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(* Author: Robert Himmelmann, TU Muenchen *)
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header {* Lebsegue measure *}
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theory Lebesgue_Measure
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imports Product_Measure Gauge_Measure Complete_Measure
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begin
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subsection {* Standard Cubes *}
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definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where
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"cube n \<equiv> {\<chi>\<chi> i. - real n .. \<chi>\<chi> i. real n}"
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lemma cube_closed[intro]: "closed (cube n)"
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unfolding cube_def by auto
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lemma cube_subset[intro]: "n \<le> N \<Longrightarrow> cube n \<subseteq> cube N"
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by (fastsimp simp: eucl_le[where 'a='a] cube_def)
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lemma cube_subset_iff:
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"cube n \<subseteq> cube N \<longleftrightarrow> n \<le> N"
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proof
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assume subset: "cube n \<subseteq> (cube N::'a set)"
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then have "((\<chi>\<chi> i. real n)::'a) \<in> cube N"
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using DIM_positive[where 'a='a]
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by (fastsimp simp: cube_def eucl_le[where 'a='a])
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then show "n \<le> N"
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by (fastsimp simp: cube_def eucl_le[where 'a='a])
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next
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assume "n \<le> N" then show "cube n \<subseteq> (cube N::'a set)" by (rule cube_subset)
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qed
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lemma ball_subset_cube:"ball (0::'a::ordered_euclidean_space) (real n) \<subseteq> cube n"
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unfolding cube_def subset_eq mem_interval apply safe unfolding euclidean_lambda_beta'
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proof- fix x::'a and i assume as:"x\<in>ball 0 (real n)" "i<DIM('a)"
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thus "- real n \<le> x $$ i" "real n \<ge> x $$ i"
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using component_le_norm[of x i] by(auto simp: dist_norm)
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qed
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lemma mem_big_cube: obtains n where "x \<in> cube n"
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proof- from real_arch_lt[of "norm x"] guess n ..
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thus ?thesis apply-apply(rule that[where n=n])
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apply(rule ball_subset_cube[unfolded subset_eq,rule_format])
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by (auto simp add:dist_norm)
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qed
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lemma Union_inter_cube:"\<Union>{s \<inter> cube n |n. n \<in> UNIV} = s"
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proof safe case goal1
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from mem_big_cube[of x] guess n . note n=this
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show ?case unfolding Union_iff apply(rule_tac x="s \<inter> cube n" in bexI)
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using n goal1 by auto
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qed
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lemma gmeasurable_cube[intro]:"gmeasurable (cube n)"
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unfolding cube_def by auto
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lemma gmeasure_le_inter_cube[intro]: fixes s::"'a::ordered_euclidean_space set"
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assumes "gmeasurable (s \<inter> cube n)" shows "gmeasure (s \<inter> cube n) \<le> gmeasure (cube n::'a set)"
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apply(rule has_gmeasure_subset[of "s\<inter>cube n" _ "cube n"])
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unfolding has_gmeasure_measure[THEN sym] using assms by auto
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lemma has_gmeasure_cube[intro]: "(cube n::('a::ordered_euclidean_space) set)
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has_gmeasure ((2 * real n) ^ (DIM('a)))"
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proof-
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have "content {\<chi>\<chi> i. - real n..(\<chi>\<chi> i. real n)::'a} = (2 * real n) ^ (DIM('a))"
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apply(subst content_closed_interval) defer
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by (auto simp add:setprod_constant)
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thus ?thesis unfolding cube_def
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using has_gmeasure_interval(1)[of "(\<chi>\<chi> i. - real n)::'a" "(\<chi>\<chi> i. real n)::'a"]
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by auto
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qed
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lemma gmeasure_cube_eq[simp]:
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"gmeasure (cube n::('a::ordered_euclidean_space) set) = (2 * real n) ^ DIM('a)"
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by (intro measure_unique) auto
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lemma gmeasure_cube_ge_n: "gmeasure (cube n::('a::ordered_euclidean_space) set) \<ge> real n"
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proof cases
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assume "n = 0" then show ?thesis by simp
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next
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assume "n \<noteq> 0"
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have "real n \<le> (2 * real n)^1" by simp
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also have "\<dots> \<le> (2 * real n)^DIM('a)"
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using DIM_positive[where 'a='a] `n \<noteq> 0`
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by (intro power_increasing) auto
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also have "\<dots> = gmeasure (cube n::'a set)" by simp
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finally show ?thesis .
