src/HOL/Probability/Lebesgue_Integration.thy
author hoelzl
Thu Sep 02 17:12:40 2010 +0200 (2010-09-02)
changeset 39092 98de40859858
parent 38705 aaee86c0e237
child 39198 f967a16dfcdd
permissions -rw-r--r--
move lemmas to correct theory files
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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 (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)
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      finally show "k \<le> j*2^j" by (auto simp del: real_of_nat_mult)
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    qed simp }
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  note f_upper = this
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  let "?g j x" = "of_nat (f x j) / 2^j :: pinfreal"
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  show ?thesis unfolding simple_function_def isoton_fun_expand unfolding isoton_def
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  proof (safe intro!: exI[of _ ?g])
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    fix j
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    have *: "?g j ` space M \<subseteq> (\<lambda>x. of_nat x / 2^j) ` {..j * 2^j}"
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      using f_upper by auto
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    thus "finite (?g j ` space M)" by (rule finite_subset) auto
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  next
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    fix j t assume "t \<in> space M"
hoelzl@38656
   275
    have **: "?g j -` {?g j t} \<inter> space M = {x \<in> space M. f t j = f x j}"
hoelzl@38656
   276
      by (auto simp: power_le_zero_eq Real_eq_Real mult_le_0_iff)
hoelzl@35582
   277
hoelzl@38656
   278
    show "?g j -` {?g j t} \<inter> space M \<in> sets M"
hoelzl@38656
   279
    proof cases
hoelzl@38656
   280
      assume "of_nat j \<le> u t"
hoelzl@38656
   281
      hence "?g j -` {?g j t} \<inter> space M = {x\<in>space M. of_nat j \<le> u x}"
hoelzl@38656
   282
        unfolding ** f_eq[symmetric] by auto
hoelzl@38656
   283
      thus "?g j -` {?g j t} \<inter> space M \<in> sets M"
hoelzl@38656
   284
        using u by auto
hoelzl@35582
   285
    next
hoelzl@38656
   286
      assume not_t: "\<not> of_nat j \<le> u t"
hoelzl@38656
   287
      hence less: "f t j < j*2^j" using f_eq[symmetric] `f t j \<le> j*2^j` by auto
hoelzl@38656
   288
      have split_vimage: "?g j -` {?g j t} \<inter> space M =
hoelzl@38656
   289
          {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
   290
        unfolding **
hoelzl@38656
   291
      proof safe
hoelzl@38656
   292
        fix x assume [simp]: "f t j = f x j"
hoelzl@38656
   293
        have *: "\<not> of_nat j \<le> u x" using not_t f_eq[symmetric] by simp
hoelzl@38656
   294
        hence "of_nat (f t j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f t j))"
hoelzl@38656
   295
          using upper lower by auto
hoelzl@38656
   296
        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
   297
          by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
hoelzl@38656
   298
        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
   299
      next
hoelzl@38656
   300
        fix x
hoelzl@38656
   301
        assume "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j"
hoelzl@38656
   302
        hence "of_nat (f t j) \<le> u x * 2 ^ j \<and> u x * 2 ^ j < of_nat (Suc (f t j))"
hoelzl@38656
   303
          by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
hoelzl@38656
   304
        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
   305
        note 2
hoelzl@38656
   306
        also have "\<dots> \<le> of_nat (j*2^j)"
hoelzl@38656
   307
          using less by (auto simp: zero_le_mult_iff simp del: real_of_nat_mult)
hoelzl@38656
   308
        finally have bound_ux: "u x < of_nat j"
hoelzl@38656
   309
          by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq)
hoelzl@38656
   310
        show "f t j = f x j"
hoelzl@38656
   311
        proof (rule antisym)
hoelzl@38656
   312
          from 1 lower[OF bound_ux]
hoelzl@38656
   313
          show "f t j \<le> f x j" by (cases "u x") (auto split: split_if_asm)
hoelzl@38656
   314
          from upper[OF bound_ux] 2
hoelzl@38656
   315
          show "f x j \<le> f t j" by (cases "u x") (auto split: split_if_asm)
hoelzl@38656
   316
        qed
hoelzl@38656
   317
      qed
hoelzl@38656
   318
      show ?thesis unfolding split_vimage using u by auto
hoelzl@35582
   319
    qed
hoelzl@38656
   320
  next
hoelzl@38656
   321
    fix j t assume "?g j t = \<omega>" thus False by (auto simp: power_le_zero_eq)
hoelzl@38656
   322
  next
hoelzl@38656
   323
    fix t
hoelzl@38656
   324
    { fix i
hoelzl@38656
   325
      have "f t i * 2 \<le> f t (Suc i)"
hoelzl@38656
   326
      proof (rule fI)
hoelzl@38656
   327
        assume "of_nat (Suc i) \<le> u t"
hoelzl@38656
   328
        hence "of_nat i \<le> u t" by (cases "u t") auto
hoelzl@38656
   329
        thus "f t i * 2 \<le> Suc i * 2 ^ Suc i" unfolding f_eq[symmetric] by simp
hoelzl@38656
   330
      next
hoelzl@38656
   331
        fix k
hoelzl@38656
   332
        assume *: "u t * 2 ^ Suc i < of_nat (Suc k)"
hoelzl@38656
   333
        show "f t i * 2 \<le> k"
hoelzl@38656
   334
        proof (rule fI)
hoelzl@38656
   335
          assume "of_nat i \<le> u t"
hoelzl@38656
   336
          hence "of_nat (i * 2^Suc i) \<le> u t * 2^Suc i"
hoelzl@38656
   337
            by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
hoelzl@38656
   338
          also have "\<dots> < of_nat (Suc k)" using * by auto
hoelzl@38656
   339
          finally show "i * 2 ^ i * 2 \<le> k"
hoelzl@38656
   340
            by (auto simp del: real_of_nat_mult)
hoelzl@38656
   341
        next
hoelzl@38656
   342
          fix j assume "of_nat j \<le> u t * 2 ^ i"
hoelzl@38656
   343
          with * show "j * 2 \<le> k" by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
hoelzl@38656
   344
        qed
hoelzl@38656
   345
      qed
hoelzl@38656
   346
      thus "?g i t \<le> ?g (Suc i) t"
hoelzl@38656
   347
        by (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps) }
hoelzl@38656
   348
    hence mono: "mono (\<lambda>i. ?g i t)" unfolding mono_iff_le_Suc by auto
hoelzl@35582
   349
hoelzl@38656
   350
    show "(SUP j. of_nat (f t j) / 2 ^ j) = u t"
hoelzl@38656
   351
    proof (rule pinfreal_SUPI)
hoelzl@38656
   352
      fix j show "of_nat (f t j) / 2 ^ j \<le> u t"
hoelzl@38656
   353
      proof (rule fI)
hoelzl@38656
   354
        assume "of_nat j \<le> u t" thus "of_nat (j * 2 ^ j) / 2 ^ j \<le> u t"
hoelzl@38656
   355
          by (cases "u t") (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps)
hoelzl@38656
   356
      next
hoelzl@38656
   357
        fix k assume "of_nat k \<le> u t * 2 ^ j"
hoelzl@38656
   358
        thus "of_nat k / 2 ^ j \<le> u t"
hoelzl@38656
   359
          by (cases "u t")
hoelzl@38656
   360
             (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps zero_le_mult_iff)
hoelzl@38656
   361
      qed
hoelzl@38656
   362
    next
hoelzl@38656
   363
      fix y :: pinfreal assume *: "\<And>j. j \<in> UNIV \<Longrightarrow> of_nat (f t j) / 2 ^ j \<le> y"
hoelzl@38656
   364
      show "u t \<le> y"
hoelzl@38656
   365
      proof (cases "u t")
hoelzl@38656
   366
        case (preal r)
hoelzl@38656
   367
        show ?thesis
hoelzl@38656
   368
        proof (rule ccontr)
hoelzl@38656
   369
          assume "\<not> u t \<le> y"
hoelzl@38656
   370
          then obtain p where p: "y = Real p" "0 \<le> p" "p < r" using preal by (cases y) auto
hoelzl@38656
   371
          with LIMSEQ_inverse_realpow_zero[of 2, unfolded LIMSEQ_iff, rule_format, of "r - p"]
hoelzl@38656
   372
          obtain n where n: "\<And>N. n \<le> N \<Longrightarrow> inverse (2^N) < r - p" by auto
hoelzl@38656
   373
          let ?N = "max n (natfloor r + 1)"
hoelzl@38656
   374
          have "u t < of_nat ?N" "n \<le> ?N"
hoelzl@38656
   375
            using ge_natfloor_plus_one_imp_gt[of r n] preal
hoelzl@38705
   376
            using real_natfloor_add_one_gt
hoelzl@38705
   377
            by (auto simp: max_def real_of_nat_Suc)
hoelzl@38656
   378
          from lower[OF this(1)]
hoelzl@38656
   379
          have "r < (real (f t ?N) + 1) / 2 ^ ?N" unfolding less_divide_eq
hoelzl@38656
   380
            using preal by (auto simp add: power_le_zero_eq zero_le_mult_iff real_of_nat_Suc split: split_if_asm)
hoelzl@38656
   381
          hence "u t < of_nat (f t ?N) / 2 ^ ?N + 1 / 2 ^ ?N"
hoelzl@38656
   382
            using preal by (auto simp: field_simps divide_real_def[symmetric])
hoelzl@38656
   383
          with n[OF `n \<le> ?N`] p preal *[of ?N]
hoelzl@38656
   384
          show False
hoelzl@38656
   385
            by (cases "f t ?N") (auto simp: power_le_zero_eq split: split_if_asm)
hoelzl@38656
   386
        qed
hoelzl@38656
   387
      next
hoelzl@38656
   388
        case infinite
hoelzl@38656
   389
        { fix j have "f t j = j*2^j" using top[of j t] infinite by simp
hoelzl@38656
   390
          hence "of_nat j \<le> y" using *[of j]
hoelzl@38656
   391
            by (cases y) (auto simp: power_le_zero_eq zero_le_mult_iff field_simps) }
hoelzl@38656
   392
        note all_less_y = this
hoelzl@38656
   393
        show ?thesis unfolding infinite
hoelzl@38656
   394
        proof (rule ccontr)
hoelzl@38656
   395
          assume "\<not> \<omega> \<le> y" then obtain r where r: "y = Real r" "0 \<le> r" by (cases y) auto
hoelzl@38656
   396
          moreover obtain n ::nat where "r < real n" using ex_less_of_nat by (auto simp: real_eq_of_nat)
hoelzl@38656
   397
          with all_less_y[of n] r show False by auto
hoelzl@38656
   398
        qed
hoelzl@38656
   399
      qed
hoelzl@38656
   400
    qed
hoelzl@35582
   401
  qed
hoelzl@35582
   402
qed
hoelzl@35582
   403
hoelzl@38656
   404
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
hoelzl@38656
   405
  fixes u :: "'a \<Rightarrow> pinfreal"
hoelzl@38656
   406
  assumes "u \<in> borel_measurable M"
hoelzl@38656
   407
  obtains (x) f where "f \<up> u" "\<And>i. simple_function (f i)" "\<And>i. \<omega>\<notin>f i`space M"
hoelzl@35582
   408
proof -
hoelzl@38656
   409
  from borel_measurable_implies_simple_function_sequence[OF assms]
hoelzl@38656
   410
  obtain f where x: "\<And>i. simple_function (f i)" "f \<up> u"
hoelzl@38656
   411
    and fin: "\<And>i. \<And>x. x\<in>space M \<Longrightarrow> f i x \<noteq> \<omega>" by auto
hoelzl@38656
   412
  { fix i from fin[of _ i] have "\<omega> \<notin> f i`space M" by fastsimp }
hoelzl@38656
   413
  with x show thesis by (auto intro!: that[of f])
hoelzl@38656
   414
qed
hoelzl@38656
   415
hoelzl@39092
   416
lemma (in sigma_algebra) simple_function_eq_borel_measurable:
hoelzl@39092
   417
  fixes f :: "'a \<Rightarrow> pinfreal"
hoelzl@39092
   418
  shows "simple_function f \<longleftrightarrow>
hoelzl@39092
   419
    finite (f`space M) \<and> f \<in> borel_measurable M"
hoelzl@39092
   420
  using simple_function_borel_measurable[of f]
hoelzl@39092
   421
    borel_measurable_simple_function[of f]
hoelzl@39092
   422
  by (fastsimp simp: simple_function_def)
hoelzl@39092
   423
hoelzl@39092
   424
lemma (in measure_space) simple_function_restricted:
hoelzl@39092
   425
  fixes f :: "'a \<Rightarrow> pinfreal" assumes "A \<in> sets M"
hoelzl@39092
   426
  shows "sigma_algebra.simple_function (restricted_space A) f \<longleftrightarrow> simple_function (\<lambda>x. f x * indicator A x)"
hoelzl@39092
   427
    (is "sigma_algebra.simple_function ?R f \<longleftrightarrow> simple_function ?f")
hoelzl@39092
   428
proof -
hoelzl@39092
   429
  interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
hoelzl@39092
   430
  have "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
hoelzl@39092
   431
  proof cases
hoelzl@39092
   432
    assume "A = space M"
hoelzl@39092
   433
    then have "f`A = ?f`space M" by (fastsimp simp: image_iff)
hoelzl@39092
   434
    then show ?thesis by simp
hoelzl@39092
   435
  next
hoelzl@39092
   436
    assume "A \<noteq> space M"
hoelzl@39092
   437
    then obtain x where x: "x \<in> space M" "x \<notin> A"
hoelzl@39092
   438
      using sets_into_space `A \<in> sets M` by auto
hoelzl@39092
   439
    have *: "?f`space M = f`A \<union> {0}"
hoelzl@39092
   440
    proof (auto simp add: image_iff)
hoelzl@39092
   441
      show "\<exists>x\<in>space M. f x = 0 \<or> indicator A x = 0"
