u x < 1 * u x" by (rule_tac pinfreal_mult_strict_right_mono) (auto simp: image_iff)
hoelzl@38656
   837
        also have "\<dots> \<le> (SUP i. f i x)" using le `f \<up> s`
hoelzl@38656
   838
          unfolding isoton_fun_expand by (auto simp: isoton_def le_fun_def)
hoelzl@38656
   839
        finally obtain i where "a * u x < f i x" unfolding SUPR_def
hoelzl@38656
   840
          by (auto simp add: less_Sup_iff)
hoelzl@38656
   841
        hence "a * u x \<le> f i x" by auto
hoelzl@38656
   842
        thus ?thesis using `x \<in> space M` by auto
hoelzl@38656
   843
      qed
hoelzl@38656
   844
    qed auto }
hoelzl@38656
   845
  note measure_conv = measure_up[OF u Int[OF u B] this]
hoelzl@38656
   846
hoelzl@38656
   847
  have "simple_integral u = (SUP i. simple_integral (?uB i))"
hoelzl@38656
   848
    unfolding simple_integral_indicator[OF B `simple_function u`]
hoelzl@38656
   849
  proof (subst SUPR_pinfreal_setsum, safe)
hoelzl@38656
   850
    fix x n assume "x \<in> space M"
hoelzl@38656
   851
    have "\<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f n x})
hoelzl@38656
   852
      \<le> \<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f (Suc n) x})"
hoelzl@38656
   853
      using B_mono Int[OF u[OF `x \<in> space M`] B] by (auto intro!: measure_mono)
hoelzl@38656
   854
    thus "u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B n)
hoelzl@38656
   855
            \<le> u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B (Suc n))"
hoelzl@38656
   856
      by (auto intro: mult_left_mono)
hoelzl@38656
   857
  next
hoelzl@38656
   858
    show "simple_integral u =
hoelzl@38656
   859
      (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (u -` {i} \<inter> space M \<inter> ?B n))"
hoelzl@38656
   860
      using measure_conv unfolding simple_integral_def isoton_def
hoelzl@38656
   861
      by (auto intro!: setsum_cong simp: pinfreal_SUP_cmult)
hoelzl@38656
   862
  qed
hoelzl@38656
   863
  moreover
hoelzl@38656
   864
  have "a * (SUP i. simple_integral (?uB i)) \<le> ?S"
hoelzl@38656
   865
    unfolding pinfreal_SUP_cmult[symmetric]
hoelzl@38705
   866
  proof (safe intro!: SUP_mono bexI)
hoelzl@38656
   867
    fix i
hoelzl@38656
   868
    have "a * simple_integral (?uB i) = simple_integral (\<lambda>x. a * ?uB i x)"
hoelzl@38656
   869
      using B `simple_function u`
hoelzl@38656
   870
      by (subst simple_integral_mult[symmetric]) (auto simp: field_simps)
hoelzl@38656
   871
    also have "\<dots> \<le> positive_integral (f i)"
hoelzl@38656
   872
    proof -
hoelzl@38656
   873
      have "\<And>x. a * ?uB i x \<le> f i x" unfolding indicator_def by auto
hoelzl@38656
   874
      hence *: "simple_function (\<lambda>x. a * ?uB i x)" using B assms(3)
hoelzl@38656
   875
        by (auto intro!: simple_integral_mono)
hoelzl@38656
   876
      show ?thesis unfolding positive_integral_eq_simple_integral[OF *, symmetric]
hoelzl@38656
   877
        by (auto intro!: positive_integral_mono simp: indicator_def)
hoelzl@38656
   878
    qed
hoelzl@38656
   879
    finally show "a * simple_integral (?uB i) \<le> positive_integral (f i)"
hoelzl@38656
   880
      by auto
hoelzl@38705
   881
  qed simp
hoelzl@38656
   882
  ultimately show "a * simple_integral u \<le> ?S" by simp
hoelzl@35582
   883
qed
hoelzl@35582
   884
hoelzl@35582
   885
text {* Beppo-Levi monotone convergence theorem *}
hoelzl@38656
   886
lemma (in measure_space) positive_integral_isoton:
hoelzl@38656
   887
  assumes "f \<up> u" "\<And>i. f i \<in> borel_measurable M"
hoelzl@38656
   888
  shows "(\<lambda>i. positive_integral (f i)) \<up> positive_integral u"
hoelzl@38656
   889
  unfolding isoton_def
hoelzl@38656
   890
proof safe
hoelzl@38656
   891
  fix i show "positive_integral (f i) \<le> positive_integral (f (Suc i))"
hoelzl@38656
   892
    apply (rule positive_integral_mono)
hoelzl@38656
   893
    using `f \<up> u` unfolding isoton_def le_fun_def by auto
hoelzl@38656
   894
next
hoelzl@38656
   895
  have "u \<in> borel_measurable M"
hoelzl@38656
   896
    using borel_measurable_SUP[of UNIV f] assms by (auto simp: isoton_def)
hoelzl@38656
   897
  have u: "u = (SUP i. f i)" using `f \<up> u` unfolding isoton_def by auto
hoelzl@35582
   898
hoelzl@38656
   899
  show "(SUP i. positive_integral (f i)) = positive_integral u"
hoelzl@38656
   900
  proof (rule antisym)
hoelzl@38656
   901
    from `f \<up> u`[THEN isoton_Sup, unfolded le_fun_def]
hoelzl@38656
   902
    show "(SUP j. positive_integral (f j)) \<le> positive_integral u"
hoelzl@38656
   903
      by (auto intro!: SUP_leI positive_integral_mono)
hoelzl@38656
   904
  next
hoelzl@38656
   905
    show "positive_integral u \<le> (SUP i. positive_integral (f i))"
hoelzl@38656
   906
      unfolding positive_integral_def[of u]
hoelzl@38656
   907
      by (auto intro!: SUP_leI positive_integral_SUP_approx[OF assms])
hoelzl@35582
   908
  qed
hoelzl@35582
   909
qed
hoelzl@35582
   910
hoelzl@38656
   911
lemma (in measure_space) SUP_simple_integral_sequences:
hoelzl@38656
   912
  assumes f: "f \<up> u" "\<And>i. simple_function (f i)"
hoelzl@38656
   913
  and g: "g \<up> u" "\<And>i. simple_function (g i)"
hoelzl@38656
   914
  shows "(SUP i. simple_integral (f i)) = (SUP i. simple_integral (g i))"
hoelzl@38656
   915
    (is "SUPR _ ?F = SUPR _ ?G")
hoelzl@38656
   916
proof -
hoelzl@38656
   917
  have "(SUP i. ?F i) = (SUP i. positive_integral (f i))"
hoelzl@38656
   918
    using assms by (simp add: positive_integral_eq_simple_integral)
hoelzl@38656
   919
  also have "\<dots> = positive_integral u"
hoelzl@38656
   920
    using positive_integral_isoton[OF f(1) borel_measurable_simple_function[OF f(2)]]
hoelzl@38656
   921
    unfolding isoton_def by simp
hoelzl@38656
   922
  also have "\<dots> = (SUP i. positive_integral (g i))"
hoelzl@38656
   923
    using positive_integral_isoton[OF g(1) borel_measurable_simple_function[OF g(2)]]
hoelzl@38656
   924
    unfolding isoton_def by simp
hoelzl@38656
   925
  also have "\<dots> = (SUP i. ?G i)"
hoelzl@38656
   926
    using assms by (simp add: positive_integral_eq_simple_integral)
hoelzl@38656
   927
  finally show ?thesis .
hoelzl@38656
   928
qed
hoelzl@38656
   929
hoelzl@38656
   930
lemma (in measure_space) positive_integral_const[simp]:
hoelzl@38656
   931
  "positive_integral (\<lambda>x. c) = c * \<mu> (space M)"
hoelzl@38656
   932
  by (subst positive_integral_eq_simple_integral) auto
hoelzl@38656
   933
hoelzl@38656
   934
lemma (in measure_space) positive_integral_isoton_simple:
hoelzl@38656
   935
  assumes "f \<up> u" and e: "\<And>i. simple_function (f i)"
hoelzl@38656
   936
  shows "(\<lambda>i. simple_integral (f i)) \<up> positive_integral u"
hoelzl@38656
   937
  using positive_integral_isoton[OF `f \<up> u` e(1)[THEN borel_measurable_simple_function]]
hoelzl@38656
   938
  unfolding positive_integral_eq_simple_integral[OF e] .