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qed
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lemma gmeasure_setsum:
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assumes "finite A" and "\<And>s t. s \<in> A \<Longrightarrow> t \<in> A \<Longrightarrow> s \<noteq> t \<Longrightarrow> f s \<inter> f t = {}"
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and "\<And>i. i \<in> A \<Longrightarrow> gmeasurable (f i)"
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shows "gmeasure (\<Union>i\<in>A. f i) = (\<Sum>i\<in>A. gmeasure (f i))"
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proof -
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have "gmeasure (\<Union>i\<in>A. f i) = gmeasure (\<Union>f ` A)" by auto
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also have "\<dots> = setsum gmeasure (f ` A)" using assms
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proof (intro measure_negligible_unions)
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fix X Y assume "X \<in> f`A" "Y \<in> f`A" "X \<noteq> Y"
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then have "X \<inter> Y = {}" using assms by auto
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then show "negligible (X \<inter> Y)" by auto
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qed auto
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also have "\<dots> = setsum gmeasure (f ` A - {{}})"
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using assms by (intro setsum_mono_zero_cong_right) auto
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also have "\<dots> = (\<Sum>i\<in>A - {i. f i = {}}. gmeasure (f i))"
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proof (intro setsum_reindex_cong inj_onI)
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fix s t assume *: "s \<in> A - {i. f i = {}}" "t \<in> A - {i. f i = {}}" "f s = f t"
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show "s = t"
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proof (rule ccontr)
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assume "s \<noteq> t" with assms(2)[of s t] * show False by auto
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qed
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qed auto
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also have "\<dots> = (\<Sum>i\<in>A. gmeasure (f i))"
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using assms by (intro setsum_mono_zero_cong_left) auto
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finally show ?thesis .
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qed
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lemma gmeasurable_finite_UNION[intro]:
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assumes "\<And>i. i \<in> S \<Longrightarrow> gmeasurable (A i)" "finite S"
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shows "gmeasurable (\<Union>i\<in>S. A i)"
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unfolding UNION_eq_Union_image using assms
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by (intro gmeasurable_finite_unions) auto
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lemma gmeasurable_countable_UNION[intro]:
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fixes A :: "nat \<Rightarrow> ('a::ordered_euclidean_space) set"
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assumes measurable: "\<And>i. gmeasurable (A i)"
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and finite: "\<And>n. gmeasure (UNION {.. n} A) \<le> B"
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shows "gmeasurable (\<Union>i. A i)"
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proof -
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have *: "\<And>n. \<Union>{A k |k. k \<le> n} = (\<Union>i\<le>n. A i)"
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"(\<Union>{A n |n. n \<in> UNIV}) = (\<Union>i. A i)" by auto
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show ?thesis
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by (rule gmeasurable_countable_unions_strong[of A B, unfolded *, OF assms])
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qed
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subsection {* Measurability *}
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definition lebesgue :: "'a::ordered_euclidean_space algebra" where
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"lebesgue = \<lparr> space = UNIV, sets = {A. \<forall>n. gmeasurable (A \<inter> cube n)} \<rparr>"
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lemma space_lebesgue[simp]:"space lebesgue = UNIV"
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unfolding lebesgue_def by auto
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lemma lebesgueD[dest]: assumes "S \<in> sets lebesgue"
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shows "\<And>n. gmeasurable (S \<inter> cube n)"
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using assms unfolding lebesgue_def by auto
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lemma lebesgueI[intro]: assumes "gmeasurable S"
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shows "S \<in> sets lebesgue" unfolding lebesgue_def cube_def
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using assms gmeasurable_interval by auto
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lemma lebesgueI2: "(\<And>n. gmeasurable (S \<inter> cube n)) \<Longrightarrow> S \<in> sets lebesgue"
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using assms unfolding lebesgue_def by auto
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interpretation lebesgue: sigma_algebra lebesgue
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proof
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show "sets lebesgue \<subseteq> Pow (space lebesgue)"
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unfolding lebesgue_def by auto