hoelzl@39092
   442
        using x by (auto intro!: bexI[of _ x])
hoelzl@39092
   443
    next
hoelzl@39092
   444
      fix x assume "x \<in> A"
hoelzl@39092
   445
      then show "\<exists>y\<in>space M. f x = f y * indicator A y"
hoelzl@39092
   446
        using `A \<in> sets M` sets_into_space by (auto intro!: bexI[of _ x])
hoelzl@39092
   447
    next
hoelzl@39092
   448
      fix x
hoelzl@39092
   449
      assume "indicator A x \<noteq> (0::pinfreal)"
hoelzl@39092
   450
      then have "x \<in> A" by (auto simp: indicator_def split: split_if_asm)
hoelzl@39092
   451
      moreover assume "x \<in> space M" "\<forall>y\<in>A. ?f x \<noteq> f y"
hoelzl@39092
   452
      ultimately show "f x = 0" by auto
hoelzl@39092
   453
    qed
hoelzl@39092
   454
    then show ?thesis by auto
hoelzl@39092
   455
  qed
hoelzl@39092
   456
  then show ?thesis
hoelzl@39092
   457
    unfolding simple_function_eq_borel_measurable
hoelzl@39092
   458
      R.simple_function_eq_borel_measurable
hoelzl@39092
   459
    unfolding borel_measurable_restricted[OF `A \<in> sets M`]
hoelzl@39092
   460
    by auto
hoelzl@39092
   461
qed
hoelzl@39092
   462
hoelzl@39092
   463
lemma (in sigma_algebra) simple_function_subalgebra:
hoelzl@39092
   464
  assumes "sigma_algebra.simple_function (M\<lparr>sets:=N\<rparr>) f"
hoelzl@39092
   465
  and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr>sets:=N\<rparr>)"
hoelzl@39092
   466
  shows "simple_function f"
hoelzl@39092
   467
  using assms
hoelzl@39092
   468
  unfolding simple_function_def
hoelzl@39092
   469
  unfolding sigma_algebra.simple_function_def[OF N_subalgebra(2)]
hoelzl@39092
   470
  by auto
hoelzl@39092
   471
hoelzl@38656
   472
section "Simple integral"
hoelzl@38656
   473
hoelzl@38656
   474
definition (in measure_space)
hoelzl@38656
   475
  "simple_integral f = (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M))"
hoelzl@35582
   476
hoelzl@38656
   477
lemma (in measure_space) simple_integral_cong:
hoelzl@38656
   478
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
hoelzl@38656
   479
  shows "simple_integral f = simple_integral g"
hoelzl@38656
   480
proof -
hoelzl@38656
   481
  have "f ` space M = g ` space M"
hoelzl@38656
   482
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
hoelzl@38656
   483
    using assms by (auto intro!: image_eqI)
hoelzl@38656
   484
  thus ?thesis unfolding simple_integral_def by simp
hoelzl@38656
   485
qed
hoelzl@38656
   486
hoelzl@38656
   487
lemma (in measure_space) simple_integral_const[simp]:
hoelzl@38656
   488
  "simple_integral (\<lambda>x. c) = c * \<mu> (space M)"
hoelzl@38656
   489
proof (cases "space M = {}")
hoelzl@38656
   490
  case True thus ?thesis unfolding simple_integral_def by simp
hoelzl@38656
   491
next
hoelzl@38656
   492
  case False hence "(\<lambda>x. c) ` space M = {c}" by auto
hoelzl@38656
   493
  thus ?thesis unfolding simple_integral_def by simp
hoelzl@35582
   494
qed
hoelzl@35582
   495
hoelzl@38656
   496
lemma (in measure_space) simple_function_partition:
hoelzl@38656
   497
  assumes "simple_function f" and "simple_function g"
hoelzl@38656
   498
  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
   499
    (is "_ = setsum _ (?p ` space M)")
hoelzl@38656
   500
proof-
hoelzl@38656
   501
  let "?sub x" = "?p ` (f -` {x} \<inter> space M)"
hoelzl@38656
   502
  let ?SIGMA = "Sigma (f`space M) ?sub"
hoelzl@35582
   503
hoelzl@38656
   504
  have [intro]:
hoelzl@38656
   505
    "finite (f ` space M)"
hoelzl@38656
   506
    "finite (g ` space M)"
hoelzl@38656
   507
    using assms unfolding simple_function_def by simp_all
hoelzl@38656
   508
hoelzl@38656
   509
  { fix A
hoelzl@38656
   510
    have "?p ` (A \<inter> space M) \<subseteq>
hoelzl@38656
   511
      (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
hoelzl@38656
   512
      by auto
hoelzl@38656
   513
    hence "finite (?p ` (A \<inter> space M))"
hoelzl@38656
   514
      by (rule finite_subset) (auto intro: finite_SigmaI finite_imageI) }
hoelzl@38656
   515
  note this[intro, simp]
hoelzl@35582
   516
hoelzl@38656
   517
  { fix x assume "x \<in> space M"
hoelzl@38656
   518
    have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
hoelzl@38656
   519
    moreover {
hoelzl@38656
   520
      fix x y
hoelzl@38656
   521
      have *: "f -` {f x} \<inter> g -` {g x} \<inter> space M
hoelzl@38656
   522
          = (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)" by auto
hoelzl@38656
   523
      assume "x \<in> space M" "y \<in> space M"
hoelzl@38656
   524
      hence "f -` {f x} \<inter> g -` {g x} \<inter> space M \<in> sets M"
hoelzl@38656
   525
        using assms unfolding simple_function_def * by auto }
hoelzl@38656
   526
    ultimately
hoelzl@38656
   527
    have "\<mu> (f -` {f x} \<inter> space M) = setsum (\<mu>) (?sub (f x))"
hoelzl@38656
   528
      by (subst measure_finitely_additive) auto }
hoelzl@38656
   529
  hence "simple_integral f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
hoelzl@38656
   530
    unfolding simple_integral_def
hoelzl@38656
   531
    by (subst setsum_Sigma[symmetric],
hoelzl@38656
   532
       auto intro!: setsum_cong simp: setsum_right_distrib[symmetric])
hoelzl@38656
   533
  also have "\<dots> = (\<Sum>A\<in>?p ` space M. contents (f`A) * \<mu> A)"
hoelzl@38656
   534
  proof -
hoelzl@38656
   535
    have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
hoelzl@38656
   536
    have "(\<lambda>A. (contents (f ` A), A)) ` ?p ` space M
hoelzl@38656
   537
      = (\<lambda>x. (f x, ?p x)) ` space M"
hoelzl@38656
   538
    proof safe
hoelzl@38656
   539
      fix x assume "x \<in> space M"
hoelzl@38656
   540
      thus "(f x, ?p x) \<in> (\<lambda>A. (contents (f`A), A)) ` ?p ` space M"
hoelzl@38656
   541
        by (auto intro!: image_eqI[of _ _ "?p x"])
hoelzl@38656
   542
    qed auto
hoelzl@38656
   543
    thus ?thesis
hoelzl@38656
   544
      apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (contents (f`A), A)"] inj_onI)
hoelzl@38656
   545
      apply (rule_tac x="xa" in image_eqI)
hoelzl@38656
   546
      by simp_all
hoelzl@38656
   547
  qed
hoelzl@38656
   548
  finally show ?thesis .
hoelzl@35582
   549
qed
hoelzl@35582
   550
hoelzl@38656
   551
lemma (in measure_space) simple_integral_add[simp]:
hoelzl@38656
   552
  assumes "simple_function f" and "simple_function g"
hoelzl@38656
   553
  shows "simple_integral (\<lambda>x. f x + g x) = simple_integral f + simple_integral g"
hoelzl@35582
   554
proof -
hoelzl@38656
   555
  { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
hoelzl@38656
   556
    assume "x \<in> space M"
hoelzl@38656
   557
    hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
hoelzl@38656
   558
        "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
hoelzl@38656
   559
      by auto }
hoelzl@38656
   560
  thus ?thesis
hoelzl@38656
   561
    unfolding
hoelzl@38656
   562
      simple_function_partition[OF simple_function_add[OF assms] simple_function_Pair[OF assms]]
hoelzl@38656
   563
      simple_function_partition[OF `simple_function f` `simple_function g`]
hoelzl@38656
   564
      simple_function_partition[OF `simple_function g` `simple_function f`]
hoelzl@38656
   565
    apply (subst (3) Int_commute)
hoelzl@38656
   566
    by (auto simp add: field_simps setsum_addf[symmetric] intro!: setsum_cong)
hoelzl@35582
   567
qed
hoelzl@35582
   568
hoelzl@38656
   569
lemma (in measure_space) simple_integral_setsum[simp]:
hoelzl@38656
   570
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
hoelzl@38656
   571
  shows "simple_integral (\<lambda>x. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. simple_integral (f i))"
hoelzl@38656
   572
proof cases
hoelzl@38656
   573
  assume "finite P"
hoelzl@38656
   574
  from this assms show ?thesis
hoelzl@38656
   575
    by induct (auto simp: simple_function_setsum simple_integral_add)
hoelzl@38656
   576
qed auto
hoelzl@38656
   577
hoelzl@38656
   578
lemma (in measure_space) simple_integral_mult[simp]:
hoelzl@38656
   579
  assumes "simple_function f"
hoelzl@38656
   580
  shows "simple_integral (\<lambda>x. c * f x) = c * simple_integral f"
hoelzl@38656
   581
proof -
hoelzl@38656
   582
  note mult = simple_function_mult[OF simple_function_const[of c] assms]
hoelzl@38656
   583
  { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
hoelzl@38656
   584
    assume "x \<in> space M"
hoelzl@38656
   585
    hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
hoelzl@38656
   586
      by auto }
hoelzl@38656
   587
  thus ?thesis
hoelzl@38656
   588
    unfolding simple_function_partition[OF mult assms]
hoelzl@38656
   589
      simple_function_partition[OF assms mult]
hoelzl@38656
   590
    by (auto simp: setsum_right_distrib field_simps intro!: setsum_cong)
hoelzl@35582
   591
qed
hoelzl@35582
   592
hoelzl@38656
   593
lemma (in measure_space) simple_integral_mono:
hoelzl@38656
   594
  assumes "simple_function f" and "simple_function g"
hoelzl@38656
   595
  and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
hoelzl@38656
   596
  shows "simple_integral f \<le> simple_integral g"
hoelzl@38656
   597
  unfolding
hoelzl@38656
   598
    simple_function_partition[OF `simple_function f` `simple_function g`]
hoelzl@38656
   599
    simple_function_partition[OF `simple_function g` `simple_function f`]
hoelzl@38656
   600
  apply (subst Int_commute)
hoelzl@38656
   601
proof (safe intro!: setsum_mono)
hoelzl@38656
   602
  fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
hoelzl@38656
   603
  assume "x \<in> space M"
hoelzl@38656
   604
  hence "f ` ?S = {f x}" "g ` ?S = {g x}" by auto
hoelzl@38656
   605
  thus "contents (f`?S) * \<mu> ?S \<le> contents (g`?S) * \<mu> ?S"
hoelzl@38656
   606
    using mono[OF `x \<in> space M`] by (auto intro!: mult_right_mono)
hoelzl@35582
   607
qed
hoelzl@35582
   608
hoelzl@38656
   609
lemma (in measure_space) simple_integral_indicator:
hoelzl@38656
   610
  assumes "A \<in> sets M"
hoelzl@38656
   611
  assumes "simple_function f"
hoelzl@38656
   612
  shows "simple_integral (\<lambda>x. f x * indicator A x) =
hoelzl@38656
   613
    (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
hoelzl@38656
   614
proof cases
hoelzl@38656
   615
  assume "A = space M"
hoelzl@38656
   616
  moreover hence "simple_integral (\<lambda>x. f x * indicator A x) = simple_integral f"
hoelzl@38656
   617
    by (auto intro!: simple_integral_cong)
hoelzl@38656
   618
  moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto
hoelzl@38656
   619
  ultimately show ?thesis by (simp add: simple_integral_def)
hoelzl@38656
   620
next
hoelzl@38656
   621
  assume "A \<noteq> space M"
hoelzl@38656
   622
  then obtain x where x: "x \<in> space M" "x \<notin> A" using sets_into_space[OF assms(1)] by auto
hoelzl@38656
   623
  have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
hoelzl@35582
   624
  proof safe
hoelzl@38656
   625
    fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
hoelzl@38656
   626
  next
hoelzl@38656
   627
    fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
hoelzl@38656
   628
      using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
hoelzl@38656
   629
  next
hoelzl@38656
   630
    show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
hoelzl@35582
   631
  qed
hoelzl@38656
   632
  have *: "simple_integral (\<lambda>x. f x * indicator A x) =
hoelzl@38656
   633
    (\<Sum>x \<in> f ` space M \<union> {0}. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
hoelzl@38656
   634
    unfolding simple_integral_def I
hoelzl@38656
   635
  proof (rule setsum_mono_zero_cong_left)
hoelzl@38656
   636
    show "finite (f ` space M \<union> {0})"
hoelzl@38656
   637
      using assms(2) unfolding simple_function_def by auto
hoelzl@38656
   638
    show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
hoelzl@38656
   639
      using sets_into_space[OF assms(1)] by auto
hoelzl@38656
   640
    have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}" by (auto simp: image_iff)
hoelzl@38656
   641
    thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
hoelzl@38656
   642
      i * \<mu> (f -` {i} \<inter> space M \<inter> A) = 0" by auto
hoelzl@38656
   643
  next
hoelzl@38656
   644
    fix x assume "x \<in> f`A \<union> {0}"
hoelzl@38656
   645
    hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
hoelzl@38656
   646
      by (auto simp: indicator_def split: split_if_asm)
hoelzl@38656
   647
    thus "x * \<mu> (?I -` {x} \<inter> space M) =
hoelzl@38656