hoelzl@38656
   939
hoelzl@38656
   940
lemma (in measure_space) positive_integral_linear:
hoelzl@38656
   941
  assumes f: "f \<in> borel_measurable M"
hoelzl@38656
   942
  and g: "g \<in> borel_measurable M"
hoelzl@38656
   943
  shows "positive_integral (\<lambda>x. a * f x + g x) =
hoelzl@38656
   944
      a * positive_integral f + positive_integral g"
hoelzl@38656
   945
    (is "positive_integral ?L = _")
hoelzl@35582
   946
proof -
hoelzl@38656
   947
  from borel_measurable_implies_simple_function_sequence'[OF f] guess u .
hoelzl@38656
   948
  note u = this positive_integral_isoton_simple[OF this(1-2)]
hoelzl@38656
   949
  from borel_measurable_implies_simple_function_sequence'[OF g] guess v .
hoelzl@38656
   950
  note v = this positive_integral_isoton_simple[OF this(1-2)]
hoelzl@38656
   951
  let "?L' i x" = "a * u i x + v i x"
hoelzl@38656
   952
hoelzl@38656
   953
  have "?L \<in> borel_measurable M"
hoelzl@38656
   954
    using assms by simp
hoelzl@38656
   955
  from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
hoelzl@38656
   956
  note positive_integral_isoton_simple[OF this(1-2)] and l = this
hoelzl@38656
   957
  moreover have
hoelzl@38656
   958
      "(SUP i. simple_integral (l i)) = (SUP i. simple_integral (?L' i))"
hoelzl@38656
   959
  proof (rule SUP_simple_integral_sequences[OF l(1-2)])
hoelzl@38656
   960
    show "?L' \<up> ?L" "\<And>i. simple_function (?L' i)"
hoelzl@38656
   961
      using u v by (auto simp: isoton_fun_expand isoton_add isoton_cmult_right)
hoelzl@38656
   962
  qed
hoelzl@38656
   963
  moreover from u v have L'_isoton:
hoelzl@38656
   964
      "(\<lambda>i. simple_integral (?L' i)) \<up> a * positive_integral f + positive_integral g"
hoelzl@38656
   965
    by (simp add: isoton_add isoton_cmult_right)
hoelzl@38656
   966
  ultimately show ?thesis by (simp add: isoton_def)
hoelzl@38656
   967
qed
hoelzl@38656
   968
hoelzl@38656
   969
lemma (in measure_space) positive_integral_cmult:
hoelzl@38656
   970
  assumes "f \<in> borel_measurable M"
hoelzl@38656
   971
  shows "positive_integral (\<lambda>x. c * f x) = c * positive_integral f"
hoelzl@38656
   972
  using positive_integral_linear[OF assms, of "\<lambda>x. 0" c] by auto
hoelzl@38656
   973
hoelzl@38656
   974
lemma (in measure_space) positive_integral_indicator[simp]:
hoelzl@38656
   975
  "A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. indicator A x) = \<mu> A"
hoelzl@38656
   976
by (subst positive_integral_eq_simple_integral) (auto simp: simple_function_indicator simple_integral_indicator)
hoelzl@38656
   977
hoelzl@38656
   978
lemma (in measure_space) positive_integral_cmult_indicator:
hoelzl@38656
   979
  "A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. c * indicator A x) = c * \<mu> A"
hoelzl@38656
   980
by (subst positive_integral_eq_simple_integral) (auto simp: simple_function_indicator simple_integral_indicator)
hoelzl@38656
   981
hoelzl@38656
   982
lemma (in measure_space) positive_integral_add:
hoelzl@38656
   983
  assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@38656
   984
  shows "positive_integral (\<lambda>x. f x + g x) = positive_integral f + positive_integral g"
hoelzl@38656
   985
  using positive_integral_linear[OF assms, of 1] by simp
hoelzl@38656
   986
hoelzl@38656
   987
lemma (in measure_space) positive_integral_setsum:
hoelzl@38656
   988
  assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M"
hoelzl@38656
   989
  shows "positive_integral (\<lambda>x. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. positive_integral (f i))"
hoelzl@38656
   990
proof cases
hoelzl@38656
   991
  assume "finite P"
hoelzl@38656
   992
  from this assms show ?thesis
hoelzl@38656
   993
  proof induct
hoelzl@38656
   994
    case (insert i P)
hoelzl@38656
   995
    have "f i \<in> borel_measurable M"
hoelzl@38656
   996
      "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M"
hoelzl@38656
   997
      using insert by (auto intro!: borel_measurable_pinfreal_setsum)
hoelzl@38656
   998
    from positive_integral_add[OF this]
hoelzl@38656
   999
    show ?case using insert by auto
hoelzl@38656
  1000
  qed simp
hoelzl@38656
  1001
qed simp
hoelzl@38656
  1002
hoelzl@38656
  1003
lemma (in measure_space) positive_integral_diff:
hoelzl@38656
  1004
  assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
hoelzl@38656
  1005
  and fin: "positive_integral g \<noteq> \<omega>"
hoelzl@38656
  1006
  and mono: "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"
hoelzl@38656
  1007
  shows "positive_integral (\<lambda>x. f x - g x) = positive_integral f - positive_integral g"
hoelzl@38656
  1008
proof -
hoelzl@38656
  1009
  have borel: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
hoelzl@38656
  1010
    using f g by (rule borel_measurable_pinfreal_diff)
hoelzl@38656
  1011
  have "positive_integral (\<lambda>x. f x - g x) + positive_integral g =
hoelzl@38656
  1012
    positive_integral f"
hoelzl@38656
  1013
    unfolding positive_integral_add[OF borel g, symmetric]
hoelzl@38656
  1014
  proof (rule positive_integral_cong)
hoelzl@38656
  1015
    fix x assume "x \<in> space M"
hoelzl@38656
  1016
    from mono[OF this] show "f x - g x + g x = f x"
hoelzl@38656
  1017
      by (cases "f x", cases "g x", simp, simp, cases "g x", auto)
hoelzl@38656
  1018
  qed
hoelzl@38656
  1019
  with mono show ?thesis
hoelzl@38656
  1020
    by (subst minus_pinfreal_eq2[OF _ fin]) (auto intro!: positive_integral_mono)
hoelzl@38656
  1021
qed
hoelzl@38656
  1022
hoelzl@38656
  1023
lemma (in measure_space) positive_integral_psuminf:
hoelzl@38656
  1024
  assumes "\<And>i. f i \<in> borel_measurable M"
hoelzl@38656
  1025
  shows "positive_integral (\<lambda>x. \<Sum>\<^isub>\<infinity> i. f i x) = (\<Sum>\<^isub>\<infinity> i. positive_integral (f i))"
hoelzl@38656
  1026
proof -
hoelzl@38656
  1027
  have "(\<lambda>i. positive_integral (\<lambda>x. \<Sum>i<i. f i x)) \<up> positive_integral (\<lambda>x. \<Sum>\<^isub>\<infinity>i. f i x)"
hoelzl@38656
  1028
    by (rule positive_integral_isoton)
hoelzl@38656
  1029
       (auto intro!: borel_measurable_pinfreal_setsum assms positive_integral_mono
hoelzl@38656
  1030
                     arg_cong[where f=Sup]
hoelzl@38656
  1031
             simp: isoton_def le_fun_def psuminf_def expand_fun_eq SUPR_def Sup_fun_def)
hoelzl@38656
  1032
  thus ?thesis
hoelzl@38656
  1033
    by (auto simp: isoton_def psuminf_def positive_integral_setsum[OF assms])
hoelzl@38656
  1034
qed
hoelzl@38656
  1035
hoelzl@38656
  1036
text {* Fatou's lemma: convergence theorem on limes inferior *}
hoelzl@38656
  1037
lemma (in measure_space) positive_integral_lim_INF:
hoelzl@38656
  1038
  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pinfreal"
hoelzl@38656
  1039
  assumes "\<And>i. u i \<in> borel_measurable M"
hoelzl@38656
  1040
  shows "positive_integral (SUP n. INF m. u (m + n)) \<le>
hoelzl@38656
  1041
    (SUP n. INF m. positive_integral (u (m + n)))"
hoelzl@38656
  1042
proof -
hoelzl@38656
  1043
  have "(SUP n. INF m. u (m + n)) \<in> borel_measurable M"
hoelzl@38656
  1044
    by (auto intro!: borel_measurable_SUP borel_measurable_INF assms)
hoelzl@38656
  1045
hoelzl@38656
  1046
  have "(\<lambda>n. INF m. u (m + n)) \<up> (SUP n. INF m. u (m + n))"
hoelzl@38705
  1047
  proof (unfold isoton_def, safe intro!: INF_mono bexI)
hoelzl@38705
  1048
    fix i m show "u (Suc m + i) \<le> u (m + Suc i)" by simp
hoelzl@38705
  1049
  qed simp
hoelzl@38656
  1050
  from positive_integral_isoton[OF this] assms
hoelzl@38656
  1051
  have "positive_integral (SUP n. INF m. u (m + n)) =
hoelzl@38656
  1052
    (SUP n. positive_integral (INF m. u (m + n)))"
hoelzl@38656
  1053
    unfolding isoton_def by (simp add: borel_measurable_INF)
hoelzl@38656
  1054
  also have "\<dots> \<le> (SUP n. INF m. positive_integral (u (m + n)))"
hoelzl@38705
  1055
    apply (rule SUP_mono)
hoelzl@38705
  1056
    apply (rule_tac x=n in bexI)
hoelzl@38705
  1057
    by (auto intro!: positive_integral_mono INFI_bound INF_leI exI simp: INFI_fun_expand)
hoelzl@38656
  1058
  finally show ?thesis .