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show "{} \<in> sets lebesgue"
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using gmeasurable_empty by auto
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{ fix A B :: "'a set" assume "A \<in> sets lebesgue" "B \<in> sets lebesgue"
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then show "A \<union> B \<in> sets lebesgue"
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by (auto intro: gmeasurable_union simp: lebesgue_def Int_Un_distrib2) }
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{ fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets lebesgue"
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show "(\<Union>i. A i) \<in> sets lebesgue"
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proof (rule lebesgueI2)
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fix n show "gmeasurable ((\<Union>i. A i) \<inter> cube n)" unfolding UN_extend_simps
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using A
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by (intro gmeasurable_countable_UNION[where B="gmeasure (cube n::'a set)"])
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(auto intro!: measure_subset gmeasure_setsum simp: UN_extend_simps simp del: gmeasure_cube_eq UN_simps)
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qed }
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{ fix A assume A: "A \<in> sets lebesgue" show "space lebesgue - A \<in> sets lebesgue"
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proof (rule lebesgueI2)
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fix n
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have *: "(space lebesgue - A) \<inter> cube n = cube n - (A \<inter> cube n)"
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unfolding lebesgue_def by auto
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show "gmeasurable ((space lebesgue - A) \<inter> cube n)" unfolding *
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using A by (auto intro!: gmeasurable_diff)
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qed }
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qed
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lemma lebesgueI_borel[intro, simp]: fixes s::"'a::ordered_euclidean_space set"
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assumes "s \<in> sets borel" shows "s \<in> sets lebesgue"
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proof- let ?S = "range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)})"
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have *:"?S \<subseteq> sets lebesgue" by auto
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have "s \<in> sigma_sets UNIV ?S" using assms
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unfolding borel_eq_atLeastAtMost by (simp add: sigma_def)
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thus ?thesis
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using lebesgue.sigma_subset[of "\<lparr> space = UNIV, sets = ?S\<rparr>", simplified, OF *]
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by (auto simp: sigma_def)
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qed
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lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set"
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assumes "negligible s" shows "s \<in> sets lebesgue"
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proof (rule lebesgueI2)
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fix n
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have *:"\<And>x. (if x \<in> cube n then indicator s x else 0) = (if x \<in> s \<inter> cube n then 1 else 0)"
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unfolding indicator_def_raw by auto
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note assms[unfolded negligible_def,rule_format,of "(\<chi>\<chi> i. - real n)::'a" "\<chi>\<chi> i. real n"]
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thus "gmeasurable (s \<inter> cube n)" apply-apply(rule gmeasurableI[of _ 0]) unfolding has_gmeasure_def
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apply(subst(asm) has_integral_restrict_univ[THEN sym]) unfolding cube_def[symmetric]
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apply(subst has_integral_restrict_univ[THEN sym]) unfolding * .
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qed
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section {* The Lebesgue Measure *}
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definition "lmeasure A = (SUP n. Real (gmeasure (A \<inter> cube n)))"
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lemma lmeasure_eq_0: assumes "negligible S" shows "lmeasure S = 0"
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proof -
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from lebesgueI_negligible[OF assms]
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have "\<And>n. gmeasurable (S \<inter> cube n)" by auto
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from gmeasurable_measure_eq_0[OF this]
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have "\<And>n. gmeasure (S \<inter> cube n) = 0" using assms by auto
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then show ?thesis unfolding lmeasure_def by simp
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qed
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lemma lmeasure_iff_LIMSEQ:
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assumes "A \<in> sets lebesgue" "0 \<le> m"
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shows "lmeasure A = Real m \<longleftrightarrow> (\<lambda>n. (gmeasure (A \<inter> cube n))) ----> m"
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unfolding lmeasure_def using assms cube_subset[where 'a='a]
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by (intro SUP_eq_LIMSEQ monoI measure_subset) force+