   648
      x * \<mu> (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
hoelzl@38656
   649
  qed
hoelzl@38656
   650
  show ?thesis unfolding *
hoelzl@38656
   651
    using assms(2) unfolding simple_function_def
hoelzl@38656
   652
    by (auto intro!: setsum_mono_zero_cong_right)
hoelzl@38656
   653
qed
hoelzl@35582
   654
hoelzl@38656
   655
lemma (in measure_space) simple_integral_indicator_only[simp]:
hoelzl@38656
   656
  assumes "A \<in> sets M"
hoelzl@38656
   657
  shows "simple_integral (indicator A) = \<mu> A"
hoelzl@38656
   658
proof cases
hoelzl@38656
   659
  assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
hoelzl@38656
   660
  thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
hoelzl@38656
   661
next
hoelzl@38656
   662
  assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::pinfreal}" by auto
hoelzl@38656
   663
  thus ?thesis
hoelzl@38656
   664
    using simple_integral_indicator[OF assms simple_function_const[of 1]]
hoelzl@38656
   665
    using sets_into_space[OF assms]
hoelzl@38656
   666
    by (auto intro!: arg_cong[where f="\<mu>"])
hoelzl@38656
   667
qed
hoelzl@35582
   668
hoelzl@38656
   669
lemma (in measure_space) simple_integral_null_set:
hoelzl@38656
   670
  assumes "simple_function u" "N \<in> null_sets"
hoelzl@38656
   671
  shows "simple_integral (\<lambda>x. u x * indicator N x) = 0"
hoelzl@38656
   672
proof -
hoelzl@38656
   673
  have "simple_integral (\<lambda>x. u x * indicator N x) \<le>
hoelzl@38656
   674
    simple_integral (\<lambda>x. \<omega> * indicator N x)"
hoelzl@38656
   675
    using assms
hoelzl@38656
   676
    by (safe intro!: simple_integral_mono simple_function_mult simple_function_indicator simple_function_const) simp
hoelzl@38656
   677
  also have "... = 0" apply(subst simple_integral_mult)
hoelzl@38656
   678
    using assms(2) by auto
hoelzl@38656
   679
  finally show ?thesis by auto
hoelzl@38656
   680
qed
hoelzl@35582
   681
hoelzl@38656
   682
lemma (in measure_space) simple_integral_cong':
hoelzl@38656
   683
  assumes f: "simple_function f" and g: "simple_function g"
hoelzl@38656
   684
  and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
hoelzl@38656
   685
  shows "simple_integral f = simple_integral g"
hoelzl@38656
   686
proof -
hoelzl@38656
   687
  let ?h = "\<lambda>h. \<lambda>x. (h x * indicator {x\<in>space M. f x = g x} x
hoelzl@38656
   688
    + h x * indicator {x\<in>space M. f x \<noteq> g x} x
hoelzl@38656
   689
    + h x * indicator (-space M) x::pinfreal)"
hoelzl@38656
   690
  have *:"\<And>h. h = ?h h" unfolding indicator_def apply rule by auto
hoelzl@38656
   691
  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
   692
  then have mea_nullset: "{x \<in> space M. f x \<noteq> g x} \<in> null_sets" using mea by auto
hoelzl@38656
   693
  have h1:"\<And>h::_=>pinfreal. simple_function h \<Longrightarrow>
hoelzl@38656
   694
    simple_function (\<lambda>x. h x * indicator {x\<in>space M. f x = g x} x)"
hoelzl@38656
   695
    apply(safe intro!: simple_function_add simple_function_mult simple_function_indicator)
hoelzl@38656
   696
    using f g by (auto simp: borel_measurable_simple_function)
hoelzl@38656
   697
  have h2:"\<And>h::_\<Rightarrow>pinfreal. simple_function h \<Longrightarrow>
hoelzl@38656
   698
    simple_function (\<lambda>x. h x * indicator {x\<in>space M. f x \<noteq> g x} x)"
hoelzl@38656
   699
    apply(safe intro!: simple_function_add simple_function_mult simple_function_indicator)
hoelzl@38656
   700
    by(rule mea_neq)
hoelzl@38656
   701
  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
   702
  note *** = simple_integral_add[OF simple_function_add[OF h1 h2] simple_function_notspace]
hoelzl@38656
   703
    simple_integral_add[OF h1 h2]
hoelzl@38656
   704
  show ?thesis apply(subst *[of g]) apply(subst *[of f])
hoelzl@38656
   705
    unfolding ***[OF f f] ***[OF g g]
hoelzl@38656
   706
  proof(rule **) case goal1 show ?case apply(rule arg_cong[where f=simple_integral]) apply rule 
hoelzl@38656
   707
      unfolding indicator_def by auto
hoelzl@38656
   708
  next note * = simple_integral_null_set[OF _ mea_nullset]
hoelzl@38656
   709
    case goal2 show ?case unfolding *[OF f] *[OF g] ..
hoelzl@38656
   710
  next case goal3 show ?case apply(rule simple_integral_cong) by auto
hoelzl@35582
   711
  qed
hoelzl@35582
   712
qed
hoelzl@35582
   713
hoelzl@39092
   714
lemma (in measure_space) simple_integral_restricted:
hoelzl@39092
   715
  assumes "A \<in> sets M"
hoelzl@39092
   716
  assumes sf: "simple_function (\<lambda>x. f x * indicator A x)"
hoelzl@39092
   717
  shows "measure_space.simple_integral (restricted_space A) \<mu> f = simple_integral (\<lambda>x. f x * indicator A x)"
hoelzl@39092
   718
    (is "_ = simple_integral ?f")
hoelzl@39092
   719
  unfolding measure_space.simple_integral_def[OF restricted_measure_space[OF `A \<in> sets M`]]
hoelzl@39092
   720
  unfolding simple_integral_def
hoelzl@39092
   721
proof (simp, safe intro!: setsum_mono_zero_cong_left)
hoelzl@39092
   722
  from sf show "finite (?f ` space M)"
hoelzl@39092
   723
    unfolding simple_function_def by auto
hoelzl@39092
   724
next
hoelzl@39092
   725
  fix x assume "x \<in> A"
hoelzl@39092
   726
  then show "f x \<in> ?f ` space M"
hoelzl@39092
   727
    using sets_into_space `A \<in> sets M` by (auto intro!: image_eqI[of _ _ x])
hoelzl@39092
   728
next
hoelzl@39092
   729
  fix x assume "x \<in> space M" "?f x \<notin> f`A"
hoelzl@39092
   730
  then have "x \<notin> A" by (auto simp: image_iff)
hoelzl@39092
   731
  then show "?f x * \<mu> (?f -` {?f x} \<inter> space M) = 0" by simp
hoelzl@39092
   732
next
hoelzl@39092
   733
  fix x assume "x \<in> A"
hoelzl@39092
   734
  then have "f x \<noteq> 0 \<Longrightarrow>
hoelzl@39092
   735
    f -` {f x} \<inter> A = ?f -` {f x} \<inter> space M"
hoelzl@39092
   736
    using `A \<in> sets M` sets_into_space
hoelzl@39092
   737
    by (auto simp: indicator_def split: split_if_asm)
hoelzl@39092
   738
  then show "f x * \<mu> (f -` {f x} \<inter> A) =
hoelzl@39092
   739
    f x * \<mu> (?f -` {f x} \<inter> space M)"
hoelzl@39092
   740
    unfolding pinfreal_mult_cancel_left by auto
hoelzl@39092
   741
qed
hoelzl@39092
   742
hoelzl@39092
   743
lemma (in measure_space) simple_integral_subalgebra[simp]:
hoelzl@39092
   744
  assumes "measure_space (M\<lparr>sets := N\<rparr>) \<mu>"
hoelzl@39092
   745
  shows "measure_space.simple_integral (M\<lparr>sets := N\<rparr>) \<mu> = simple_integral"
hoelzl@39092
   746
  unfolding simple_integral_def_raw
hoelzl@39092
   747
  unfolding measure_space.simple_integral_def_raw[OF assms] by simp
hoelzl@39092
   748
hoelzl@35692
   749
section "Continuous posititve integration"
hoelzl@35692
   750
hoelzl@38656
   751
definition (in measure_space)
hoelzl@38656
   752
  "positive_integral f =
hoelzl@38656
   753
    (SUP g : {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}. simple_integral g)"
hoelzl@35582
   754
hoelzl@38656
   755
lemma (in measure_space) positive_integral_alt1:
hoelzl@38656
   756
  "positive_integral f =
hoelzl@38656
   757
    (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
   758
  unfolding positive_integral_def SUPR_def
hoelzl@38656
   759
proof (safe intro!: arg_cong[where f=Sup])
hoelzl@38656
   760
  fix g let ?g = "\<lambda>x. if x \<in> space M then g x else f x"
hoelzl@38656
   761
  assume "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
hoelzl@38656
   762
  hence "?g \<le> f" "simple_function ?g" "simple_integral g = simple_integral ?g"
hoelzl@38656
   763
    "\<omega> \<notin> g`space M"
hoelzl@38656
   764
    unfolding le_fun_def by (auto cong: simple_function_cong simple_integral_cong)
hoelzl@38656
   765
  thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}"
hoelzl@38656
   766
    by auto
hoelzl@38656
   767
next
hoelzl@38656
   768
  fix g assume "simple_function g" "g \<le> f" "\<omega> \<notin> g`space M"
hoelzl@38656
   769
  hence "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
hoelzl@38656
   770
    by (auto simp add: le_fun_def image_iff)
hoelzl@38656
   771
  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
   772
    by auto
hoelzl@35582
   773
qed
hoelzl@35582
   774
hoelzl@38656
   775
lemma (in measure_space) positive_integral_alt:
hoelzl@38656
   776
  "positive_integral f =
hoelzl@38656
   777
    (SUP g : {g. simple_function g \<and> g \<le> f}. simple_integral g)"
hoelzl@38656
   778
  apply(rule order_class.antisym) unfolding positive_integral_def 
hoelzl@38656
   779
  apply(rule SUPR_subset) apply blast apply(rule SUPR_mono_lim)
hoelzl@38656
   780
proof safe fix u assume u:"simple_function u" and uf:"u \<le> f"
hoelzl@38656
   781
  let ?u = "\<lambda>n x. if u x = \<omega> then Real (real (n::nat)) else u x"
hoelzl@38656
   782
  have su:"\<And>n. simple_function (?u n)" using simple_function_compose1[OF u] .
hoelzl@38656
   783
  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
   784
    (\<lambda>n. simple_integral (b n)) ----> simple_integral u"
hoelzl@38656
   785
    apply(rule_tac x="?u" in exI, safe) apply(rule su)
hoelzl@38656
   786
  proof- fix n::nat have "?u n \<le> u" unfolding le_fun_def by auto
hoelzl@38656
   787
    also note uf finally show "?u n \<le> f" .
hoelzl@38656
   788
    let ?s = "{x \<in> space M. u x = \<omega>}"
hoelzl@38656
   789
    show "(\<lambda>n. simple_integral (?u n)) ----> simple_integral u"
hoelzl@38656
   790
    proof(cases "\<mu> ?s = 0")
hoelzl@38656
   791
      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
   792
      have *:"\<And>n. simple_integral (?u n) = simple_integral u"
hoelzl@38656
   793
        apply(rule simple_integral_cong'[OF su u]) unfolding * True ..
hoelzl@38656
   794
      show ?thesis unfolding * by auto 
hoelzl@38656
   795
    next case False note m0=this
hoelzl@38656
   796
      have s:"{x \<in> space M. u x = \<omega>} \<in> sets M" using u  by (auto simp: borel_measurable_simple_function)
hoelzl@38656
   797
      have "\<omega> = simple_integral (\<lambda>x. \<omega> * indicator {x\<in>space M. u x = \<omega>} x)"
hoelzl@38656
   798
        apply(subst simple_integral_mult) using s
hoelzl@38656
   799
        unfolding simple_integral_indicator_only[OF s] using False by auto
hoelzl@38656
   800
      also have "... \<le> simple_integral u"
hoelzl@38656
   801
        apply (rule simple_integral_mono)
hoelzl@38656
   802
        apply (rule simple_function_mult)
hoelzl@38656
   803
        apply (rule simple_function_const)
hoelzl@38656
   804
        apply(rule ) prefer 3 apply(subst indicator_def)
hoelzl@38656
   805
        using s u by auto
hoelzl@38656
   806
      finally have *:"simple_integral u = \<omega>" by auto
hoelzl@38656
   807
      show ?thesis unfolding * Lim_omega_pos
hoelzl@38656
   808
      proof safe case goal1
hoelzl@38656
   809
        from real_arch_simple[of "B / real (\<mu> ?s)"] guess N0 .. note N=this
hoelzl@38656
   810
        def N \<equiv> "Suc N0" have N:"real N \<ge> B / real (\<mu> ?s)" "N > 0"
hoelzl@38656
   811
          unfolding N_def using N by auto
hoelzl@38656
   812
        show ?case apply-apply(rule_tac x=N in exI,safe) 
hoelzl@38656
   813
        proof- case goal1
hoelzl@38656
   814
          have "Real B \<le> Real (real N) * \<mu> ?s"
hoelzl@38656
   815
          proof(cases "\<mu> ?s = \<omega>")
hoelzl@38656
   816
            case True thus ?thesis using `B>0` N by auto
hoelzl@38656
   817
          next case False
hoelzl@38656
   818
            have *:"B \<le> real N * real (\<mu> ?s)" 
hoelzl@38656
   819
              using N(1) apply-apply(subst (asm) pos_divide_le_eq)
hoelzl@38656
   820
              apply rule using m0 False by auto
hoelzl@38656
   821
            show ?thesis apply(subst Real_real'[THEN sym,OF False])
hoelzl@38656
   822
              apply(subst pinfreal_times,subst if_P) defer
hoelzl@38656
   823
              apply(subst pinfreal_less_eq,subst if_P) defer
hoelzl@38656
   824
              using * N `B>0` by(auto intro: mult_nonneg_nonneg)
hoelzl@38656
   825
          qed
hoelzl@38656
   826
          also have "... \<le> Real (real n) * \<mu> ?s"
hoelzl@38656
   827
            apply(rule mult_right_mono) using goal1 by auto
hoelzl@38656
   828
          also have "... = simple_integral (\<lambda>x. Real (real n) * indicator ?s x)" 
hoelzl@38656
   829
            apply(subst simple_integral_mult) apply(rule simple_function_indicator[OF s])
hoelzl@38656
   830
            unfolding simple_integral_indicator_only[OF s] ..