hoelzl@35582
  1059
qed
hoelzl@35582
  1060
hoelzl@38656
  1061
lemma (in measure_space) measure_space_density:
hoelzl@38656
  1062
  assumes borel: "u \<in> borel_measurable M"
hoelzl@38656
  1063
  shows "measure_space M (\<lambda>A. positive_integral (\<lambda>x. u x * indicator A x))" (is "measure_space M ?v")
hoelzl@38656
  1064
proof
hoelzl@38656
  1065
  show "?v {} = 0" by simp
hoelzl@38656
  1066
  show "countably_additive M ?v"
hoelzl@38656
  1067
    unfolding countably_additive_def
hoelzl@38656
  1068
  proof safe
hoelzl@38656
  1069
    fix A :: "nat \<Rightarrow> 'a set"
hoelzl@38656
  1070
    assume "range A \<subseteq> sets M"
hoelzl@38656
  1071
    hence "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
hoelzl@38656
  1072
      using borel by (auto intro: borel_measurable_indicator)
hoelzl@38656
  1073
    moreover assume "disjoint_family A"
hoelzl@38656
  1074
    note psuminf_indicator[OF this]
hoelzl@38656
  1075
    ultimately show "(\<Sum>\<^isub>\<infinity>n. ?v (A n)) = ?v (\<Union>x. A x)"
hoelzl@38656
  1076
      by (simp add: positive_integral_psuminf[symmetric])
hoelzl@38656
  1077
  qed
hoelzl@38656
  1078
qed
hoelzl@35582
  1079
hoelzl@38656
  1080
lemma (in measure_space) positive_integral_null_set:
hoelzl@38656
  1081
  assumes borel: "u \<in> borel_measurable M" and "N \<in> null_sets"
hoelzl@38656
  1082
  shows "positive_integral (\<lambda>x. u x * indicator N x) = 0" (is "?I = 0")
hoelzl@38656
  1083
proof -
hoelzl@38656
  1084
  have "N \<in> sets M" using `N \<in> null_sets` by auto
hoelzl@38656
  1085
  have "(\<lambda>i x. min (of_nat i) (u x) * indicator N x) \<up> (\<lambda>x. u x * indicator N x)"
hoelzl@38656
  1086
    unfolding isoton_fun_expand
hoelzl@38656
  1087
  proof (safe intro!: isoton_cmult_left, unfold isoton_def, safe)
hoelzl@38656
  1088
    fix j i show "min (of_nat j) (u i) \<le> min (of_nat (Suc j)) (u i)"
hoelzl@38656
  1089
      by (rule min_max.inf_mono) auto
hoelzl@38656
  1090
  next
hoelzl@38656
  1091
    fix i show "(SUP j. min (of_nat j) (u i)) = u i"
hoelzl@38656
  1092
    proof (cases "u i")
hoelzl@38656
  1093
      case infinite
hoelzl@38656
  1094
      moreover hence "\<And>j. min (of_nat j) (u i) = of_nat j"
hoelzl@38656
  1095
        by (auto simp: min_def)
hoelzl@38656
  1096
      ultimately show ?thesis by (simp add: Sup_\<omega>)
hoelzl@35582
  1097
    next
hoelzl@38656
  1098
      case (preal r)
hoelzl@38656
  1099
      obtain j where "r \<le> of_nat j" using ex_le_of_nat ..
hoelzl@38656
  1100
      hence "u i \<le> of_nat j" using preal by (auto simp: real_of_nat_def)
hoelzl@38656
  1101
      show ?thesis
hoelzl@38656
  1102
      proof (rule pinfreal_SUPI)
hoelzl@38656
  1103
        fix y assume "\<And>j. j \<in> UNIV \<Longrightarrow> min (of_nat j) (u i) \<le> y"
hoelzl@38656
  1104
        note this[of j]
hoelzl@38656
  1105
        moreover have "min (of_nat j) (u i) = u i"
hoelzl@38656
  1106
          using `u i \<le> of_nat j` by (auto simp: min_def)
hoelzl@38656
  1107
        ultimately show "u i \<le> y" by simp
hoelzl@35582
  1108
      qed simp
hoelzl@35582
  1109
    qed
hoelzl@35582
  1110
  qed
hoelzl@38656
  1111
  from positive_integral_isoton[OF this]
hoelzl@38656
  1112
  have "?I = (SUP i. positive_integral (\<lambda>x. min (of_nat i) (u x) * indicator N x))"
hoelzl@38656
  1113
    unfolding isoton_def using borel `N \<in> sets M` by (simp add: borel_measurable_indicator)
hoelzl@38656
  1114
  also have "\<dots> \<le> (SUP i. positive_integral (\<lambda>x. of_nat i * indicator N x))"
hoelzl@38705
  1115
  proof (rule SUP_mono, rule bexI, rule positive_integral_mono)
hoelzl@38656
  1116
    fix x i show "min (of_nat i) (u x) * indicator N x \<le> of_nat i * indicator N x"
hoelzl@38656
  1117
      by (cases "x \<in> N") auto
hoelzl@38705
  1118
  qed simp
hoelzl@38656
  1119
  also have "\<dots> = 0"
hoelzl@38656
  1120
    using `N \<in> null_sets` by (simp add: positive_integral_cmult_indicator)
hoelzl@38656
  1121
  finally show ?thesis by simp
hoelzl@38656
  1122
qed
hoelzl@35582
  1123
hoelzl@38656
  1124
lemma (in measure_space) positive_integral_Markov_inequality:
hoelzl@38656
  1125
  assumes borel: "u \<in> borel_measurable M" and "A \<in> sets M" and c: "c \<noteq> \<omega>"
hoelzl@38656
  1126
  shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * positive_integral (\<lambda>x. u x * indicator A x)"
hoelzl@38656
  1127
    (is "\<mu> ?A \<le> _ * ?PI")
hoelzl@38656
  1128
proof -
hoelzl@38656
  1129
  have "?A \<in> sets M"
hoelzl@38656
  1130
    using `A \<in> sets M` borel by auto
hoelzl@38656
  1131
  hence "\<mu> ?A = positive_integral (\<lambda>x. indicator ?A x)"
hoelzl@38656
  1132
    using positive_integral_indicator by simp
hoelzl@38656
  1133
  also have "\<dots> \<le> positive_integral (\<lambda>x. c * (u x * indicator A x))"
hoelzl@38656
  1134
  proof (rule positive_integral_mono)
hoelzl@38656
  1135
    fix x assume "x \<in> space M"
hoelzl@38656
  1136
    show "indicator ?A x \<le> c * (u x * indicator A x)"
hoelzl@38656
  1137
      by (cases "x \<in> ?A") auto
hoelzl@38656
  1138
  qed
hoelzl@38656
  1139
  also have "\<dots> = c * positive_integral (\<lambda>x. u x * indicator A x)"
hoelzl@38656
  1140
    using assms
hoelzl@38656
  1141
    by (auto intro!: positive_integral_cmult borel_measurable_indicator)
hoelzl@38656
  1142
  finally show ?thesis .