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interpretation lebesgue: measure_space lebesgue lmeasure
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proof
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show "lmeasure {} = 0"
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by (auto intro!: lmeasure_eq_0)
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show "countably_additive lebesgue lmeasure"
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proof (unfold countably_additive_def, intro allI impI conjI)
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fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets lebesgue" "disjoint_family A"
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then have A: "\<And>i. A i \<in> sets lebesgue" by auto
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show "(\<Sum>\<^isub>\<infinity>n. lmeasure (A n)) = lmeasure (\<Union>i. A i)" unfolding lmeasure_def
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proof (subst psuminf_SUP_eq)
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{ fix i n
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have "gmeasure (A i \<inter> cube n) \<le> gmeasure (A i \<inter> cube (Suc n))"
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using A cube_subset[of n "Suc n"] by (auto intro!: measure_subset)
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then show "Real (gmeasure (A i \<inter> cube n)) \<le> Real (gmeasure (A i \<inter> cube (Suc n)))"
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by auto }
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show "(SUP n. \<Sum>\<^isub>\<infinity>i. Real (gmeasure (A i \<inter> cube n))) = (SUP n. Real (gmeasure ((\<Union>i. A i) \<inter> cube n)))"
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proof (intro arg_cong[where f="SUPR UNIV"] ext)
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fix n
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have sums: "(\<lambda>i. gmeasure (A i \<inter> cube n)) sums gmeasure (\<Union>{A i \<inter> cube n |i. i \<in> UNIV})"
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proof (rule has_gmeasure_countable_negligible_unions(2))
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fix i show "gmeasurable (A i \<inter> cube n)" using A by auto
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next
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fix i m :: nat assume "m \<noteq> i"
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then have "A m \<inter> cube n \<inter> (A i \<inter> cube n) = {}"
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using `disjoint_family A` unfolding disjoint_family_on_def by auto
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then show "negligible (A m \<inter> cube n \<inter> (A i \<inter> cube n))" by auto
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next
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fix i
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have "(\<Sum>k = 0..i. gmeasure (A k \<inter> cube n)) = gmeasure (\<Union>k\<le>i . A k \<inter> cube n)"
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unfolding atLeast0AtMost using A
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proof (intro gmeasure_setsum[symmetric])
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fix s t :: nat assume "s \<noteq> t" then have "A t \<inter> A s = {}"
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using `disjoint_family A` unfolding disjoint_family_on_def by auto
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then show "A s \<inter> cube n \<inter> (A t \<inter> cube n) = {}" by auto
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qed auto
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also have "\<dots> \<le> gmeasure (cube n :: 'b set)" using A
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by (intro measure_subset gmeasurable_finite_UNION) auto
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finally show "(\<Sum>k = 0..i. gmeasure (A k \<inter> cube n)) \<le> gmeasure (cube n :: 'b set)" .
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qed
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show "(\<Sum>\<^isub>\<infinity>i. Real (gmeasure (A i \<inter> cube n))) = Real (gmeasure ((\<Union>i. A i) \<inter> cube n))"
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unfolding psuminf_def
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apply (subst setsum_Real)
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apply (simp add: measure_pos_le)
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proof (rule SUP_eq_LIMSEQ[THEN iffD2])
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have "(\<Union>{A i \<inter> cube n |i. i \<in> UNIV}) = (\<Union>i. A i) \<inter> cube n" by auto
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with sums show "(\<lambda>i. \<Sum>k<i. gmeasure (A k \<inter> cube n)) ----> gmeasure ((\<Union>i. A i) \<inter> cube n)"
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unfolding sums_def atLeast0LessThan by simp
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qed (auto intro!: monoI setsum_nonneg setsum_mono2)
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qed
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qed
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qed
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qed
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lemma lmeasure_finite_has_gmeasure: assumes "s \<in> sets lebesgue" "lmeasure s = Real m" "0 \<le> m"
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shows "s has_gmeasure m"
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proof-
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have *:"(\<lambda>n. (gmeasure (s \<inter> cube n))) ----> m"
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using `lmeasure s = Real m` unfolding lmeasure_iff_LIMSEQ[OF `s \<in> sets lebesgue` `0 \<le> m`] .