hoelzl@38656
   831
          also have "... \<le> simple_integral (\<lambda>x. if u x = \<omega> then Real (real n) else u x)"
hoelzl@38656
   832
            apply(rule simple_integral_mono) apply(rule simple_function_mult)
hoelzl@38656
   833
            apply(rule simple_function_const)
hoelzl@38656
   834
            apply(rule simple_function_indicator) apply(rule s su)+ by auto
hoelzl@38656
   835
          finally show ?case .
hoelzl@38656
   836
        qed qed qed
hoelzl@38656
   837
    fix x assume x:"\<omega> = (if u x = \<omega> then Real (real n) else u x)" "x \<in> space M"
hoelzl@38656
   838
    hence "u x = \<omega>" apply-apply(rule ccontr) by auto
hoelzl@38656
   839
    hence "\<omega> = Real (real n)" using x by auto
hoelzl@38656
   840
    thus False by auto
hoelzl@35582
   841
  qed
hoelzl@35582
   842
qed
hoelzl@35582
   843
hoelzl@38656
   844
lemma (in measure_space) positive_integral_cong:
hoelzl@38656
   845
  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
hoelzl@38656
   846
  shows "positive_integral f = positive_integral g"
hoelzl@38656
   847
proof -
hoelzl@38656
   848
  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
   849
    using assms by auto
hoelzl@38656
   850
  thus ?thesis unfolding positive_integral_alt1 by auto
hoelzl@38656
   851
qed
hoelzl@38656
   852
hoelzl@38656
   853
lemma (in measure_space) positive_integral_eq_simple_integral:
hoelzl@38656
   854
  assumes "simple_function f"
hoelzl@38656
   855
  shows "positive_integral f = simple_integral f"
hoelzl@38656
   856
  unfolding positive_integral_alt
hoelzl@38656
   857
proof (safe intro!: pinfreal_SUPI)
hoelzl@38656
   858
  fix g assume "simple_function g" "g \<le> f"
hoelzl@38656
   859
  with assms show "simple_integral g \<le> simple_integral f"
hoelzl@38656
   860
    by (auto intro!: simple_integral_mono simp: le_fun_def)
hoelzl@38656
   861
next
hoelzl@38656
   862
  fix y assume "\<forall>x. x\<in>{g. simple_function g \<and> g \<le> f} \<longrightarrow> simple_integral x \<le> y"
hoelzl@38656
   863
  with assms show "simple_integral f \<le> y" by auto
hoelzl@38656
   864
qed
hoelzl@35582
   865
hoelzl@38656
   866
lemma (in measure_space) positive_integral_mono:
hoelzl@38656
   867
  assumes mono: "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x"
hoelzl@38656
   868
  shows "positive_integral u \<le> positive_integral v"
hoelzl@38656
   869
  unfolding positive_integral_alt1
hoelzl@38656
   870
proof (safe intro!: SUPR_mono)
hoelzl@38656
   871
  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
   872
  with mono have "\<forall>x\<in>space M. a x \<le> v x \<and> a x \<noteq> \<omega>" by fastsimp
hoelzl@38656
   873
  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
   874
    by (auto intro!: bexI[of _ a])
hoelzl@38656
   875
qed
hoelzl@38656
   876
hoelzl@38656
   877
lemma (in measure_space) positive_integral_SUP_approx:
hoelzl@38656
   878
  assumes "f \<up> s"
hoelzl@38656
   879
  and f: "\<And>i. f i \<in> borel_measurable M"
hoelzl@38656
   880
  and "simple_function u"
hoelzl@38656
   881
  and le: "u \<le> s" and real: "\<omega> \<notin> u`space M"
hoelzl@38656
   882
  shows "simple_integral u \<le> (SUP i. positive_integral (f i))" (is "_ \<le> ?S")
hoelzl@38656
   883
proof (rule pinfreal_le_mult_one_interval)
hoelzl@38656
   884
  fix a :: pinfreal assume "0 < a" "a < 1"
hoelzl@38656
   885
  hence "a \<noteq> 0" by auto
hoelzl@38656
   886
  let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
hoelzl@38656
   887
  have B: "\<And>i. ?B i \<in> sets M"
hoelzl@38656
   888
    using f `simple_function u` by (auto simp: borel_measurable_simple_function)
hoelzl@38656
   889
hoelzl@38656
   890
  let "?uB i x" = "u x * indicator (?B i) x"
hoelzl@38656
   891
hoelzl@38656
   892
  { fix i have "?B i \<subseteq> ?B (Suc i)"
hoelzl@38656
   893
    proof safe
hoelzl@38656
   894
      fix i x assume "a * u x \<le> f i x"
hoelzl@38656
   895
      also have "\<dots> \<le> f (Suc i) x"
hoelzl@38656
   896
        using `f \<up> s` unfolding isoton_def le_fun_def by auto
hoelzl@38656
   897
      finally show "a * u x \<le> f (Suc i) x" .
hoelzl@38656
   898
    qed }
hoelzl@38656
   899
  note B_mono = this
hoelzl@35582
   900
hoelzl@38656
   901
  have u: "\<And>x. x \<in> space M \<Longrightarrow> u -` {u x} \<inter> space M \<in> sets M"
hoelzl@38656
   902
    using `simple_function u` by (auto simp add: simple_function_def)
hoelzl@38656
   903
hoelzl@38656
   904
  { fix i
hoelzl@38656
   905
    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
   906
    proof safe
hoelzl@38656
   907
      fix x assume "x \<in> space M"
hoelzl@38656
   908
      show "x \<in> (\<Union>i. (u -` {u x} \<inter> space M) \<inter> ?B i)"
hoelzl@38656
   909
      proof cases
hoelzl@38656
   910
        assume "u x = 0" thus ?thesis using `x \<in> space M` by simp
hoelzl@38656
   911
      next
hoelzl@38656
   912
        assume "u x \<noteq> 0"
hoelzl@38656
   913
        with `a < 1` real `x \<in> space M`
hoelzl@38656
   914
        have "a * u x < 1 * u x" by (rule_tac pinfreal_mult_strict_right_mono) (auto simp: image_iff)
hoelzl@38656
   915
        also have "\<dots> \<le> (SUP i. f i x)" using le `f \<up> s`
hoelzl@38656
   916
          unfolding isoton_fun_expand by (auto simp: isoton_def le_fun_def)
hoelzl@38656
   917
        finally obtain i where "a * u x < f i x" unfolding SUPR_def
hoelzl@38656
   918
          by (auto simp add: less_Sup_iff)
hoelzl@38656
   919
        hence "a * u x \<le> f i x" by auto
hoelzl@38656
   920
        thus ?thesis using `x \<in> space M` by auto
hoelzl@38656
   921
      qed
hoelzl@38656
   922
    qed auto }
hoelzl@38656
   923
  note measure_conv = measure_up[OF u Int[OF u B] this]
hoelzl@38656
   924
hoelzl@38656
   925
  have "simple_integral u = (SUP i. simple_integral (?uB i))"
hoelzl@38656
   926
    unfolding simple_integral_indicator[OF B `simple_function u`]
hoelzl@38656
   927
  proof (subst SUPR_pinfreal_setsum, safe)
hoelzl@38656
   928
    fix x n assume "x \<in> space M"
hoelzl@38656
   929
    have "\<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f n x})
hoelzl@38656
   930
      \<le> \<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f (Suc n) x})"
hoelzl@38656
   931
      using B_mono Int[OF u[OF `x \<in> space M`] B] by (auto intro!: measure_mono)
hoelzl@38656
   932
    thus "u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B n)
hoelzl@38656
   933
            \<le> u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B (Suc n))"
hoelzl@38656
   934
      by (auto intro: mult_left_mono)
hoelzl@38656
   935
  next
hoelzl@38656
   936
    show "simple_integral u =
hoelzl@38656
   937
      (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (u -` {i} \<inter> space M \<inter> ?B n))"
hoelzl@38656
   938
      using measure_conv unfolding simple_integral_def isoton_def
hoelzl@38656
   939
      by (auto intro!: setsum_cong simp: pinfreal_SUP_cmult)
hoelzl@38656
   940
  qed
hoelzl@38656
   941
  moreover
hoelzl@38656
   942
  have "a * (SUP i. simple_integral (?uB i)) \<le> ?S"
hoelzl@38656
   943
    unfolding pinfreal_SUP_cmult[symmetric]
hoelzl@38705
   944
  proof (safe intro!: SUP_mono bexI)
hoelzl@38656
   945
    fix i
hoelzl@38656
   946
    have "a * simple_integral (?uB i) = simple_integral (\<lambda>x. a * ?uB i x)"
hoelzl@38656
   947
      using B `simple_function u`
hoelzl@38656
   948
      by (subst simple_integral_mult[symmetric]) (auto simp: field_simps)
hoelzl@38656
   949
    also have "\<dots> \<le> positive_integral (f i)"
hoelzl@38656
   950
    proof -
hoelzl@38656
   951
      have "\<And>x. a * ?uB i x \<le> f i x" unfolding indicator_def by auto
hoelzl@38656
   952
      hence *: "simple_function (\<lambda>x. a * ?uB i x)" using B assms(3)
hoelzl@38656
   953
        by (auto intro!: simple_integral_mono)
hoelzl@38656
   954
      show ?thesis unfolding positive_integral_eq_simple_integral[OF *, symmetric]
hoelzl@38656
   955
        by (auto intro!: positive_integral_mono simp: indicator_def)
hoelzl@38656
   956
    qed
hoelzl@38656
   957
    finally show "a * simple_integral (?uB i) \<le> positive_integral (f i)"
hoelzl@38656
   958
      by auto
hoelzl@38705
   959
  qed simp
hoelzl@38656
   960
  ultimately show "a * simple_integral u \<le> ?S" by simp
hoelzl@35582
   961
qed
hoelzl@35582
   962
hoelzl@35582
   963
text {* Beppo-Levi monotone convergence theorem *}
hoelzl@38656
   964
lemma (in measure_space) positive_integral_isoton:
hoelzl@38656
   965
  assumes "f \<up> u" "\<And>i. f i \<in> borel_measurable M"
hoelzl@38656
   966
  shows "(\<lambda>i. positive_integral (f i)) \<up> positive_integral u"
hoelzl@38656
   967
  unfolding isoton_def
hoelzl@38656
   968
proof safe
hoelzl@38656
   969
  fix i show "positive_integral (f i) \<le> positive_integral (f (Suc i))"
hoelzl@38656
   970
    apply (rule positive_integral_mono)
hoelzl@38656
   971
    using `f \<up> u` unfolding isoton_def le_fun_def by auto
hoelzl@38656
   972
next
hoelzl@38656
   973
  have "u \<in> borel_measurable M"
hoelzl@38656
   974
    using borel_measurable_SUP[of UNIV f] assms by (auto simp: isoton_def)
hoelzl@38656
   975
  have u: "u = (SUP i. f i)" using `f \<up> u` unfolding isoton_def by auto
hoelzl@35582
   976
hoelzl@38656
   977
  show "(SUP i. positive_integral (f i)) = positive_integral u"
hoelzl@38656
   978
  proof (rule antisym)
hoelzl@38656
   979
    from `f \<up> u`[THEN isoton_Sup, unfolded le_fun_def]
hoelzl@38656
   980
    show "(SUP j. positive_integral (f j)) \<le> positive_integral u"
hoelzl@38656
   981
      by (auto intro!: SUP_leI positive_integral_mono)
hoelzl@38656
   982
  next
hoelzl@38656
   983
    show "positive_integral u \<le> (SUP i. positive_integral (f i))"
hoelzl@38656
   984
      unfolding positive_integral_def[of u]
hoelzl@38656
   985
      by (auto intro!: SUP_leI positive_integral_SUP_approx[OF assms])
hoelzl@35582
   986
  qed
hoelzl@35582
   987
qed
hoelzl@35582
   988
hoelzl@38656
   989
lemma (in measure_space) SUP_simple_integral_sequences:
hoelzl@38656
   990
  assumes f: "f \<up> u" "\<And>i. simple_function (f i)"
hoelzl@38656
   991
  and g: "g \<up> u" "\<And>i. simple_function (g i)"
hoelzl@38656
   992
  shows "(SUP i. simple_integral (f i)) = (SUP i. simple_integral (g i))"
hoelzl@38656
   993
    (is "SUPR _ ?F = SUPR _ ?G")
hoelzl@38656
   994
proof -
hoelzl@38656
   995
  have "(SUP i. ?F i) = (SUP i. positive_integral (f i))"
hoelzl@38656
   996
    using assms by (simp add: positive_integral_eq_simple_integral)
hoelzl@38656
   997
  also have "\<dots> = positive_integral u"
hoelzl@38656
   998
    using positive_integral_isoton[OF f(1) borel_measurable_simple_function[OF f(2)]]
hoelzl@38656
   999
    unfolding isoton_def by simp
hoelzl@38656
  1000
  also have "\<dots> = (SUP i. positive_integral (g i))"
hoelzl@38656
  1001
    using positive_integral_isoton[OF g(1) borel_measurable_simple_function[OF g(2)]]
hoelzl@38656
  1002
    unfolding isoton_def by simp
hoelzl@38656
  1003
  also have "\<dots> = (SUP i. ?G i)"
hoelzl@38656
  1004
    using assms by (simp add: positive_integral_eq_simple_integral)
hoelzl@38656
  1005
  finally show ?thesis .