hoelzl@35582
  1143
qed
hoelzl@35582
  1144
hoelzl@38656
  1145
lemma (in measure_space) positive_integral_0_iff:
hoelzl@38656
  1146
  assumes borel: "u \<in> borel_measurable M"
hoelzl@38656
  1147
  shows "positive_integral u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
hoelzl@38656
  1148
    (is "_ \<longleftrightarrow> \<mu> ?A = 0")
hoelzl@35582
  1149
proof -
hoelzl@38656
  1150
  have A: "?A \<in> sets M" using borel by auto
hoelzl@38656
  1151
  have u: "positive_integral (\<lambda>x. u x * indicator ?A x) = positive_integral u"
hoelzl@38656
  1152
    by (auto intro!: positive_integral_cong simp: indicator_def)
hoelzl@35582
  1153
hoelzl@38656
  1154
  show ?thesis
hoelzl@38656
  1155
  proof
hoelzl@38656
  1156
    assume "\<mu> ?A = 0"
hoelzl@38656
  1157
    hence "?A \<in> null_sets" using `?A \<in> sets M` by auto
hoelzl@38656
  1158
    from positive_integral_null_set[OF borel this]
hoelzl@38656
  1159
    have "0 = positive_integral (\<lambda>x. u x * indicator ?A x)" by simp
hoelzl@38656
  1160
    thus "positive_integral u = 0" unfolding u by simp
hoelzl@38656
  1161
  next
hoelzl@38656
  1162
    assume *: "positive_integral u = 0"
hoelzl@38656
  1163
    let "?M n" = "{x \<in> space M. 1 \<le> of_nat n * u x}"
hoelzl@38656
  1164
    have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
hoelzl@38656
  1165
    proof -
hoelzl@38656
  1166
      { fix n
hoelzl@38656
  1167
        from positive_integral_Markov_inequality[OF borel `?A \<in> sets M`, of "of_nat n"]
hoelzl@38656
  1168
        have "\<mu> (?M n \<inter> ?A) = 0" unfolding * u by simp }
hoelzl@38656
  1169
      thus ?thesis by simp
hoelzl@35582
  1170
    qed
hoelzl@38656
  1171
    also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
hoelzl@38656
  1172
    proof (safe intro!: continuity_from_below)
hoelzl@38656
  1173
      fix n show "?M n \<inter> ?A \<in> sets M"
hoelzl@38656
  1174
        using borel by (auto intro!: Int)
hoelzl@38656
  1175
    next
hoelzl@38656
  1176
      fix n x assume "1 \<le> of_nat n * u x"
hoelzl@38656
  1177
      also have "\<dots> \<le> of_nat (Suc n) * u x"
hoelzl@38656
  1178
        by (subst (1 2) mult_commute) (auto intro!: pinfreal_mult_cancel)
hoelzl@38656
  1179
      finally show "1 \<le> of_nat (Suc n) * u x" .
hoelzl@38656
  1180
    qed
hoelzl@38656
  1181
    also have "\<dots> = \<mu> ?A"
hoelzl@38656
  1182
    proof (safe intro!: arg_cong[where f="\<mu>"])
hoelzl@38656
  1183
      fix x assume "u x \<noteq> 0" and [simp, intro]: "x \<in> space M"
hoelzl@38656
  1184
      show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
hoelzl@38656
  1185
      proof (cases "u x")
hoelzl@38656
  1186
        case (preal r)
hoelzl@38656
  1187
        obtain j where "1 / r \<le> of_nat j" using ex_le_of_nat ..
hoelzl@38656
  1188
        hence "1 / r * r \<le> of_nat j * r" using preal unfolding mult_le_cancel_right by auto
hoelzl@38656
  1189
        hence "1 \<le> of_nat j * r" using preal `u x \<noteq> 0` by auto
hoelzl@38656
  1190
        thus ?thesis using `u x \<noteq> 0` preal by (auto simp: real_of_nat_def[symmetric])
hoelzl@38656
  1191
      qed auto
hoelzl@38656
  1192
    qed
hoelzl@38656
  1193
    finally show "\<mu> ?A = 0" by simp
hoelzl@35582
  1194
  qed
hoelzl@35582
  1195
qed
hoelzl@35582
  1196
hoelzl@38656
  1197
lemma (in measure_space) positive_integral_cong_on_null_sets:
hoelzl@38656
  1198
  assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
hoelzl@38656
  1199
  and measure: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
hoelzl@38656
  1200
  shows "positive_integral f = positive_integral g"
hoelzl@35582
  1201
proof -
hoelzl@38656
  1202
  let ?N = "{x\<in>space M. f x \<noteq> g x}" and ?E = "{x\<in>space M. f x = g x}"
hoelzl@38656
  1203
  let "?A h x" = "h x * indicator ?E x :: pinfreal"
hoelzl@38656
  1204
  let "?B h x" = "h x * indicator ?N x :: pinfreal"
hoelzl@38656
  1205
hoelzl@38656
  1206
  have A: "positive_integral (?A f) = positive_integral (?A g)"
hoelzl@38656
  1207
    by (auto intro!: positive_integral_cong simp: indicator_def)
hoelzl@38656
  1208
hoelzl@38656
  1209
  have [intro]: "?N \<in> sets M" "?E \<in> sets M" using f g by auto
hoelzl@38656
  1210
  hence "?N \<in> null_sets" using measure by auto
hoelzl@38656
  1211
  hence B: "positive_integral (?B f) = positive_integral (?B g)"
hoelzl@38656
  1212
    using f g by (simp add: positive_integral_null_set)
hoelzl@38656
  1213
hoelzl@38656
  1214
  have "positive_integral f = positive_integral (\<lambda>x. ?A f x + ?B f x)"
hoelzl@38656
  1215
    by (auto intro!: positive_integral_cong simp: indicator_def)
hoelzl@38656
  1216
  also have "\<dots> = positive_integral (?A f) + positive_integral (?B f)"
hoelzl@38656
  1217
    using f g by (auto intro!: positive_integral_add borel_measurable_indicator)
hoelzl@38656
  1218
  also have "\<dots> = positive_integral (\<lambda>x. ?A g x + ?B g x)"
hoelzl@38656
  1219
    unfolding A B using f g by (auto intro!: positive_integral_add[symmetric] borel_measurable_indicator)
hoelzl@38656
  1220
  also have "\<dots> = positive_integral g"
hoelzl@38656
  1221
    by (auto intro!: positive_integral_cong simp: indicator_def)
hoelzl@38656
  1222
  finally show ?thesis by simp
hoelzl@35582
  1223
qed
hoelzl@35582
  1224
hoelzl@35692
  1225
section "Lebesgue Integral"
hoelzl@35692
  1226
hoelzl@38656
  1227
definition (in measure_space) integrable where
hoelzl@38656
  1228
  "integrable f \<longleftrightarrow> f \<in> borel_measurable M \<and>
hoelzl@38656
  1229
    positive_integral (\<lambda>x. Real (f x)) \<noteq> \<omega> \<and>
hoelzl@38656
  1230
    positive_integral (\<lambda>x. Real (- f x)) \<noteq> \<omega>"
hoelzl@35692
  1231
hoelzl@38656
  1232
lemma (in measure_space) integrableD[dest]:
hoelzl@38656
  1233
  assumes "integrable f"
hoelzl@38656
  1234
  shows "f \<in> borel_measurable M"
hoelzl@38656
  1235
  "positive_integral (\<lambda>x. Real (f x)) \<noteq> \<omega>"
hoelzl@38656
  1236
  "positive_integral (\<lambda>x. Real (- f x)) \<noteq> \<omega>"
hoelzl@38656
  1237
  using assms unfolding integrable_def by auto
hoelzl@35692
  1238
hoelzl@38656
  1239
definition (in measure_space) integral where
hoelzl@38656
  1240
  "integral f =
hoelzl@38656
  1241
    real (positive_integral (\<lambda>x. Real (f x))) -
hoelzl@38656
  1242
    real (positive_integral (\<lambda>x. Real (- f x)))"
hoelzl@38656
  1243
hoelzl@38656
  1244
lemma (in measure_space) integral_cong:
hoelzl@35582
  1245
  assumes cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
hoelzl@35582
  1246
  shows "integral f = integral g"
hoelzl@38656
  1247
  using assms by (simp cong: positive_integral_cong add: integral_def)
hoelzl@35582
  1248
hoelzl@38656
  1249
lemma (in measure_space) integrable_cong:
hoelzl@38656
  1250
  "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable f \<longleftrightarrow> integrable g"
hoelzl@38656
  1251
  by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
hoelzl@38656
  1252
hoelzl@38656
  1253
lemma (in measure_space) integral_eq_positive_integral:
hoelzl@38656
  1254
  assumes "\<And>x. 0 \<le> f x"
hoelzl@38656
  1255
  shows "integral f = real (positive_integral (\<lambda>x. Real (f x)))"
hoelzl@35582
  1256
proof -
hoelzl@38656
  1257
  have "\<And>x. Real (- f x) = 0" using assms by simp
hoelzl@38656
  1258
  thus ?thesis by (simp del: Real_eq_0 add: integral_def)
hoelzl@35582
  1259
qed
hoelzl@35582
  1260
hoelzl@38656
  1261
lemma (in measure_space) integral_minus[intro, simp]:
hoelzl@38656
  1262
  assumes "integrable f"
hoelzl@38656
  1263
  shows "integrable (\<lambda>x. - f x)" "integral (\<lambda>x. - f x) = - integral f"
hoelzl@38656
  1264
  using assms by (auto simp: integrable_def integral_def)
hoelzl@38656
  1265
hoelzl@38656
  1266
lemma (in measure_space) integral_of_positive_diff:
hoelzl@38656
  1267
  assumes integrable: "integrable u" "integrable v"
hoelzl@38656
  1268
  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@38656
  1269
  shows "integrable f" and "integral f = integral u - integral v"
hoelzl@35582
  1270
proof -
hoelzl@38656
  1271
  let ?PI = positive_integral
hoelzl@38656
  1272
  let "?f x" = "Real (f x)"
hoelzl@38656
  1273
  let "?mf x" = "Real (- f x)"
hoelzl@38656
  1274
  let "?u x" = "Real (u x)"
hoelzl@38656
  1275
  let "?v x" = "Real (v x)"
hoelzl@38656
  1276
hoelzl@38656
  1277
  from borel_measurable_diff[of u v] integrable
hoelzl@38656
  1278
  have f_borel: "?f \<in> borel_measurable M" and
hoelzl@38656
  1279
    mf_borel: "?mf \<in> borel_measurable M" and
hoelzl@38656
  1280
    v_borel: "?v \<in> borel_measurable M" and
hoelzl@38656
  1281
    u_borel: "?u \<in> borel_measurable M" and
hoelzl@38656
  1282
    "f \<in> borel_measurable M"
hoelzl@38656
  1283
    by (auto simp: f_def[symmetric] integrable_def)
hoelzl@35582
  1284
hoelzl@38656
  1285
  have "?PI (\<lambda>x. Real (u x - v x)) \<le> ?PI ?u"
hoelzl@38656
  1286
    using pos by (auto intro!: positive_integral_mono)
hoelzl@38656
  1287
  moreover have "?PI (\<lambda>x. Real (v x - u x)) \<le> ?PI ?v"
hoelzl@38656
  1288
    using pos by (auto intro!: positive_integral_mono)
hoelzl@38656
  1289
  ultimately show f: "integrable f"
hoelzl@38656
  1290
    using `integrable u` `integrable v` `f \<in> borel_measurable M`
hoelzl@38656
  1291
    by (auto simp: integrable_def f_def)
hoelzl@38656
  1292
  hence mf: "integrable (\<lambda>x. - f x)" ..
hoelzl@38656
  1293
hoelzl@38656
  1294
  have *: "\<And>x. Real (- v x) = 0" "\<And>x. Real (- u x) = 0"
hoelzl@38656
  1295
    using pos by auto
hoelzl@35582
  1296
hoelzl@38656
  1297
  have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
hoelzl@38656
  1298
    unfolding f_def using pos by simp
hoelzl@38656
  1299
  hence "?PI (\<lambda>x. ?u x + ?mf x) = ?PI (\<lambda>x. ?v x + ?f x)" by simp
hoelzl@38656
  1300
  hence "real (?PI ?u + ?PI ?mf) = real (?PI ?v + ?PI ?f)"
hoelzl@38656
  1301
    using positive_integral_add[OF u_borel mf_borel]
hoelzl@38656
  1302
    using positive_integral_add[OF v_borel f_borel]
hoelzl@38656
  1303
    by auto
hoelzl@38656
  1304
  then show "integral f = integral u - integral v"
hoelzl@38656
  1305
    using f mf `integrable u` `integrable v`
hoelzl@38656
  1306
    unfolding integral_def integrable_def *
hoelzl@38656
  1307
    by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?v", cases "?PI ?u")
hoelzl@38656
  1308
       (auto simp add: field_simps)
hoelzl@35582
  1309
qed
hoelzl@35582
  1310
hoelzl@38656
  1311
lemma (in measure_space) integral_linear:
hoelzl@38656
  1312
  assumes "integrable f" "integrable g" and "0 \<le> a"
hoelzl@38656
  1313
  shows "integrable (\<lambda>t. a * f t + g t)"
hoelzl@38656
  1314
  and "integral (\<lambda>t. a * f t + g t) = a * integral f + integral g"
hoelzl@38656
  1315
proof -
hoelzl@38656
  1316
  let ?PI = positive_integral
hoelzl@38656
  1317
  let "?f x" = "Real (f x)"
hoelzl@38656
  1318
  let "?g x" = "Real (g x)"
hoelzl@38656
  1319
  let "?mf x" = "Real (- f x)"
hoelzl@38656
  1320
  let "?mg x" = "Real (- g x)"
hoelzl@38656
  1321
  let "?p t" = "max 0 (a * f t) + max 0 (g t)"
hoelzl@38656
  1322
  let "?n t" = "max 0 (- (a * f t)) + max 0 (- g t)"
hoelzl@38656
  1323
hoelzl@38656
  1324
  have pos: "?f \<in> borel_measurable M" "?g \<in> borel_measurable M"
hoelzl@38656
  1325
    and neg: "?mf \<in> borel_measurable M" "?mg \<in> borel_measurable M"
hoelzl@38656
  1326
    and p: "?p \<in> borel_measurable M"
hoelzl@38656
  1327
    and n: "?n \<in> borel_measurable M"
hoelzl@38656
  1328
    using assms by (simp_all add: integrable_def)
hoelzl@35582
  1329
hoelzl@38656
  1330
  have *: "\<And>x. Real (?p x) = Real a * ?f x + ?g x"
hoelzl@38656
  1331
          "\<And>x. Real (?n x) = Real a * ?mf x + ?mg x"
hoelzl@38656
  1332
          "\<And>x. Real (- ?p x) = 0"
hoelzl@38656
  1333
          "\<And>x. Real (- ?n x) = 0"
hoelzl@38656
  1334
    using `0 \<le> a` by (auto simp: max_def min_def zero_le_mult_iff mult_le_0_iff add_nonpos_nonpos)
hoelzl@38656
  1335
hoelzl@38656
  1336
  note linear =
hoelzl@38656
  1337
    positive_integral_linear[OF pos]
hoelzl@38656
  1338
    positive_integral_linear[OF neg]
hoelzl@35582
  1339
hoelzl@38656
  1340
  have "integrable ?p" "integrable ?n"
hoelzl@38656
  1341
      "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
hoelzl@38656
  1342
    using assms p n unfolding integrable_def * linear by auto
hoelzl@38656
  1343
  note diff = integral_of_positive_diff[OF this]
hoelzl@38656
  1344
hoelzl@38656
  1345
  show "integrable (\<lambda>t. a * f t + g t)" by (rule diff)
hoelzl@38656
  1346
hoelzl@38656
  1347
  from assms show "integral (\<lambda>t. a * f t + g t) = a * integral f + integral g"
hoelzl@38656
  1348
    unfolding diff(2) unfolding integral_def * linear integrable_def
hoelzl@38656
  1349
    by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?g", cases "?PI ?mg")
hoelzl@38656
  1350
       (auto simp add: field_simps zero_le_mult_iff)
hoelzl@38656
  1351
qed
hoelzl@38656
  1352
hoelzl@38656
  1353
lemma (in measure_space) integral_add[simp, intro]:
hoelzl@38656
  1354
  assumes "integrable f" "integrable g"
hoelzl@35582
  1355
  shows "integrable (\<lambda>t. f t + g t)"
hoelzl@35582
  1356
  and "integral (\<lambda>t. f t + g t) = integral f + integral g"
hoelzl@38656
  1357
  using assms integral_linear[where a=1] by auto
hoelzl@38656
  1358
hoelzl@38656
  1359
lemma (in measure_space) integral_zero[simp, intro]:
hoelzl@38656
  1360
  shows "integrable (\<lambda>x. 0)"