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have s: "\<And>n. gmeasurable (s \<inter> cube n)" using assms by auto
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have "(\<lambda>x. if x \<in> s then 1 else (0::real)) integrable_on UNIV \<and>
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(\<lambda>k. integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)))
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----> integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)"
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proof(rule monotone_convergence_increasing)
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have "lmeasure s \<le> Real m" using `lmeasure s = Real m` by simp
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then have "\<forall>n. gmeasure (s \<inter> cube n) \<le> m"
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unfolding lmeasure_def complete_lattice_class.SUP_le_iff
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using `0 \<le> m` by (auto simp: measure_pos_le)
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thus *:"bounded {integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)) |k. True}"
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unfolding integral_measure_univ[OF s] bounded_def apply-
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apply(rule_tac x=0 in exI,rule_tac x=m in exI) unfolding dist_real_def
|
|
291 |
by (auto simp: measure_pos_le)
|
|
292 |
show "\<forall>k. (\<lambda>x. if x \<in> s \<inter> cube k then (1::real) else 0) integrable_on UNIV"
|
|
293 |
unfolding integrable_restrict_univ
|
40859
|
294 |
using s unfolding gmeasurable_def has_gmeasure_def by auto
|
38656
|
295 |
have *:"\<And>n. n \<le> Suc n" by auto
|
|
296 |
show "\<forall>k. \<forall>x\<in>UNIV. (if x \<in> s \<inter> cube k then 1 else 0) \<le> (if x \<in> s \<inter> cube (Suc k) then 1 else (0::real))"
|
|
297 |
using cube_subset[OF *] by fastsimp
|
|
298 |
show "\<forall>x\<in>UNIV. (\<lambda>k. if x \<in> s \<inter> cube k then 1 else 0) ----> (if x \<in> s then 1 else (0::real))"
|
40859
|
299 |
unfolding Lim_sequentially
|
38656
|
300 |
proof safe case goal1 from real_arch_lt[of "norm x"] guess N .. note N = this
|
|
301 |
show ?case apply(rule_tac x=N in exI)
|
|
302 |
proof safe case goal1
|
|
303 |
have "x \<in> cube n" using cube_subset[OF goal1] N
|
40859
|
304 |
using ball_subset_cube[of N] by(auto simp: dist_norm)
|
38656
|
305 |
thus ?case using `e>0` by auto
|
|
306 |
qed
|
|
307 |
qed
|
|
308 |
qed note ** = conjunctD2[OF this]
|
|
309 |
hence *:"m = integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)" apply-
|
40859
|
310 |
apply(rule LIMSEQ_unique[OF _ **(2)]) unfolding measure_integral_univ[THEN sym,OF s] using * .
|
38656
|
311 |
show ?thesis unfolding has_gmeasure * apply(rule integrable_integral) using ** by auto
|
|
312 |
qed
|
|
313 |
|
40859
|
314 |
lemma lmeasure_finite_gmeasurable: assumes "s \<in> sets lebesgue" "lmeasure s \<noteq> \<omega>"
|
38656
|
315 |
shows "gmeasurable s"
|
40859
|
316 |
proof (cases "lmeasure s")
|
|
317 |
case (preal m) from lmeasure_finite_has_gmeasure[OF `s \<in> sets lebesgue` this]
|
|
318 |
show ?thesis unfolding gmeasurable_def by auto
|
|
319 |
qed (insert assms, auto)
|
38656
|
320 |
|
40859
|
321 |
lemma has_gmeasure_lmeasure: assumes "s has_gmeasure m"
|
|
322 |
shows "lmeasure s = Real m"
|
|
323 |
proof-
|
|
324 |
have gmea:"gmeasurable s" using assms by auto
|
|
325 |
then have s: "s \<in> sets lebesgue" by auto
|
38656
|
326 |
have m:"m \<ge> 0" using assms by auto
|
|
327 |
have *:"m = gmeasure (\<Union>{s \<inter> cube n |n. n \<in> UNIV})" unfolding Union_inter_cube
|
|
328 |
using assms by(rule measure_unique[THEN sym])
|
40859
|
329 |
show ?thesis
|
|
330 |
unfolding lmeasure_iff_LIMSEQ[OF s `0 \<le> m`] unfolding *
|
38656
|
331 |
apply(rule gmeasurable_nested_unions[THEN conjunct2, where B1="gmeasure s"])
|
|
332 |
proof- fix n::nat show *:"gmeasurable (s \<inter> cube n)"
|
|
333 |
using gmeasurable_inter[OF gmea gmeasurable_cube] .