hoelzl@38656
  1006
qed
hoelzl@38656
  1007
hoelzl@38656
  1008
lemma (in measure_space) positive_integral_const[simp]:
hoelzl@38656
  1009
  "positive_integral (\<lambda>x. c) = c * \<mu> (space M)"
hoelzl@38656
  1010
  by (subst positive_integral_eq_simple_integral) auto
hoelzl@38656
  1011
hoelzl@38656
  1012
lemma (in measure_space) positive_integral_isoton_simple:
hoelzl@38656
  1013
  assumes "f \<up> u" and e: "\<And>i. simple_function (f i)"
hoelzl@38656
  1014
  shows "(\<lambda>i. simple_integral (f i)) \<up> positive_integral u"
hoelzl@38656
  1015
  using positive_integral_isoton[OF `f \<up> u` e(1)[THEN borel_measurable_simple_function]]
hoelzl@38656
  1016
  unfolding positive_integral_eq_simple_integral[OF e] .
hoelzl@38656
  1017
hoelzl@38656
  1018
lemma (in measure_space) positive_integral_linear:
hoelzl@38656
  1019
  assumes f: "f \<in> borel_measurable M"
hoelzl@38656
  1020
  and g: "g \<in> borel_measurable M"
hoelzl@38656
  1021
  shows "positive_integral (\<lambda>x. a * f x + g x) =
hoelzl@38656
  1022
      a * positive_integral f + positive_integral g"
hoelzl@38656
  1023
    (is "positive_integral ?L = _")
hoelzl@35582
  1024
proof -
hoelzl@38656
  1025
  from borel_measurable_implies_simple_function_sequence'[OF f] guess u .
hoelzl@38656
  1026
  note u = this positive_integral_isoton_simple[OF this(1-2)]
hoelzl@38656
  1027
  from borel_measurable_implies_simple_function_sequence'[OF g] guess v .
hoelzl@38656
  1028
  note v = this positive_integral_isoton_simple[OF this(1-2)]
hoelzl@38656
  1029
  let "?L' i x" = "a * u i x + v i x"
hoelzl@38656
  1030
hoelzl@38656
  1031
  have "?L \<in> borel_measurable M"
hoelzl@38656
  1032
    using assms by simp
hoelzl@38656
  1033
  from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
hoelzl@38656
  1034
  note positive_integral_isoton_simple[OF this(1-2)] and l = this
hoelzl@38656
  1035
  moreover have
hoelzl@38656
  1036
      "(SUP i. simple_integral (l i)) = (SUP i. simple_integral (?L' i))"
hoelzl@38656
  1037
  proof (rule SUP_simple_integral_sequences[OF l(1-2)])
hoelzl@38656
  1038
    show "?L' \<up> ?L" "\<And>i. simple_function (?L' i)"
hoelzl@38656
  1039
      using u v by (auto simp: isoton_fun_expand isoton_add isoton_cmult_right)
hoelzl@38656
  1040
  qed
hoelzl@38656
  1041
  moreover from u v have L'_isoton:
hoelzl@38656
  1042
      "(\<lambda>i. simple_integral (?L' i)) \<up> a * positive_integral f + positive_integral g"
hoelzl@38656
  1043
    by (simp add: isoton_add isoton_cmult_right)
hoelzl@38656
  1044
  ultimately show ?thesis by (simp add: isoton_def)
hoelzl@38656
  1045
qed
hoelzl@38656
  1046
hoelzl@38656
  1047
lemma (in measure_space) positive_integral_cmult:
hoelzl@38656
  1048
  assumes "f \<in> borel_measurable M"
hoelzl@38656
  1049
  shows "positive_integral (\<lambda>x. c * f x) = c * positive_integral f"
hoelzl@38656
  1050
  using positive_integral_linear[OF assms, of "\<lambda>x. 0" c] by auto
hoelzl@38656
  1051
hoelzl@38656
  1052
lemma (in measure_space) positive_integral_indicator[simp]:
hoelzl@38656
  1053
  "A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. indicator A x) = \<mu> A"
hoelzl@38656
  1054
by (subst positive_integral_eq_simple_integral) (auto simp: simple_function_indicator simple_integral_indicator)
hoelzl@38656
  1055
hoelzl@38656
  1056
lemma (in measure_space) positive_integral_cmult_indicator:
hoelzl@38656
  1057
  "A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. c * indicator A x) = c * \<mu> A"
hoelzl@38656
  1058
by (subst positive_integral_eq_simple_integral) (auto simp: simple_function_indicator simple_integral_indicator)
hoelzl@38656
  1059
hoelzl@38656
  1060
lemma (in measure_space) positive_integral_add:
hoelzl@38656
  1061
  assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@38656
  1062
  shows "positive_integral (\<lambda>x. f x + g x) = positive_integral f + positive_integral g"
hoelzl@38656
  1063
  using positive_integral_linear[OF assms, of 1] by simp
hoelzl@38656
  1064
hoelzl@38656
  1065
lemma (in measure_space) positive_integral_setsum:
hoelzl@38656
  1066
  assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M"
hoelzl@38656
  1067
  shows "positive_integral (\<lambda>x. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. positive_integral (f i))"
hoelzl@38656
  1068
proof cases
hoelzl@38656
  1069
  assume "finite P"
hoelzl@38656
  1070
  from this assms show ?thesis
hoelzl@38656
  1071
  proof induct
hoelzl@38656
  1072
    case (insert i P)
hoelzl@38656
  1073
    have "f i \<in> borel_measurable M"
hoelzl@38656
  1074
      "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M"
hoelzl@38656
  1075
      using insert by (auto intro!: borel_measurable_pinfreal_setsum)
hoelzl@38656
  1076
    from positive_integral_add[OF this]
hoelzl@38656
  1077
    show ?case using insert by auto
hoelzl@38656
  1078
  qed simp
hoelzl@38656
  1079
qed simp
hoelzl@38656
  1080
hoelzl@38656
  1081
lemma (in measure_space) positive_integral_diff:
hoelzl@38656
  1082
  assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
hoelzl@38656
  1083
  and fin: "positive_integral g \<noteq> \<omega>"
hoelzl@38656
  1084
  and mono: "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"
hoelzl@38656
  1085
  shows "positive_integral (\<lambda>x. f x - g x) = positive_integral f - positive_integral g"
hoelzl@38656
  1086
proof -
hoelzl@38656
  1087
  have borel: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
hoelzl@38656
  1088
    using f g by (rule borel_measurable_pinfreal_diff)
hoelzl@38656
  1089
  have "positive_integral (\<lambda>x. f x - g x) + positive_integral g =
hoelzl@38656
  1090
    positive_integral f"
hoelzl@38656
  1091
    unfolding positive_integral_add[OF borel g, symmetric]
hoelzl@38656
  1092
  proof (rule positive_integral_cong)
hoelzl@38656
  1093
    fix x assume "x \<in> space M"
hoelzl@38656
  1094
    from mono[OF this] show "f x - g x + g x = f x"
hoelzl@38656
  1095
      by (cases "f x", cases "g x", simp, simp, cases "g x", auto)
hoelzl@38656
  1096
  qed
hoelzl@38656
  1097
  with mono show ?thesis
hoelzl@38656
  1098
    by (subst minus_pinfreal_eq2[OF _ fin]) (auto intro!: positive_integral_mono)
hoelzl@38656
  1099
qed
hoelzl@38656
  1100
hoelzl@38656
  1101
lemma (in measure_space) positive_integral_psuminf:
hoelzl@38656
  1102
  assumes "\<And>i. f i \<in> borel_measurable M"
hoelzl@38656
  1103
  shows "positive_integral (\<lambda>x. \<Sum>\<^isub>\<infinity> i. f i x) = (\<Sum>\<^isub>\<infinity> i. positive_integral (f i))"
hoelzl@38656
  1104
proof -
hoelzl@38656
  1105
  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
  1106
    by (rule positive_integral_isoton)
hoelzl@38656
  1107
       (auto intro!: borel_measurable_pinfreal_setsum assms positive_integral_mono
hoelzl@38656
  1108
                     arg_cong[where f=Sup]
hoelzl@38656
  1109
             simp: isoton_def le_fun_def psuminf_def expand_fun_eq SUPR_def Sup_fun_def)
hoelzl@38656
  1110
  thus ?thesis
hoelzl@38656
  1111
    by (auto simp: isoton_def psuminf_def positive_integral_setsum[OF assms])
hoelzl@38656
  1112
qed
hoelzl@38656
  1113
hoelzl@38656
  1114
text {* Fatou's lemma: convergence theorem on limes inferior *}
hoelzl@38656
  1115
lemma (in measure_space) positive_integral_lim_INF:
hoelzl@38656
  1116
  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pinfreal"
hoelzl@38656
  1117
  assumes "\<And>i. u i \<in> borel_measurable M"
hoelzl@38656
  1118
  shows "positive_integral (SUP n. INF m. u (m + n)) \<le>
hoelzl@38656
  1119
    (SUP n. INF m. positive_integral (u (m + n)))"
hoelzl@38656
  1120
proof -
hoelzl@38656
  1121
  have "(SUP n. INF m. u (m + n)) \<in> borel_measurable M"
hoelzl@38656
  1122
    by (auto intro!: borel_measurable_SUP borel_measurable_INF assms)
hoelzl@38656
  1123
hoelzl@38656
  1124
  have "(\<lambda>n. INF m. u (m + n)) \<up> (SUP n. INF m. u (m + n))"
hoelzl@38705
  1125
  proof (unfold isoton_def, safe intro!: INF_mono bexI)
hoelzl@38705
  1126
    fix i m show "u (Suc m + i) \<le> u (m + Suc i)" by simp
hoelzl@38705
  1127
  qed simp
hoelzl@38656
  1128
  from positive_integral_isoton[OF this] assms
hoelzl@38656
  1129
  have "positive_integral (SUP n. INF m. u (m + n)) =
hoelzl@38656
  1130
    (SUP n. positive_integral (INF m. u (m + n)))"
hoelzl@38656
  1131
    unfolding isoton_def by (simp add: borel_measurable_INF)
hoelzl@38656
  1132
  also have "\<dots> \<le> (SUP n. INF m. positive_integral (u (m + n)))"
hoelzl@38705
  1133
    apply (rule SUP_mono)
hoelzl@38705
  1134
    apply (rule_tac x=n in bexI)
hoelzl@38705
  1135
    by (auto intro!: positive_integral_mono INFI_bound INF_leI exI simp: INFI_fun_expand)
hoelzl@38656
  1136
  finally show ?thesis .
hoelzl@35582
  1137
qed
hoelzl@35582
  1138
hoelzl@38656
  1139
lemma (in measure_space) measure_space_density:
hoelzl@38656
  1140
  assumes borel: "u \<in> borel_measurable M"
hoelzl@38656
  1141
  shows "measure_space M (\<lambda>A. positive_integral (\<lambda>x. u x * indicator A x))" (is "measure_space M ?v")
hoelzl@38656
  1142
proof
hoelzl@38656
  1143
  show "?v {} = 0" by simp
hoelzl@38656
  1144
  show "countably_additive M ?v"
hoelzl@38656
  1145
    unfolding countably_additive_def
hoelzl@38656
  1146
  proof safe
hoelzl@38656
  1147
    fix A :: "nat \<Rightarrow> 'a set"
hoelzl@38656
  1148
    assume "range A \<subseteq> sets M"
hoelzl@38656
  1149
    hence "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
hoelzl@38656
  1150
      using borel by (auto intro: borel_measurable_indicator)
hoelzl@38656
  1151
    moreover assume "disjoint_family A"
hoelzl@38656
  1152
    note psuminf_indicator[OF this]
hoelzl@38656
  1153
    ultimately show "(\<Sum>\<^isub>\<infinity>n. ?v (A n)) = ?v (\<Union>x. A x)"
hoelzl@38656
  1154
      by (simp add: positive_integral_psuminf[symmetric])
hoelzl@38656
  1155
  qed
hoelzl@38656
  1156
qed
hoelzl@35582
  1157
hoelzl@39092
  1158
lemma (in measure_space) positive_integral_translated_density:
hoelzl@39092
  1159
  assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
hoelzl@39092
  1160
  shows "measure_space.positive_integral M (\<lambda>A. positive_integral (\<lambda>x. f x * indicator A x)) g =
hoelzl@39092
  1161
    positive_integral (\<lambda>x. f x * g x)" (is "measure_space.positive_integral M ?T _ = _")
hoelzl@39092
  1162
proof -
hoelzl@39092
  1163
  from measure_space_density[OF assms(1)]
hoelzl@39092
  1164
  interpret T: measure_space M ?T .
hoelzl@39092
  1165
  from borel_measurable_implies_simple_function_sequence[OF assms(2)]
hoelzl@39092
  1166
  obtain G where G: "\<And>i. simple_function (G i)" "G \<up> g" by blast
hoelzl@39092
  1167
  note G_borel = borel_measurable_simple_function[OF this(1)]
hoelzl@39092
  1168
  from T.positive_integral_isoton[OF `G \<up> g` G_borel]
hoelzl@39092
  1169
  have *: "(\<lambda>i. T.positive_integral (G i)) \<up> T.positive_integral g" .