hoelzl@38656
  1361
  and "integral (\<lambda>x. 0) = 0"
hoelzl@38656
  1362
  unfolding integrable_def integral_def
hoelzl@38656
  1363
  by (auto simp add: borel_measurable_const)
hoelzl@35582
  1364
hoelzl@38656
  1365
lemma (in measure_space) integral_cmult[simp, intro]:
hoelzl@38656
  1366
  assumes "integrable f"
hoelzl@38656
  1367
  shows "integrable (\<lambda>t. a * f t)" (is ?P)
hoelzl@38656
  1368
  and "integral (\<lambda>t. a * f t) = a * integral f" (is ?I)
hoelzl@38656
  1369
proof -
hoelzl@38656
  1370
  have "integrable (\<lambda>t. a * f t) \<and> integral (\<lambda>t. a * f t) = a * integral f"
hoelzl@38656
  1371
  proof (cases rule: le_cases)
hoelzl@38656
  1372
    assume "0 \<le> a" show ?thesis
hoelzl@38656
  1373
      using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
hoelzl@38656
  1374
      by (simp add: integral_zero)
hoelzl@38656
  1375
  next
hoelzl@38656
  1376
    assume "a \<le> 0" hence "0 \<le> - a" by auto
hoelzl@38656
  1377
    have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
hoelzl@38656
  1378
    show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
hoelzl@38656
  1379
        integral_minus(1)[of "\<lambda>t. - a * f t"]
hoelzl@38656
  1380
      unfolding * integral_zero by simp
hoelzl@38656
  1381
  qed
hoelzl@38656
  1382
  thus ?P ?I by auto
hoelzl@35582
  1383
qed
hoelzl@35582
  1384
hoelzl@38656
  1385
lemma (in measure_space) integral_mono:
hoelzl@38656
  1386
  assumes fg: "integrable f" "integrable g"
hoelzl@35582
  1387
  and mono: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
hoelzl@35582
  1388
  shows "integral f \<le> integral g"
hoelzl@38656
  1389
  using fg unfolding integral_def integrable_def diff_minus
hoelzl@38656
  1390
proof (safe intro!: add_mono real_of_pinfreal_mono le_imp_neg_le positive_integral_mono)
hoelzl@38656
  1391
  fix x assume "x \<in> space M" from mono[OF this]
hoelzl@38656
  1392
  show "Real (f x) \<le> Real (g x)" "Real (- g x) \<le> Real (- f x)" by auto
hoelzl@35582
  1393
qed
hoelzl@35582
  1394
hoelzl@38656
  1395
lemma (in measure_space) integral_diff[simp, intro]:
hoelzl@38656
  1396
  assumes f: "integrable f" and g: "integrable g"
hoelzl@38656
  1397
  shows "integrable (\<lambda>t. f t - g t)"
hoelzl@38656
  1398
  and "integral (\<lambda>t. f t - g t) = integral f - integral g"
hoelzl@38656
  1399
  using integral_add[OF f integral_minus(1)[OF g]]
hoelzl@38656
  1400
  unfolding diff_minus integral_minus(2)[OF g]
hoelzl@38656
  1401
  by auto
hoelzl@38656
  1402
hoelzl@38656
  1403
lemma (in measure_space) integral_indicator[simp, intro]:
hoelzl@38656
  1404
  assumes "a \<in> sets M" and "\<mu> a \<noteq> \<omega>"
hoelzl@38656
  1405
  shows "integral (indicator a) = real (\<mu> a)" (is ?int)
hoelzl@38656
  1406
  and "integrable (indicator a)" (is ?able)
hoelzl@35582
  1407
proof -
hoelzl@38656
  1408
  have *:
hoelzl@38656
  1409
    "\<And>A x. Real (indicator A x) = indicator A x"
hoelzl@38656
  1410
    "\<And>A x. Real (- indicator A x) = 0" unfolding indicator_def by auto
hoelzl@38656
  1411
  show ?int ?able
hoelzl@38656
  1412
    using assms unfolding integral_def integrable_def
hoelzl@38656
  1413
    by (auto simp: * positive_integral_indicator borel_measurable_indicator)
hoelzl@35582
  1414
qed
hoelzl@35582
  1415
hoelzl@38656
  1416
lemma (in measure_space) integral_cmul_indicator:
hoelzl@38656
  1417
  assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<omega>"
hoelzl@38656
  1418
  shows "integrable (\<lambda>x. c * indicator A x)" (is ?P)
hoelzl@38656
  1419
  and "integral (\<lambda>x. c * indicator A x) = c * real (\<mu> A)" (is ?I)
hoelzl@38656
  1420
proof -
hoelzl@38656
  1421
  show ?P
hoelzl@38656
  1422
  proof (cases "c = 0")
hoelzl@38656
  1423
    case False with assms show ?thesis by simp
hoelzl@38656
  1424
  qed simp
hoelzl@35582
  1425
hoelzl@38656
  1426
  show ?I
hoelzl@38656
  1427
  proof (cases "c = 0")
hoelzl@38656
  1428
    case False with assms show ?thesis by simp
hoelzl@38656
  1429
  qed simp
hoelzl@38656
  1430
qed
hoelzl@35582
  1431
hoelzl@38656
  1432
lemma (in measure_space) integral_setsum[simp, intro]:
hoelzl@35582
  1433
  assumes "\<And>n. n \<in> S \<Longrightarrow> integrable (f n)"
hoelzl@35582
  1434
  shows "integral (\<lambda>x. \<Sum> i \<in> S. f i x) = (\<Sum> i \<in> S. integral (f i))" (is "?int S")
hoelzl@38656
  1435
    and "integrable (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
hoelzl@35582
  1436
proof -
hoelzl@38656
  1437
  have "?int S \<and> ?I S"
hoelzl@38656
  1438
  proof (cases "finite S")
hoelzl@38656
  1439
    assume "finite S"
hoelzl@38656
  1440
    from this assms show ?thesis by (induct S) simp_all
hoelzl@38656
  1441
  qed simp
hoelzl@35582
  1442
  thus "?int S" and "?I S" by auto
hoelzl@35582
  1443
qed
hoelzl@35582
  1444
hoelzl@36624
  1445
lemma (in measure_space) integrable_abs:
hoelzl@36624
  1446
  assumes "integrable f"
hoelzl@36624
  1447
  shows "integrable (\<lambda> x. \<bar>f x\<bar>)"
hoelzl@36624
  1448
proof -
hoelzl@38656
  1449
  have *:
hoelzl@38656
  1450
    "\<And>x. Real \<bar>f x\<bar> = Real (f x) + Real (- f x)"
hoelzl@38656
  1451
    "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
hoelzl@38656
  1452
  have abs: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M" and
hoelzl@38656
  1453
    f: "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
hoelzl@38656
  1454
        "(\<lambda>x. Real (- f x)) \<in> borel_measurable M"
hoelzl@38656
  1455
    using assms unfolding integrable_def by auto
hoelzl@38656
  1456
  from abs assms show ?thesis unfolding integrable_def *
hoelzl@38656
  1457
    using positive_integral_linear[OF f, of 1] by simp
hoelzl@38656
  1458
qed
hoelzl@38656
  1459
hoelzl@38656
  1460
lemma (in measure_space) integrable_bound:
hoelzl@38656
  1461
  assumes "integrable f"
hoelzl@38656
  1462
  and f: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
hoelzl@38656
  1463
    "\<And>x. x \<in> space M \<Longrightarrow> \<bar>g x\<bar> \<le> f x"
hoelzl@38656
  1464
  assumes borel: "g \<in> borel_measurable M"
hoelzl@38656
  1465
  shows "integrable g"
hoelzl@38656
  1466
proof -
hoelzl@38656
  1467
  have "positive_integral (\<lambda>x. Real (g x)) \<le> positive_integral (\<lambda>x. Real \<bar>g x\<bar>)"
hoelzl@38656
  1468
    by (auto intro!: positive_integral_mono)
hoelzl@38656
  1469
  also have "\<dots> \<le> positive_integral (\<lambda>x. Real (f x))"
hoelzl@38656
  1470
    using f by (auto intro!: positive_integral_mono)
hoelzl@38656
  1471
  also have "\<dots> < \<omega>"
hoelzl@38656
  1472
    using `integrable f` unfolding integrable_def by (auto simp: pinfreal_less_\<omega>)
hoelzl@38656
  1473
  finally have pos: "positive_integral (\<lambda>x. Real (g x)) < \<omega>" .