|
|
334 |
show "gmeasure (s \<inter> cube n) \<le> gmeasure s" apply(rule measure_subset)
|
|
335 |
apply(rule * gmea)+ by auto
|
|
336 |
show "s \<inter> cube n \<subseteq> s \<inter> cube (Suc n)" using cube_subset[of n "Suc n"] by auto
|
|
337 |
qed
|
|
338 |
qed
|
|
339 |
|
40859
|
340 |
lemma has_gmeasure_iff_lmeasure:
|
|
341 |
"A has_gmeasure m \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m)"
|
|
342 |
proof
|
|
343 |
assume "A has_gmeasure m"
|
|
344 |
with has_gmeasure_lmeasure[OF this]
|
|
345 |
have "gmeasurable A" "0 \<le> m" "lmeasure A = Real m" by auto
|
|
346 |
then show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m" by auto
|
|
347 |
next
|
|
348 |
assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m"
|
|
349 |
then show "A has_gmeasure m" by (intro lmeasure_finite_has_gmeasure) auto
|
38656
|
350 |
qed
|
|
351 |
|
40859
|
352 |
lemma gmeasure_lmeasure: assumes "gmeasurable s" shows "lmeasure s = Real (gmeasure s)"
|
|
353 |
proof -
|
|
354 |
note has_gmeasure_measureI[OF assms]
|
|
355 |
note has_gmeasure_lmeasure[OF this]
|
|
356 |
thus ?thesis .
|
|
357 |
qed
|
38656
|
358 |
|
|
359 |
lemma lebesgue_simple_function_indicator:
|
|
360 |
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
|
|
361 |
assumes f:"lebesgue.simple_function f"
|
|
362 |
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))"
|
|
363 |
apply(rule,subst lebesgue.simple_function_indicator_representation[OF f]) by auto
|
|
364 |
|
|
365 |
lemma lmeasure_gmeasure:
|
|
366 |
"gmeasurable s \<Longrightarrow> gmeasure s = real (lmeasure s)"
|
40859
|
367 |
by (subst gmeasure_lmeasure) auto
|
38656
|
368 |
|
|
369 |
lemma lmeasure_finite: assumes "gmeasurable s" shows "lmeasure s \<noteq> \<omega>"
|
|
370 |
using gmeasure_lmeasure[OF assms] by auto
|
|
371 |
|
40859
|
372 |
lemma negligible_iff_lebesgue_null_sets:
|
|
373 |
"negligible A \<longleftrightarrow> A \<in> lebesgue.null_sets"
|
|
374 |
proof
|
|
375 |
assume "negligible A"
|
|
376 |
from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0]
|
|
377 |
show "A \<in> lebesgue.null_sets" by auto
|
|
378 |
next
|
|
379 |
assume A: "A \<in> lebesgue.null_sets"
|
|
380 |
then have *:"gmeasurable A" using lmeasure_finite_gmeasurable[of A] by auto
|
|
381 |
show "negligible A"
|
|
382 |
unfolding gmeasurable_measure_eq_0[OF *, symmetric]
|
|
383 |
unfolding lmeasure_gmeasure[OF *] using A by auto
|
|
384 |
qed
|
|
385 |
|
|
386 |
lemma
|
|
387 |
fixes a b ::"'a::ordered_euclidean_space"
|
|
388 |
shows lmeasure_atLeastAtMost[simp]: "lmeasure {a..b} = Real (content {a..b})"
|
|
389 |
and lmeasure_greaterThanLessThan[simp]: "lmeasure {a <..< b} = Real (content {a..b})"
|
|
390 |
using has_gmeasure_interval[of a b] by (auto intro!: has_gmeasure_lmeasure)
|
|
391 |
|
|
392 |
lemma lmeasure_cube:
|
|
393 |
"lmeasure (cube n::('a::ordered_euclidean_space) set) = (Real ((2 * real n) ^ (DIM('a))))"
|
|
394 |
by (intro has_gmeasure_lmeasure) auto
|
|
395 |
|
|
396 |
lemma lmeasure_UNIV[intro]: "lmeasure UNIV = \<omega>"
|
|
397 |
unfolding lmeasure_def SUP_\<omega>
|
|
398 |
proof (intro allI impI)
|
|
399 |
fix x assume "x < \<omega>"
|
|
400 |
then obtain r where r: "x = Real r" "0 \<le> r" by (cases x) auto
|
|
401 |
then obtain n where n: "r < of_nat n" using ex_less_of_nat by auto
|
|
402 |
show "\<exists>i\<in>UNIV. x < Real (gmeasure (UNIV \<inter> cube i))"
|
|
403 |
proof (intro bexI[of _ n])
|
|
404 |
have "x < Real (of_nat n)" using n r by auto
|
|
405 |
also have "Real (of_nat n) \<le> Real (gmeasure (UNIV \<inter> cube n))"
|
|
406 |
using gmeasure_cube_ge_n[of n] by (auto simp: real_eq_of_nat[symmetric])
|
|
407 |
finally show "x < Real (gmeasure (UNIV \<inter> cube n))" .