hoelzl@39092
  1170
  { fix i
hoelzl@39092
  1171
    have [simp]: "finite (G i ` space M)"
hoelzl@39092
  1172
      using G(1) unfolding simple_function_def by auto
hoelzl@39092
  1173
    have "T.positive_integral (G i) = T.simple_integral (G i)"
hoelzl@39092
  1174
      using G T.positive_integral_eq_simple_integral by simp
hoelzl@39092
  1175
    also have "\<dots> = positive_integral (\<lambda>x. f x * (\<Sum>y\<in>G i`space M. y * indicator (G i -` {y} \<inter> space M) x))"
hoelzl@39092
  1176
      apply (simp add: T.simple_integral_def)
hoelzl@39092
  1177
      apply (subst positive_integral_cmult[symmetric])
hoelzl@39092
  1178
      using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
hoelzl@39092
  1179
      apply (subst positive_integral_setsum[symmetric])
hoelzl@39092
  1180
      using G_borel assms(1) apply (fastsimp intro: borel_measurable_indicator borel_measurable_vimage)
hoelzl@39092
  1181
      by (simp add: setsum_right_distrib field_simps)
hoelzl@39092
  1182
    also have "\<dots> = positive_integral (\<lambda>x. f x * G i x)"
hoelzl@39092
  1183
      by (auto intro!: positive_integral_cong
hoelzl@39092
  1184
               simp: indicator_def if_distrib setsum_cases)
hoelzl@39092
  1185
    finally have "T.positive_integral (G i) = positive_integral (\<lambda>x. f x * G i x)" . }
hoelzl@39092
  1186
  with * have eq_Tg: "(\<lambda>i. positive_integral (\<lambda>x. f x * G i x)) \<up> T.positive_integral g" by simp
hoelzl@39092
  1187
  from G(2) have "(\<lambda>i x. f x * G i x) \<up> (\<lambda>x. f x * g x)"
hoelzl@39092
  1188
    unfolding isoton_fun_expand by (auto intro!: isoton_cmult_right)
hoelzl@39092
  1189
  then have "(\<lambda>i. positive_integral (\<lambda>x. f x * G i x)) \<up> positive_integral (\<lambda>x. f x * g x)"
hoelzl@39092
  1190
    using assms(1) G_borel by (auto intro!: positive_integral_isoton borel_measurable_pinfreal_times)
hoelzl@39092
  1191
  with eq_Tg show "T.positive_integral g = positive_integral (\<lambda>x. f x * g x)"
hoelzl@39092
  1192
    unfolding isoton_def by simp
hoelzl@39092
  1193
qed
hoelzl@39092
  1194
hoelzl@38656
  1195
lemma (in measure_space) positive_integral_null_set:
hoelzl@38656
  1196
  assumes borel: "u \<in> borel_measurable M" and "N \<in> null_sets"
hoelzl@38656
  1197
  shows "positive_integral (\<lambda>x. u x * indicator N x) = 0" (is "?I = 0")
hoelzl@38656
  1198
proof -
hoelzl@38656
  1199
  have "N \<in> sets M" using `N \<in> null_sets` by auto
hoelzl@38656
  1200
  have "(\<lambda>i x. min (of_nat i) (u x) * indicator N x) \<up> (\<lambda>x. u x * indicator N x)"
hoelzl@38656
  1201
    unfolding isoton_fun_expand
hoelzl@38656
  1202
  proof (safe intro!: isoton_cmult_left, unfold isoton_def, safe)
hoelzl@38656
  1203
    fix j i show "min (of_nat j) (u i) \<le> min (of_nat (Suc j)) (u i)"
hoelzl@38656
  1204
      by (rule min_max.inf_mono) auto
hoelzl@38656
  1205
  next
hoelzl@38656
  1206
    fix i show "(SUP j. min (of_nat j) (u i)) = u i"
hoelzl@38656
  1207
    proof (cases "u i")
hoelzl@38656
  1208
      case infinite
hoelzl@38656
  1209
      moreover hence "\<And>j. min (of_nat j) (u i) = of_nat j"
hoelzl@38656
  1210
        by (auto simp: min_def)
hoelzl@38656
  1211
      ultimately show ?thesis by (simp add: Sup_\<omega>)
hoelzl@35582
  1212
    next
hoelzl@38656
  1213
      case (preal r)
hoelzl@38656
  1214
      obtain j where "r \<le> of_nat j" using ex_le_of_nat ..
hoelzl@38656
  1215
      hence "u i \<le> of_nat j" using preal by (auto simp: real_of_nat_def)
hoelzl@38656
  1216
      show ?thesis
hoelzl@38656
  1217
      proof (rule pinfreal_SUPI)
hoelzl@38656
  1218
        fix y assume "\<And>j. j \<in> UNIV \<Longrightarrow> min (of_nat j) (u i) \<le> y"
hoelzl@38656
  1219
        note this[of j]
hoelzl@38656
  1220
        moreover have "min (of_nat j) (u i) = u i"
hoelzl@38656
  1221
          using `u i \<le> of_nat j` by (auto simp: min_def)
hoelzl@38656
  1222
        ultimately show "u i \<le> y" by simp
hoelzl@35582
  1223
      qed simp
hoelzl@35582
  1224
    qed
hoelzl@35582
  1225
  qed
hoelzl@38656
  1226
  from positive_integral_isoton[OF this]
hoelzl@38656
  1227
  have "?I = (SUP i. positive_integral (\<lambda>x. min (of_nat i) (u x) * indicator N x))"
hoelzl@38656
  1228
    unfolding isoton_def using borel `N \<in> sets M` by (simp add: borel_measurable_indicator)
hoelzl@38656
  1229
  also have "\<dots> \<le> (SUP i. positive_integral (\<lambda>x. of_nat i * indicator N x))"
hoelzl@38705
  1230
  proof (rule SUP_mono, rule bexI, rule positive_integral_mono)
hoelzl@38656
  1231
    fix x i show "min (of_nat i) (u x) * indicator N x \<le> of_nat i * indicator N x"
hoelzl@38656
  1232
      by (cases "x \<in> N") auto
hoelzl@38705
  1233
  qed simp
hoelzl@38656
  1234
  also have "\<dots> = 0"
hoelzl@38656
  1235
    using `N \<in> null_sets` by (simp add: positive_integral_cmult_indicator)
hoelzl@38656
  1236
  finally show ?thesis by simp
hoelzl@38656
  1237
qed
hoelzl@35582
  1238
hoelzl@38656
  1239
lemma (in measure_space) positive_integral_Markov_inequality:
hoelzl@38656
  1240
  assumes borel: "u \<in> borel_measurable M" and "A \<in> sets M" and c: "c \<noteq> \<omega>"
hoelzl@38656
  1241
  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
  1242
    (is "\<mu> ?A \<le> _ * ?PI")
hoelzl@38656
  1243
proof -
hoelzl@38656
  1244
  have "?A \<in> sets M"
hoelzl@38656
  1245
    using `A \<in> sets M` borel by auto
hoelzl@38656
  1246
  hence "\<mu> ?A = positive_integral (\<lambda>x. indicator ?A x)"
hoelzl@38656
  1247
    using positive_integral_indicator by simp
hoelzl@38656
  1248
  also have "\<dots> \<le> positive_integral (\<lambda>x. c * (u x * indicator A x))"
hoelzl@38656
  1249
  proof (rule positive_integral_mono)
hoelzl@38656
  1250
    fix x assume "x \<in> space M"
hoelzl@38656
  1251
    show "indicator ?A x \<le> c * (u x * indicator A x)"
hoelzl@38656
  1252
      by (cases "x \<in> ?A") auto
hoelzl@38656
  1253
  qed
hoelzl@38656
  1254
  also have "\<dots> = c * positive_integral (\<lambda>x. u x * indicator A x)"
hoelzl@38656
  1255
    using assms
hoelzl@38656
  1256
    by (auto intro!: positive_integral_cmult borel_measurable_indicator)
hoelzl@38656
  1257
  finally show ?thesis .
hoelzl@35582
  1258
qed
hoelzl@35582
  1259
hoelzl@38656
  1260
lemma (in measure_space) positive_integral_0_iff:
hoelzl@38656
  1261
  assumes borel: "u \<in> borel_measurable M"
hoelzl@38656
  1262
  shows "positive_integral u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
hoelzl@38656
  1263
    (is "_ \<longleftrightarrow> \<mu> ?A = 0")
hoelzl@35582
  1264
proof -
hoelzl@38656
  1265
  have A: "?A \<in> sets M" using borel by auto
hoelzl@38656
  1266
  have u: "positive_integral (\<lambda>x. u x * indicator ?A x) = positive_integral u"
hoelzl@38656
  1267
    by (auto intro!: positive_integral_cong simp: indicator_def)
hoelzl@35582
  1268
hoelzl@38656
  1269
  show ?thesis
hoelzl@38656
  1270
  proof
hoelzl@38656
  1271
    assume "\<mu> ?A = 0"
hoelzl@38656
  1272
    hence "?A \<in> null_sets" using `?A \<in> sets M` by auto
hoelzl@38656
  1273
    from positive_integral_null_set[OF borel this]
hoelzl@38656
  1274
    have "0 = positive_integral (\<lambda>x. u x * indicator ?A x)" by simp
hoelzl@38656
  1275
    thus "positive_integral u = 0" unfolding u by simp
hoelzl@38656
  1276
  next
hoelzl@38656
  1277
    assume *: "positive_integral u = 0"
hoelzl@38656
  1278
    let "?M n" = "{x \<in> space M. 1 \<le> of_nat n * u x}"
hoelzl@38656
  1279
    have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
hoelzl@38656
  1280
    proof -
hoelzl@38656
  1281
      { fix n
hoelzl@38656
  1282
        from positive_integral_Markov_inequality[OF borel `?A \<in> sets M`, of "of_nat n"]
hoelzl@38656
  1283
        have "\<mu> (?M n \<inter> ?A) = 0" unfolding * u by simp }
hoelzl@38656
  1284
      thus ?thesis by simp
hoelzl@35582
  1285
    qed
hoelzl@38656
  1286
    also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
hoelzl@38656
  1287
    proof (safe intro!: continuity_from_below)
hoelzl@38656
  1288
      fix n show "?M n \<inter> ?A \<in> sets M"
hoelzl@38656
  1289
        using borel by (auto intro!: Int)
hoelzl@38656
  1290
    next
hoelzl@38656
  1291
      fix n x assume "1 \<le> of_nat n * u x"
hoelzl@38656
  1292
      also have "\<dots> \<le> of_nat (Suc n) * u x"
hoelzl@38656
  1293
        by (subst (1 2) mult_commute) (auto intro!: pinfreal_mult_cancel)
hoelzl@38656
  1294
      finally show "1 \<le> of_nat (Suc n) * u x" .
hoelzl@38656
  1295
    qed
hoelzl@38656
  1296
    also have "\<dots> = \<mu> ?A"
hoelzl@38656
  1297
    proof (safe intro!: arg_cong[where f="\<mu>"])
hoelzl@38656
  1298
      fix x assume "u x \<noteq> 0" and [simp, intro]: "x \<in> space M"
hoelzl@38656
  1299
      show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
hoelzl@38656
  1300
      proof (cases "u x")
hoelzl@38656
  1301
        case (preal r)
hoelzl@38656
  1302
        obtain j where "1 / r \<le> of_nat j" using ex_le_of_nat ..
hoelzl@38656
  1303
        hence "1 / r * r \<le> of_nat j * r" using preal unfolding mult_le_cancel_right by auto
hoelzl@38656
  1304
        hence "1 \<le> of_nat j * r" using preal `u x \<noteq> 0` by auto
hoelzl@38656
  1305
        thus ?thesis using `u x \<noteq> 0` preal by (auto simp: real_of_nat_def[symmetric])
hoelzl@38656
  1306
      qed auto
hoelzl@38656
  1307
    qed
hoelzl@38656
  1308
    finally show "\<mu> ?A = 0" by simp
hoelzl@35582
  1309
  qed
hoelzl@35582
  1310
qed
hoelzl@35582
  1311
hoelzl@38656
  1312
lemma (in measure_space) positive_integral_cong_on_null_sets:
hoelzl@38656
  1313
  assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
hoelzl@38656
  1314
  and measure: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
hoelzl@38656
  1315
  shows "positive_integral f = positive_integral g"
hoelzl@35582
  1316
proof -
hoelzl@38656
  1317
  let ?N = "{x\<in>space M. f x \<noteq> g x}" and ?E = "{x\<in>space M. f x = g x}"
hoelzl@38656
  1318
  let "?A h x" = "h x * indicator ?E x :: pinfreal"
hoelzl@38656
  1319
  let "?B h x" = "h x * indicator ?N x :: pinfreal"
hoelzl@38656
  1320
hoelzl@38656
  1321
  have A: "positive_integral (?A f) = positive_integral (?A g)"
hoelzl@38656
  1322
    by (auto intro!: positive_integral_cong simp: indicator_def)
hoelzl@38656
  1323
hoelzl@38656
  1324
  have [intro]: "?N \<in> sets M" "?E \<in> sets M" using f g by auto
hoelzl@38656
  1325
  hence "?N \<in> null_sets" using measure by auto
hoelzl@38656
  1326
  hence B: "positive_integral (?B f) = positive_integral (?B g)"
hoelzl@38656
  1327
    using f g by (simp add: positive_integral_null_set)
hoelzl@38656
  1328
hoelzl@38656
  1329
  have "positive_integral f = positive_integral (\<lambda>x. ?A f x + ?B f x)"
hoelzl@38656
  1330
    by (auto intro!: positive_integral_cong simp: indicator_def)
hoelzl@38656
  1331
  also have "\<dots> = positive_integral (?A f) + positive_integral (?B f)"
hoelzl@38656
  1332
    using f g by (auto intro!: positive_integral_add borel_measurable_indicator)
hoelzl@38656
  1333
  also have "\<dots> = positive_integral (\<lambda>x. ?A g x + ?B g x)"
hoelzl@38656
  1334
    unfolding A B using f g by (auto intro!: positive_integral_add[symmetric] borel_measurable_indicator)
hoelzl@38656
  1335
  also have "\<dots> = positive_integral g"
hoelzl@38656
  1336
    by (auto intro!: positive_integral_cong simp: indicator_def)
hoelzl@38656
  1337
  finally show ?thesis by simp
hoelzl@35582
  1338
qed
hoelzl@35582
  1339
hoelzl@39092
  1340
lemma (in measure_space) positive_integral_restricted:
hoelzl@39092
  1341
  assumes "A \<in> sets M"
hoelzl@39092
  1342
  shows "measure_space.positive_integral (restricted_space A) \<mu> f = positive_integral (\<lambda>x. f x * indicator A x)"
hoelzl@39092
  1343
    (is "measure_space.positive_integral ?R \<mu> f = positive_integral ?f")
hoelzl@39092
  1344
proof -
hoelzl@39092
  1345
  have msR: "measure_space ?R \<mu>" by (rule restricted_measure_space[OF `A \<in> sets M`])
hoelzl@39092
  1346
  then interpret R: measure_space ?R \<mu> .