hoelzl@38656
  1474
hoelzl@38656
  1475
  have "positive_integral (\<lambda>x. Real (- g x)) \<le> positive_integral (\<lambda>x. Real (\<bar>g x\<bar>))"
hoelzl@38656
  1476
    by (auto intro!: positive_integral_mono)
hoelzl@38656
  1477
  also have "\<dots> \<le> positive_integral (\<lambda>x. Real (f x))"
hoelzl@38656
  1478
    using f by (auto intro!: positive_integral_mono)
hoelzl@38656
  1479
  also have "\<dots> < \<omega>"
hoelzl@38656
  1480
    using `integrable f` unfolding integrable_def by (auto simp: pinfreal_less_\<omega>)
hoelzl@38656
  1481
  finally have neg: "positive_integral (\<lambda>x. Real (- g x)) < \<omega>" .
hoelzl@38656
  1482
hoelzl@38656
  1483
  from neg pos borel show ?thesis
hoelzl@36624
  1484
    unfolding integrable_def by auto
hoelzl@38656
  1485
qed
hoelzl@38656
  1486
hoelzl@38656
  1487
lemma (in measure_space) integrable_abs_iff:
hoelzl@38656
  1488
  "f \<in> borel_measurable M \<Longrightarrow> integrable (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable f"
hoelzl@38656
  1489
  by (auto intro!: integrable_bound[where g=f] integrable_abs)
hoelzl@38656
  1490
hoelzl@38656
  1491
lemma (in measure_space) integrable_max:
hoelzl@38656
  1492
  assumes int: "integrable f" "integrable g"
hoelzl@38656
  1493
  shows "integrable (\<lambda> x. max (f x) (g x))"
hoelzl@38656
  1494
proof (rule integrable_bound)
hoelzl@38656
  1495
  show "integrable (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
hoelzl@38656
  1496
    using int by (simp add: integrable_abs)
hoelzl@38656
  1497
  show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
hoelzl@38656
  1498
    using int unfolding integrable_def by auto
hoelzl@38656
  1499
next
hoelzl@38656
  1500
  fix x assume "x \<in> space M"
hoelzl@38656
  1501
  show "0 \<le> \<bar>f x\<bar> + \<bar>g x\<bar>" "\<bar>max (f x) (g x)\<bar> \<le> \<bar>f x\<bar> + \<bar>g x\<bar>"
hoelzl@38656
  1502
    by auto
hoelzl@38656
  1503
qed
hoelzl@38656
  1504
hoelzl@38656
  1505
lemma (in measure_space) integrable_min:
hoelzl@38656
  1506
  assumes int: "integrable f" "integrable g"
hoelzl@38656
  1507
  shows "integrable (\<lambda> x. min (f x) (g x))"
hoelzl@38656
  1508
proof (rule integrable_bound)
hoelzl@38656
  1509
  show "integrable (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
hoelzl@38656
  1510
    using int by (simp add: integrable_abs)
hoelzl@38656
  1511
  show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
hoelzl@38656
  1512
    using int unfolding integrable_def by auto
hoelzl@38656
  1513
next
hoelzl@38656
  1514
  fix x assume "x \<in> space M"
hoelzl@38656
  1515
  show "0 \<le> \<bar>f x\<bar> + \<bar>g x\<bar>" "\<bar>min (f x) (g x)\<bar> \<le> \<bar>f x\<bar> + \<bar>g x\<bar>"
hoelzl@38656
  1516
    by auto
hoelzl@38656
  1517
qed
hoelzl@38656
  1518
hoelzl@38656
  1519
lemma (in measure_space) integral_triangle_inequality:
hoelzl@38656
  1520
  assumes "integrable f"
hoelzl@38656
  1521
  shows "\<bar>integral f\<bar> \<le> integral (\<lambda>x. \<bar>f x\<bar>)"
hoelzl@38656
  1522
proof -
hoelzl@38656
  1523
  have "\<bar>integral f\<bar> = max (integral f) (- integral f)" by auto
hoelzl@38656
  1524
  also have "\<dots> \<le> integral (\<lambda>x. \<bar>f x\<bar>)"
hoelzl@38656
  1525
      using assms integral_minus(2)[of f, symmetric]
hoelzl@38656
  1526
      by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
hoelzl@38656
  1527
  finally show ?thesis .
hoelzl@36624
  1528
qed
hoelzl@36624
  1529
hoelzl@38656
  1530
lemma (in measure_space) integral_positive:
hoelzl@38656
  1531
  assumes "integrable f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
hoelzl@38656
  1532
  shows "0 \<le> integral f"
hoelzl@38656
  1533
proof -
hoelzl@38656
  1534
  have "0 = integral (\<lambda>x. 0)" by (auto simp: integral_zero)
hoelzl@38656
  1535
  also have "\<dots> \<le> integral f"
hoelzl@38656
  1536
    using assms by (rule integral_mono[OF integral_zero(1)])
hoelzl@38656
  1537
  finally show ?thesis .