|
|
408 |
qed auto
|
|
409 |
qed
|
|
410 |
|
|
411 |
lemma atLeastAtMost_singleton_euclidean[simp]:
|
|
412 |
fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
|
|
413 |
by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a])
|
|
414 |
|
|
415 |
lemma content_singleton[simp]: "content {a} = 0"
|
|
416 |
proof -
|
|
417 |
have "content {a .. a} = 0"
|
|
418 |
by (subst content_closed_interval) auto
|
|
419 |
then show ?thesis by simp
|
|
420 |
qed
|
|
421 |
|
|
422 |
lemma lmeasure_singleton[simp]:
|
|
423 |
fixes a :: "'a::ordered_euclidean_space" shows "lmeasure {a} = 0"
|
|
424 |
using has_gmeasure_interval[of a a] unfolding zero_pinfreal_def
|
|
425 |
by (intro has_gmeasure_lmeasure)
|
|
426 |
(simp add: content_closed_interval DIM_positive)
|
|
427 |
|
|
428 |
declare content_real[simp]
|
|
429 |
|
|
430 |
lemma
|
|
431 |
fixes a b :: real
|
|
432 |
shows lmeasure_real_greaterThanAtMost[simp]:
|
|
433 |
"lmeasure {a <.. b} = Real (if a \<le> b then b - a else 0)"
|
|
434 |
proof cases
|
|
435 |
assume "a < b"
|
|
436 |
then have "lmeasure {a <.. b} = lmeasure {a <..< b} + lmeasure {b}"
|
|
437 |
by (subst lebesgue.measure_additive)
|
|
438 |
(auto intro!: lebesgueI_borel arg_cong[where f=lmeasure])
|
|
439 |
then show ?thesis by auto
|
|
440 |
qed auto
|
|
441 |
|
|
442 |
lemma
|
|
443 |
fixes a b :: real
|
|
444 |
shows lmeasure_real_atLeastLessThan[simp]:
|
|
445 |
"lmeasure {a ..< b} = Real (if a \<le> b then b - a else 0)" (is ?eqlt)
|
|
446 |
proof cases
|
|
447 |
assume "a < b"
|
|
448 |
then have "lmeasure {a ..< b} = lmeasure {a} + lmeasure {a <..< b}"
|
|
449 |
by (subst lebesgue.measure_additive)
|
|
450 |
(auto intro!: lebesgueI_borel arg_cong[where f=lmeasure])
|
|
451 |
then show ?thesis by auto
|
|
452 |
qed auto
|
|
453 |
|
|
454 |
interpretation borel: measure_space borel lmeasure
|
|
455 |
proof
|
|
456 |
show "countably_additive borel lmeasure"
|
|
457 |
using lebesgue.ca unfolding countably_additive_def
|
|
458 |
apply safe apply (erule_tac x=A in allE) by auto
|
|
459 |
qed auto
|
|
460 |
|
|
461 |
interpretation borel: sigma_finite_measure borel lmeasure
|
|
462 |
proof (default, intro conjI exI[of _ "\<lambda>n. cube n"])
|
|
463 |
show "range cube \<subseteq> sets borel" by (auto intro: borel_closed)
|
|
464 |
{ fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto }
|
|
465 |
thus "(\<Union>i. cube i) = space borel" by auto
|
|
466 |
show "\<forall>i. lmeasure (cube i) \<noteq> \<omega>" unfolding lmeasure_cube by auto
|
|
467 |
qed
|
|
468 |
|
|
469 |
interpretation lebesgue: sigma_finite_measure lebesgue lmeasure
|
|
470 |
proof
|
|
471 |
from borel.sigma_finite guess A ..