hoelzl@39092
  1347
  have saR: "sigma_algebra ?R" by fact
hoelzl@39092
  1348
  have *: "R.positive_integral f = R.positive_integral ?f"
hoelzl@39092
  1349
    by (auto intro!: R.positive_integral_cong)
hoelzl@39092
  1350
  show ?thesis
hoelzl@39092
  1351
    unfolding * R.positive_integral_def positive_integral_def
hoelzl@39092
  1352
    unfolding simple_function_restricted[OF `A \<in> sets M`]
hoelzl@39092
  1353
    apply (simp add: SUPR_def)
hoelzl@39092
  1354
    apply (rule arg_cong[where f=Sup])
hoelzl@39092
  1355
  proof (auto simp: image_iff simple_integral_restricted[OF `A \<in> sets M`])
hoelzl@39092
  1356
    fix g assume "simple_function (\<lambda>x. g x * indicator A x)"
hoelzl@39092
  1357
      "g \<le> f" "\<forall>x\<in>A. \<omega> \<noteq> g x"
hoelzl@39092
  1358
    then show "\<exists>x. simple_function x \<and> x \<le> (\<lambda>x. f x * indicator A x) \<and> (\<forall>y\<in>space M. \<omega> \<noteq> x y) \<and>
hoelzl@39092
  1359
      simple_integral (\<lambda>x. g x * indicator A x) = simple_integral x"
hoelzl@39092
  1360
      apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
hoelzl@39092
  1361
      by (auto simp: indicator_def le_fun_def)
hoelzl@39092
  1362
  next
hoelzl@39092
  1363
    fix g assume g: "simple_function g" "g \<le> (\<lambda>x. f x * indicator A x)"
hoelzl@39092
  1364
      "\<forall>x\<in>space M. \<omega> \<noteq> g x"
hoelzl@39092
  1365
    then have *: "(\<lambda>x. g x * indicator A x) = g"
hoelzl@39092
  1366
      "\<And>x. g x * indicator A x = g x"
hoelzl@39092
  1367
      "\<And>x. g x \<le> f x"
hoelzl@39092
  1368
      by (auto simp: le_fun_def expand_fun_eq indicator_def split: split_if_asm)
hoelzl@39092
  1369
    from g show "\<exists>x. simple_function (\<lambda>xa. x xa * indicator A xa) \<and> x \<le> f \<and> (\<forall>xa\<in>A. \<omega> \<noteq> x xa) \<and>
hoelzl@39092
  1370
      simple_integral g = simple_integral (\<lambda>xa. x xa * indicator A xa)"
hoelzl@39092
  1371
      using `A \<in> sets M`[THEN sets_into_space]
hoelzl@39092
  1372
      apply (rule_tac exI[of _ "\<lambda>x. g x * indicator A x"])
hoelzl@39092
  1373
      by (fastsimp simp: le_fun_def *)
hoelzl@39092
  1374
  qed
hoelzl@39092
  1375
qed
hoelzl@39092
  1376
hoelzl@39092
  1377
lemma (in measure_space) positive_integral_subalgebra[simp]:
hoelzl@39092
  1378
  assumes borel: "f \<in> borel_measurable (M\<lparr>sets := N\<rparr>)"
hoelzl@39092
  1379
  and N_subalgebra: "N \<subseteq> sets M" "sigma_algebra (M\<lparr>sets := N\<rparr>)"
hoelzl@39092
  1380
  shows "measure_space.positive_integral (M\<lparr>sets := N\<rparr>) \<mu> f = positive_integral f"
hoelzl@39092
  1381
proof -
hoelzl@39092
  1382
  note msN = measure_space_subalgebra[OF N_subalgebra]
hoelzl@39092
  1383
  then interpret N: measure_space "M\<lparr>sets:=N\<rparr>" \<mu> .
hoelzl@39092
  1384
  from N.borel_measurable_implies_simple_function_sequence[OF borel]
hoelzl@39092
  1385
  obtain fs where Nsf: "\<And>i. N.simple_function (fs i)" and "fs \<up> f" by blast
hoelzl@39092
  1386
  then have sf: "\<And>i. simple_function (fs i)"
hoelzl@39092
  1387
    using simple_function_subalgebra[OF _ N_subalgebra] by blast
hoelzl@39092
  1388
  from positive_integral_isoton_simple[OF `fs \<up> f` sf] N.positive_integral_isoton_simple[OF `fs \<up> f` Nsf]
hoelzl@39092
  1389
  show ?thesis unfolding simple_integral_subalgebra[OF msN] isoton_def by simp
hoelzl@39092
  1390
qed
hoelzl@39092
  1391
hoelzl@35692
  1392
section "Lebesgue Integral"
hoelzl@35692
  1393
hoelzl@38656
  1394
definition (in measure_space) integrable where
hoelzl@38656
  1395
  "integrable f \<longleftrightarrow> f \<in> borel_measurable M \<and>
hoelzl@38656
  1396
    positive_integral (\<lambda>x. Real (f x)) \<noteq> \<omega> \<and>
hoelzl@38656
  1397
    positive_integral (\<lambda>x. Real (- f x)) \<noteq> \<omega>"
hoelzl@35692
  1398
hoelzl@38656
  1399
lemma (in measure_space) integrableD[dest]:
hoelzl@38656
  1400
  assumes "integrable f"
hoelzl@38656
  1401
  shows "f \<in> borel_measurable M"
hoelzl@38656
  1402
  "positive_integral (\<lambda>x. Real (f x)) \<noteq> \<omega>"
hoelzl@38656
  1403
  "positive_integral (\<lambda>x. Real (- f x)) \<noteq> \<omega>"
hoelzl@38656
  1404
  using assms unfolding integrable_def by auto
hoelzl@35692
  1405
hoelzl@38656
  1406
definition (in measure_space) integral where
hoelzl@38656
  1407
  "integral f =
hoelzl@38656
  1408
    real (positive_integral (\<lambda>x. Real (f x))) -
hoelzl@38656
  1409
    real (positive_integral (\<lambda>x. Real (- f x)))"
hoelzl@38656
  1410
hoelzl@38656
  1411
lemma (in measure_space) integral_cong:
hoelzl@35582
  1412
  assumes cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
hoelzl@35582
  1413
  shows "integral f = integral g"
hoelzl@38656
  1414
  using assms by (simp cong: positive_integral_cong add: integral_def)
hoelzl@35582
  1415
hoelzl@38656
  1416
lemma (in measure_space) integrable_cong:
hoelzl@38656
  1417
  "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable f \<longleftrightarrow> integrable g"
hoelzl@38656
  1418
  by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
hoelzl@38656
  1419
hoelzl@38656
  1420
lemma (in measure_space) integral_eq_positive_integral:
hoelzl@38656
  1421
  assumes "\<And>x. 0 \<le> f x"
hoelzl@38656
  1422
  shows "integral f = real (positive_integral (\<lambda>x. Real (f x)))"
hoelzl@35582
  1423
proof -
hoelzl@38656
  1424
  have "\<And>x. Real (- f x) = 0" using assms by simp
hoelzl@38656
  1425
  thus ?thesis by (simp del: Real_eq_0 add: integral_def)
hoelzl@35582
  1426
qed
hoelzl@35582
  1427
hoelzl@38656
  1428
lemma (in measure_space) integral_minus[intro, simp]:
hoelzl@38656
  1429
  assumes "integrable f"
hoelzl@38656
  1430
  shows "integrable (\<lambda>x. - f x)" "integral (\<lambda>x. - f x) = - integral f"
hoelzl@38656
  1431
  using assms by (auto simp: integrable_def integral_def)
hoelzl@38656
  1432
hoelzl@38656
  1433
lemma (in measure_space) integral_of_positive_diff:
hoelzl@38656
  1434
  assumes integrable: "integrable u" "integrable v"
hoelzl@38656
  1435
  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
  1436
  shows "integrable f" and "integral f = integral u - integral v"
hoelzl@35582
  1437
proof -
hoelzl@38656
  1438
  let ?PI = positive_integral
hoelzl@38656
  1439
  let "?f x" = "Real (f x)"
hoelzl@38656
  1440
  let "?mf x" = "Real (- f x)"
hoelzl@38656
  1441
  let "?u x" = "Real (u x)"
hoelzl@38656
  1442
  let "?v x" = "Real (v x)"
hoelzl@38656
  1443
hoelzl@38656
  1444
  from borel_measurable_diff[of u v] integrable
hoelzl@38656
  1445
  have f_borel: "?f \<in> borel_measurable M" and
hoelzl@38656
  1446
    mf_borel: "?mf \<in> borel_measurable M" and
hoelzl@38656
  1447
    v_borel: "?v \<in> borel_measurable M" and
hoelzl@38656
  1448
    u_borel: "?u \<in> borel_measurable M" and
hoelzl@38656
  1449
    "f \<in> borel_measurable M"
hoelzl@38656
  1450
    by (auto simp: f_def[symmetric] integrable_def)
hoelzl@35582
  1451
hoelzl@38656
  1452
  have "?PI (\<lambda>x. Real (u x - v x)) \<le> ?PI ?u"
hoelzl@38656
  1453
    using pos by (auto intro!: positive_integral_mono)
hoelzl@38656
  1454
  moreover have "?PI (\<lambda>x. Real (v x - u x)) \<le> ?PI ?v"
hoelzl@38656
  1455
    using pos by (auto intro!: positive_integral_mono)
hoelzl@38656
  1456
  ultimately show f: "integrable f"
hoelzl@38656
  1457
    using `integrable u` `integrable v` `f \<in> borel_measurable M`
hoelzl@38656
  1458
    by (auto simp: integrable_def f_def)
hoelzl@38656
  1459
  hence mf: "integrable (\<lambda>x. - f x)" ..
hoelzl@38656
  1460
hoelzl@38656
  1461
  have *: "\<And>x. Real (- v x) = 0" "\<And>x. Real (- u x) = 0"
hoelzl@38656
  1462
    using pos by auto
hoelzl@35582
  1463
hoelzl@38656
  1464
  have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
hoelzl@38656
  1465
    unfolding f_def using pos by simp
hoelzl@38656
  1466
  hence "?PI (\<lambda>x. ?u x + ?mf x) = ?PI (\<lambda>x. ?v x + ?f x)" by simp
hoelzl@38656
  1467
  hence "real (?PI ?u + ?PI ?mf) = real (?PI ?v + ?PI ?f)"
hoelzl@38656
  1468
    using positive_integral_add[OF u_borel mf_borel]
hoelzl@38656
  1469
    using positive_integral_add[OF v_borel f_borel]
hoelzl@38656
  1470
    by auto
hoelzl@38656
  1471
  then show "integral f = integral u - integral v"
hoelzl@38656
  1472
    using f mf `integrable u` `integrable v`
hoelzl@38656
  1473
    unfolding integral_def integrable_def *
hoelzl@38656
  1474
    by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?v", cases "?PI ?u")
hoelzl@38656
  1475
       (auto simp add: field_simps)
hoelzl@35582
  1476
qed
hoelzl@35582
  1477
hoelzl@38656
  1478
lemma (in measure_space) integral_linear:
hoelzl@38656
  1479
  assumes "integrable f" "integrable g" and "0 \<le> a"
hoelzl@38656
  1480
  shows "integrable (\<lambda>t. a * f t + g t)"
hoelzl@38656
  1481
  and "integral (\<lambda>t. a * f t + g t) = a * integral f + integral g"
hoelzl@38656
  1482
proof -
hoelzl@38656
  1483
  let ?PI = positive_integral
hoelzl@38656
  1484
  let "?f x" = "Real (f x)"
hoelzl@38656
  1485
  let "?g x" = "Real (g x)"
hoelzl@38656
  1486
  let "?mf x" = "Real (- f x)"
hoelzl@38656
  1487
  let "?mg x" = "Real (- g x)"
hoelzl@38656
  1488
  let "?p t" = "max 0 (a * f t) + max 0 (g t)"
hoelzl@38656
  1489
  let "?n t" = "max 0 (- (a * f t)) + max 0 (- g t)"
hoelzl@38656
  1490
hoelzl@38656
  1491
  have pos: "?f \<in> borel_measurable M" "?g \<in> borel_measurable M"
hoelzl@38656
  1492
    and neg: "?mf \<in> borel_measurable M" "?mg \<in> borel_measurable M"
hoelzl@38656
  1493
    and p: "?p \<in> borel_measurable M"
hoelzl@38656
  1494
    and n: "?n \<in> borel_measurable M"
hoelzl@38656
  1495
    using assms by (simp_all add: integrable_def)
hoelzl@35582
  1496
hoelzl@38656
  1497
  have *: "\<And>x. Real (?p x) = Real a * ?f x + ?g x"
hoelzl@38656
  1498
          "\<And>x. Real (?n x) = Real a * ?mf x + ?mg x"
hoelzl@38656
  1499
          "\<And>x. Real (- ?p x) = 0"
hoelzl@38656
  1500
          "\<And>x. Real (- ?n x) = 0"
hoelzl@38656
  1501
    using `0 \<le> a` by (auto simp: max_def min_def zero_le_mult_iff mult_le_0_iff add_nonpos_nonpos)
hoelzl@38656
  1502
hoelzl@38656
  1503
  note linear =
hoelzl@38656
  1504
    positive_integral_linear[OF pos]
hoelzl@38656
  1505
    positive_integral_linear[OF neg]
hoelzl@35582
  1506
hoelzl@38656
  1507
  have "integrable ?p" "integrable ?n"
hoelzl@38656
  1508
      "\<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
  1509
    using assms p n unfolding integrable_def * linear by auto
hoelzl@38656
  1510
  note diff = integral_of_positive_diff[OF this]
hoelzl@38656
  1511
hoelzl@38656
  1512
  show "integrable (\<lambda>t. a * f t + g t)" by (rule diff)
hoelzl@38656
  1513
hoelzl@38656
  1514
  from assms show "integral (\<lambda>t. a * f t + g t) = a * integral f + integral g"
hoelzl@38656
  1515
    unfolding diff(2) unfolding integral_def * linear integrable_def
hoelzl@38656
  1516
    by (cases "?PI ?f", cases "?PI ?mf", cases "?PI ?g", cases "?PI ?mg")
hoelzl@38656
  1517
       (auto simp add: field_simps zero_le_mult_iff)
hoelzl@38656
  1518
qed