hoelzl@38656
  1538
qed
hoelzl@38656
  1539
hoelzl@38656
  1540
lemma (in measure_space) integral_monotone_convergence_pos:
hoelzl@38656
  1541
  assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
hoelzl@38656
  1542
  and pos: "\<And>x i. 0 \<le> f i x"
hoelzl@38656
  1543
  and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
hoelzl@38656
  1544
  and ilim: "(\<lambda>i. integral (f i)) ----> x"
hoelzl@38656
  1545
  shows "integrable u"
hoelzl@38656
  1546
  and "integral u = x"
hoelzl@35582
  1547
proof -
hoelzl@38656
  1548
  { fix x have "0 \<le> u x"
hoelzl@38656
  1549
      using mono pos[of 0 x] incseq_le[OF _ lim, of x 0]
hoelzl@38656
  1550
      by (simp add: mono_def incseq_def) }
hoelzl@38656
  1551
  note pos_u = this
hoelzl@38656
  1552
hoelzl@38656
  1553
  hence [simp]: "\<And>i x. Real (- f i x) = 0" "\<And>x. Real (- u x) = 0"
hoelzl@38656
  1554
    using pos by auto
hoelzl@38656
  1555
hoelzl@38656
  1556
  have SUP_F: "\<And>x. (SUP n. Real (f n x)) = Real (u x)"
hoelzl@38656
  1557
    using mono pos pos_u lim by (rule SUP_eq_LIMSEQ[THEN iffD2])
hoelzl@38656
  1558
hoelzl@38656
  1559
  have borel_f: "\<And>i. (\<lambda>x. Real (f i x)) \<in> borel_measurable M"
hoelzl@38656
  1560
    using i unfolding integrable_def by auto
hoelzl@38656
  1561
  hence "(SUP i. (\<lambda>x. Real (f i x))) \<in> borel_measurable M"
hoelzl@35582
  1562
    by auto
hoelzl@38656
  1563
  hence borel_u: "u \<in> borel_measurable M"
hoelzl@38656
  1564
    using pos_u by (auto simp: borel_measurable_Real_eq SUPR_fun_expand SUP_F)
hoelzl@38656
  1565
hoelzl@38656
  1566
  have integral_eq: "\<And>n. positive_integral (\<lambda>x. Real (f n x)) = Real (integral (f n))"
hoelzl@38656
  1567
    using i unfolding integral_def integrable_def by (auto simp: Real_real)
hoelzl@38656
  1568
hoelzl@38656
  1569
  have pos_integral: "\<And>n. 0 \<le> integral (f n)"
hoelzl@38656
  1570
    using pos i by (auto simp: integral_positive)
hoelzl@38656
  1571
  hence "0 \<le> x"
hoelzl@38656
  1572
    using LIMSEQ_le_const[OF ilim, of 0] by auto
hoelzl@38656
  1573
hoelzl@38656
  1574
  have "(\<lambda>i. positive_integral (\<lambda>x. Real (f i x))) \<up> positive_integral (\<lambda>x. Real (u x))"
hoelzl@38656
  1575
  proof (rule positive_integral_isoton)
hoelzl@38656
  1576
    from SUP_F mono pos
hoelzl@38656
  1577
    show "(\<lambda>i x. Real (f i x)) \<up> (\<lambda>x. Real (u x))"
hoelzl@38656
  1578
      unfolding isoton_fun_expand by (auto simp: isoton_def mono_def)
hoelzl@38656
  1579
  qed (rule borel_f)
hoelzl@38656
  1580
  hence pI: "positive_integral (\<lambda>x. Real (u x)) =
hoelzl@38656
  1581
      (SUP n. positive_integral (\<lambda>x. Real (f n x)))"
hoelzl@38656
  1582
    unfolding isoton_def by simp
hoelzl@38656
  1583
  also have "\<dots> = Real x" unfolding integral_eq
hoelzl@38656
  1584
  proof (rule SUP_eq_LIMSEQ[THEN iffD2])
hoelzl@38656
  1585
    show "mono (\<lambda>n. integral (f n))"
hoelzl@38656
  1586
      using mono i by (auto simp: mono_def intro!: integral_mono)
hoelzl@38656
  1587
    show "\<And>n. 0 \<le> integral (f n)" using pos_integral .
hoelzl@38656
  1588
    show "0 \<le> x" using `0 \<le> x` .
hoelzl@38656
  1589
    show "(\<lambda>n. integral (f n)) ----> x" using ilim .
hoelzl@38656
  1590
  qed
hoelzl@38656
  1591
  finally show  "integrable u" "integral u = x" using borel_u `0 \<le> x`
hoelzl@38656
  1592
    unfolding integrable_def integral_def by auto
hoelzl@38656
  1593
qed
hoelzl@38656
  1594
hoelzl@38656
  1595
lemma (in measure_space) integral_monotone_convergence:
hoelzl@38656
  1596
  assumes f: "\<And>i. integrable (f i)" and "mono f"
hoelzl@38656
  1597
  and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
hoelzl@38656
  1598
  and ilim: "(\<lambda>i. integral (f i)) ----> x"
hoelzl@38656
  1599
  shows "integrable u"
hoelzl@38656
  1600
  and "integral u = x"
hoelzl@38656
  1601
proof -
hoelzl@38656
  1602
  have 1: "\<And>i. integrable (\<lambda>x. f i x - f 0 x)"
hoelzl@38656
  1603
      using f by (auto intro!: integral_diff)
hoelzl@38656
  1604
  have 2: "\<And>x. mono (\<lambda>n. f n x - f 0 x)" using `mono f`
hoelzl@38656
  1605
      unfolding mono_def le_fun_def by auto
hoelzl@38656
  1606
  have 3: "\<And>x n. 0 \<le> f n x - f 0 x" using `mono f`
hoelzl@38656
  1607
      unfolding mono_def le_fun_def by (auto simp: field_simps)
hoelzl@38656
  1608
  have 4: "\<And>x. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
hoelzl@38656
  1609
    using lim by (auto intro!: LIMSEQ_diff)
hoelzl@38656
  1610
  have 5: "(\<lambda>i. integral (\<lambda>x. f i x - f 0 x)) ----> x - integral (f 0)"
hoelzl@38656
  1611
    using f ilim by (auto intro!: LIMSEQ_diff simp: integral_diff)
hoelzl@38656
  1612
  note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5]
hoelzl@38656
  1613
  have "integrable (\<lambda>x. (u x - f 0 x) + f 0 x)"
hoelzl@38656
  1614
    using diff(1) f by (rule integral_add(1))
hoelzl@38656
  1615
  with diff(2) f show "integrable u" "integral u = x"
hoelzl@38656
  1616
    by (auto simp: integral_diff)
hoelzl@38656
  1617
qed
hoelzl@38656
  1618
hoelzl@38656
  1619
lemma (in measure_space) integral_0_iff:
hoelzl@38656
  1620
  assumes "integrable f"
hoelzl@38656
  1621
  shows "integral (\<lambda>x. \<bar>f x\<bar>) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
hoelzl@38656
  1622
proof -
hoelzl@38656
  1623
  have *: "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
hoelzl@38656
  1624
  have "integrable (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
hoelzl@38656
  1625
  hence "(\<lambda>x. Real (\<bar>f x\<bar>)) \<in> borel_measurable M"
hoelzl@38656
  1626
    "positive_integral (\<lambda>x. Real \<bar>f x\<bar>) \<noteq> \<omega>" unfolding integrable_def by auto
hoelzl@38656
  1627
  from positive_integral_0_iff[OF this(1)] this(2)
hoelzl@38656
  1628
  show ?thesis unfolding integral_def *
hoelzl@38656
  1629
    by (simp add: real_of_pinfreal_eq_0)
hoelzl@35582
  1630
qed
hoelzl@35582
  1631
hoelzl@38656
  1632
lemma LIMSEQ_max:
hoelzl@38656
  1633
  "u ----> (x::real) \<Longrightarrow> (\<lambda>i. max (u i) 0) ----> max x 0"
hoelzl@38656
  1634
  by (fastsimp intro!: LIMSEQ_I dest!: LIMSEQ_D)
hoelzl@38656
  1635
hoelzl@38656
  1636
lemma (in sigma_algebra) borel_measurable_LIMSEQ:
hoelzl@38656
  1637
  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@38656
  1638
  assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' x"
hoelzl@38656
  1639
  and u: "\<And>i. u i \<in> borel_measurable M"
hoelzl@38656
  1640
  shows "u' \<in> borel_measurable M"
hoelzl@38656
  1641
proof -
hoelzl@38656
  1642
  let "?pu x i" = "max (u i x) 0"
hoelzl@38656
  1643
  let "?nu x i" = "max (- u i x) 0"
hoelzl@38656
  1644
hoelzl@38656
  1645
  { fix x assume x: "x \<in> space M"
hoelzl@38656
  1646
    have "(?pu x) ----> max (u' x) 0"
hoelzl@38656
  1647
      "(?nu x) ----> max (- u' x) 0"
hoelzl@38656
  1648
      using u'[OF x] by (auto intro!: LIMSEQ_max LIMSEQ_minus)
hoelzl@38656
  1649
    from LIMSEQ_imp_lim_INF[OF _ this(1)] LIMSEQ_imp_lim_INF[OF _ this(2)]