hoelzl@38656
  1519
hoelzl@38656
  1520
lemma (in measure_space) integral_add[simp, intro]:
hoelzl@38656
  1521
  assumes "integrable f" "integrable g"
hoelzl@35582
  1522
  shows "integrable (\<lambda>t. f t + g t)"
hoelzl@35582
  1523
  and "integral (\<lambda>t. f t + g t) = integral f + integral g"
hoelzl@38656
  1524
  using assms integral_linear[where a=1] by auto
hoelzl@38656
  1525
hoelzl@38656
  1526
lemma (in measure_space) integral_zero[simp, intro]:
hoelzl@38656
  1527
  shows "integrable (\<lambda>x. 0)"
hoelzl@38656
  1528
  and "integral (\<lambda>x. 0) = 0"
hoelzl@38656
  1529
  unfolding integrable_def integral_def
hoelzl@38656
  1530
  by (auto simp add: borel_measurable_const)
hoelzl@35582
  1531
hoelzl@38656
  1532
lemma (in measure_space) integral_cmult[simp, intro]:
hoelzl@38656
  1533
  assumes "integrable f"
hoelzl@38656
  1534
  shows "integrable (\<lambda>t. a * f t)" (is ?P)
hoelzl@38656
  1535
  and "integral (\<lambda>t. a * f t) = a * integral f" (is ?I)
hoelzl@38656
  1536
proof -
hoelzl@38656
  1537
  have "integrable (\<lambda>t. a * f t) \<and> integral (\<lambda>t. a * f t) = a * integral f"
hoelzl@38656
  1538
  proof (cases rule: le_cases)
hoelzl@38656
  1539
    assume "0 \<le> a" show ?thesis
hoelzl@38656
  1540
      using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
hoelzl@38656
  1541
      by (simp add: integral_zero)
hoelzl@38656
  1542
  next
hoelzl@38656
  1543
    assume "a \<le> 0" hence "0 \<le> - a" by auto
hoelzl@38656
  1544
    have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
hoelzl@38656
  1545
    show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
hoelzl@38656
  1546
        integral_minus(1)[of "\<lambda>t. - a * f t"]
hoelzl@38656
  1547
      unfolding * integral_zero by simp
hoelzl@38656
  1548
  qed
hoelzl@38656
  1549
  thus ?P ?I by auto
hoelzl@35582
  1550
qed
hoelzl@35582
  1551
hoelzl@38656
  1552
lemma (in measure_space) integral_mono:
hoelzl@38656
  1553
  assumes fg: "integrable f" "integrable g"
hoelzl@35582
  1554
  and mono: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
hoelzl@35582
  1555
  shows "integral f \<le> integral g"
hoelzl@38656
  1556
  using fg unfolding integral_def integrable_def diff_minus
hoelzl@38656
  1557
proof (safe intro!: add_mono real_of_pinfreal_mono le_imp_neg_le positive_integral_mono)
hoelzl@38656
  1558
  fix x assume "x \<in> space M" from mono[OF this]
hoelzl@38656
  1559
  show "Real (f x) \<le> Real (g x)" "Real (- g x) \<le> Real (- f x)" by auto
hoelzl@35582
  1560
qed
hoelzl@35582
  1561
hoelzl@38656
  1562
lemma (in measure_space) integral_diff[simp, intro]:
hoelzl@38656
  1563
  assumes f: "integrable f" and g: "integrable g"
hoelzl@38656
  1564
  shows "integrable (\<lambda>t. f t - g t)"
hoelzl@38656
  1565
  and "integral (\<lambda>t. f t - g t) = integral f - integral g"
hoelzl@38656
  1566
  using integral_add[OF f integral_minus(1)[OF g]]
hoelzl@38656
  1567
  unfolding diff_minus integral_minus(2)[OF g]
hoelzl@38656
  1568
  by auto
hoelzl@38656
  1569
hoelzl@38656
  1570
lemma (in measure_space) integral_indicator[simp, intro]:
hoelzl@38656
  1571
  assumes "a \<in> sets M" and "\<mu> a \<noteq> \<omega>"
hoelzl@38656
  1572
  shows "integral (indicator a) = real (\<mu> a)" (is ?int)
hoelzl@38656
  1573
  and "integrable (indicator a)" (is ?able)
hoelzl@35582
  1574
proof -
hoelzl@38656
  1575
  have *:
hoelzl@38656
  1576
    "\<And>A x. Real (indicator A x) = indicator A x"
hoelzl@38656
  1577
    "\<And>A x. Real (- indicator A x) = 0" unfolding indicator_def by auto
hoelzl@38656
  1578
  show ?int ?able
hoelzl@38656
  1579
    using assms unfolding integral_def integrable_def
hoelzl@38656
  1580
    by (auto simp: * positive_integral_indicator borel_measurable_indicator)
hoelzl@35582
  1581
qed
hoelzl@35582
  1582
hoelzl@38656
  1583
lemma (in measure_space) integral_cmul_indicator:
hoelzl@38656
  1584
  assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<omega>"
hoelzl@38656
  1585
  shows "integrable (\<lambda>x. c * indicator A x)" (is ?P)
hoelzl@38656
  1586
  and "integral (\<lambda>x. c * indicator A x) = c * real (\<mu> A)" (is ?I)
hoelzl@38656
  1587
proof -
hoelzl@38656
  1588
  show ?P
hoelzl@38656
  1589
  proof (cases "c = 0")
hoelzl@38656
  1590
    case False with assms show ?thesis by simp
hoelzl@38656
  1591
  qed simp
hoelzl@35582
  1592
hoelzl@38656
  1593
  show ?I
hoelzl@38656
  1594
  proof (cases "c = 0")
hoelzl@38656
  1595
    case False with assms show ?thesis by simp
hoelzl@38656
  1596
  qed simp
hoelzl@38656
  1597
qed
hoelzl@35582
  1598
hoelzl@38656
  1599
lemma (in measure_space) integral_setsum[simp, intro]:
hoelzl@35582
  1600
  assumes "\<And>n. n \<in> S \<Longrightarrow> integrable (f n)"
hoelzl@35582
  1601
  shows "integral (\<lambda>x. \<Sum> i \<in> S. f i x) = (\<Sum> i \<in> S. integral (f i))" (is "?int S")
hoelzl@38656
  1602
    and "integrable (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
hoelzl@35582
  1603
proof -
hoelzl@38656
  1604
  have "?int S \<and> ?I S"
hoelzl@38656
  1605
  proof (cases "finite S")
hoelzl@38656
  1606
    assume "finite S"
hoelzl@38656
  1607
    from this assms show ?thesis by (induct S) simp_all
hoelzl@38656
  1608
  qed simp
hoelzl@35582
  1609
  thus "?int S" and "?I S" by auto
hoelzl@35582
  1610
qed
hoelzl@35582
  1611
hoelzl@36624
  1612
lemma (in measure_space) integrable_abs:
hoelzl@36624
  1613
  assumes "integrable f"
hoelzl@36624
  1614
  shows "integrable (\<lambda> x. \<bar>f x\<bar>)"
hoelzl@36624
  1615
proof -
hoelzl@38656
  1616
  have *:
hoelzl@38656
  1617
    "\<And>x. Real \<bar>f x\<bar> = Real (f x) + Real (- f x)"
hoelzl@38656
  1618
    "\<And>x. Real (- \<bar>f x\<bar>) = 0" by auto
hoelzl@38656
  1619
  have abs: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M" and
hoelzl@38656
  1620
    f: "(\<lambda>x. Real (f x)) \<in> borel_measurable M"
hoelzl@38656
  1621
        "(\<lambda>x. Real (- f x)) \<in> borel_measurable M"
hoelzl@38656
  1622
    using assms unfolding integrable_def by auto
hoelzl@38656
  1623
  from abs assms show ?thesis unfolding integrable_def *
hoelzl@38656
  1624
    using positive_integral_linear[OF f, of 1] by simp
hoelzl@38656
  1625
qed
hoelzl@38656
  1626
hoelzl@38656
  1627
lemma (in measure_space) integrable_bound:
hoelzl@38656
  1628
  assumes "integrable f"
hoelzl@38656
  1629
  and f: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
hoelzl@38656
  1630
    "\<And>x. x \<in> space M \<Longrightarrow> \<bar>g x\<bar> \<le> f x"
hoelzl@38656
  1631
  assumes borel: "g \<in> borel_measurable M"
hoelzl@38656
  1632
  shows "integrable g"
hoelzl@38656
  1633
proof -
hoelzl@38656
  1634
  have "positive_integral (\<lambda>x. Real (g x)) \<le> positive_integral (\<lambda>x. Real \<bar>g x\<bar>)"
hoelzl@38656
  1635
    by (auto intro!: positive_integral_mono)
hoelzl@38656
  1636
  also have "\<dots> \<le> positive_integral (\<lambda>x. Real (f x))"
hoelzl@38656
  1637
    using f by (auto intro!: positive_integral_mono)
hoelzl@38656
  1638
  also have "\<dots> < \<omega>"
hoelzl@38656
  1639
    using `integrable f` unfolding integrable_def by (auto simp: pinfreal_less_\<omega>)
hoelzl@38656
  1640
  finally have pos: "positive_integral (\<lambda>x. Real (g x)) < \<omega>" .
hoelzl@38656
  1641
hoelzl@38656
  1642
  have "positive_integral (\<lambda>x. Real (- g x)) \<le> positive_integral (\<lambda>x. Real (\<bar>g x\<bar>))"
hoelzl@38656
  1643
    by (auto intro!: positive_integral_mono)
hoelzl@38656
  1644
  also have "\<dots> \<le> positive_integral (\<lambda>x. Real (f x))"
hoelzl@38656
  1645
    using f by (auto intro!: positive_integral_mono)
hoelzl@38656
  1646
  also have "\<dots> < \<omega>"
hoelzl@38656
  1647
    using `integrable f` unfolding integrable_def by (auto simp: pinfreal_less_\<omega>)
hoelzl@38656
  1648
  finally have neg: "positive_integral (\<lambda>x. Real (- g x)) < \<omega>" .
hoelzl@38656
  1649
hoelzl@38656
  1650
  from neg pos borel show ?thesis
hoelzl@36624
  1651
    unfolding integrable_def by auto
hoelzl@38656
  1652
qed
hoelzl@38656
  1653
hoelzl@38656
  1654
lemma (in measure_space) integrable_abs_iff:
hoelzl@38656
  1655
  "f \<in> borel_measurable M \<Longrightarrow> integrable (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable f"
hoelzl@38656
  1656
  by (auto intro!: integrable_bound[where g=f] integrable_abs)
hoelzl@38656
  1657
hoelzl@38656
  1658
lemma (in measure_space) integrable_max:
hoelzl@38656
  1659
  assumes int: "integrable f" "integrable g"
hoelzl@38656
  1660
  shows "integrable (\<lambda> x. max (f x) (g x))"
hoelzl@38656
  1661
proof (rule integrable_bound)
hoelzl@38656
  1662
  show "integrable (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
hoelzl@38656
  1663
    using int by (simp add: integrable_abs)
hoelzl@38656
  1664
  show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
hoelzl@38656
  1665
    using int unfolding integrable_def by auto
hoelzl@38656
  1666
next
hoelzl@38656
  1667
  fix x assume "x \<in> space M"
hoelzl@38656
  1668
  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
  1669
    by auto
hoelzl@38656
  1670
qed
hoelzl@38656
  1671
hoelzl@38656
  1672
lemma (in measure_space) integrable_min:
hoelzl@38656
  1673
  assumes int: "integrable f" "integrable g"
hoelzl@38656
  1674
  shows "integrable (\<lambda> x. min (f x) (g x))"
hoelzl@38656
  1675
proof (rule integrable_bound)
hoelzl@38656
  1676
  show "integrable (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
hoelzl@38656
  1677
    using int by (simp add: integrable_abs)
hoelzl@38656
  1678
  show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
hoelzl@38656
  1679
    using int unfolding integrable_def by auto
hoelzl@38656
  1680
next
hoelzl@38656
  1681
  fix x assume "x \<in> space M"
hoelzl@38656
  1682
  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
  1683
    by auto
hoelzl@38656
  1684
qed
hoelzl@38656
  1685
hoelzl@38656
  1686
lemma (in measure_space) integral_triangle_inequality:
hoelzl@38656
  1687
  assumes "integrable f"
hoelzl@38656
  1688
  shows "\<bar>integral f\<bar> \<le> integral (\<lambda>x. \<bar>f x\<bar>)"
hoelzl@38656
  1689
proof -
hoelzl@38656
  1690
  have "\<bar>integral f\<bar> = max (integral f) (- integral f)" by auto
hoelzl@38656
  1691
  also have "\<dots> \<le> integral (\<lambda>x. \<bar>f x\<bar>)"
hoelzl@38656
  1692
      using assms integral_minus(2)[of f, symmetric]
hoelzl@38656
  1693
      by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
hoelzl@38656
  1694
  finally show ?thesis .
hoelzl@36624
  1695
qed
hoelzl@36624
  1696
hoelzl@38656
  1697
lemma (in measure_space) integral_positive:
hoelzl@38656
  1698
  assumes "integrable f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
hoelzl@38656
  1699
  shows "0 \<le> integral f"
hoelzl@38656
  1700
proof -
hoelzl@38656
  1701
  have "0 = integral (\<lambda>x. 0)" by (auto simp: integral_zero)
hoelzl@38656
  1702
  also have "\<dots> \<le> integral f"
hoelzl@38656
  1703
    using assms by (rule integral_mono[OF integral_zero(1)])
hoelzl@38656
  1704
  finally show ?thesis .
hoelzl@38656
  1705
qed
hoelzl@38656
  1706