|
|
472 |
moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast
|
|
473 |
ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lmeasure (A i) \<noteq> \<omega>)"
|
|
474 |
by auto
|
|
475 |
qed
|
|
476 |
|
|
477 |
lemma simple_function_has_integral:
|
|
478 |
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
|
|
479 |
assumes f:"lebesgue.simple_function f"
|
|
480 |
and f':"\<forall>x. f x \<noteq> \<omega>"
|
|
481 |
and om:"\<forall>x\<in>range f. lmeasure (f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
|
|
482 |
shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
|
|
483 |
unfolding lebesgue.simple_integral_def
|
|
484 |
apply(subst lebesgue_simple_function_indicator[OF f])
|
|
485 |
proof- case goal1
|
|
486 |
have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f -` {y}) x \<noteq> \<omega>"
|
|
487 |
"\<forall>x\<in>range f. x * lmeasure (f -` {x} \<inter> UNIV) \<noteq> \<omega>"
|
|
488 |
using f' om unfolding indicator_def by auto
|
|
489 |
show ?case unfolding space_lebesgue real_of_pinfreal_setsum'[OF *(1),THEN sym]
|
|
490 |
unfolding real_of_pinfreal_setsum'[OF *(2),THEN sym]
|
|
491 |
unfolding real_of_pinfreal_setsum space_lebesgue
|
|
492 |
apply(rule has_integral_setsum)
|
|
493 |
proof safe show "finite (range f)" using f by (auto dest: lebesgue.simple_functionD)
|
|
494 |
fix y::'a show "((\<lambda>x. real (f y * indicator (f -` {f y}) x)) has_integral
|
|
495 |
real (f y * lmeasure (f -` {f y} \<inter> UNIV))) UNIV"
|
|
496 |
proof(cases "f y = 0") case False
|
|
497 |
have mea:"gmeasurable (f -` {f y})" apply(rule lmeasure_finite_gmeasurable)
|
|
498 |
using assms unfolding lebesgue.simple_function_def using False by auto
|
|
499 |
have *:"\<And>x. real (indicator (f -` {f y}) x::pinfreal) = (if x \<in> f -` {f y} then 1 else 0)" by auto
|
|
500 |
show ?thesis unfolding real_of_pinfreal_mult[THEN sym]
|
|
501 |
apply(rule has_integral_cmul[where 'b=real, unfolded real_scaleR_def])
|
|
502 |
unfolding Int_UNIV_right lmeasure_gmeasure[OF mea,THEN sym]
|
|
503 |
unfolding measure_integral_univ[OF mea] * apply(rule integrable_integral)
|
|
504 |
unfolding gmeasurable_integrable[THEN sym] using mea .
|
|
505 |
qed auto
|
|
506 |
qed qed
|
|
507 |
|
|
508 |
lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
|
|
509 |
unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
|
|
510 |
using assms by auto
|
|
511 |
|
|
512 |
lemma simple_function_has_integral':
|
|
513 |
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pinfreal"
|
|
514 |
assumes f:"lebesgue.simple_function f"
|
|
515 |
and i: "lebesgue.simple_integral f \<noteq> \<omega>"
|
|
516 |
shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV"
|
|
517 |
proof- let ?f = "\<lambda>x. if f x = \<omega> then 0 else f x"
|
|
518 |
{ fix x have "real (f x) = real (?f x)" by (cases "f x") auto } note * = this
|
|
519 |
have **:"{x. f x \<noteq> ?f x} = f -` {\<omega>}" by auto
|
|
520 |
have **:"lmeasure {x\<in>space lebesgue. f x \<noteq> ?f x} = 0"
|
|
521 |
using lebesgue.simple_integral_omega[OF assms] by(auto simp add:**)
|
|
522 |
show ?thesis apply(subst lebesgue.simple_integral_cong'[OF f _ **])
|
|
523 |
apply(rule lebesgue.simple_function_compose1[OF f])
|
|
524 |
unfolding * defer apply(rule simple_function_has_integral)
|
|
525 |
proof-
|
|
526 |
show "lebesgue.simple_function ?f"
|
|
527 |
using lebesgue.simple_function_compose1[OF f] .
|
|
528 |
show "\<forall>x. ?f x \<noteq> \<omega>" by auto
|
|
529 |
show "\<forall>x\<in>range ?f. lmeasure (?f -` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0"
|
|
530 |
proof (safe, simp, safe, rule ccontr)
|
|
531 |
fix y assume "f y \<noteq> \<omega>" "f y \<noteq> 0"
|
|
532 |
hence "(\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y} = f -` {f y}"
|
|
533 |
by (auto split: split_if_asm)
|
|
534 |
moreover assume "lmeasure ((\<lambda>x. if f x = \<omega> then 0 else f x) -` {if f y = \<omega> then 0 else f y}) = \<omega>"
|
|
535 |
ultimately have "lmeasure (f -` {f y}) = \<omega>" by simp
|
|
536 |
moreover
|
|
537 |
have "f y * lmeasure (f -` {f y}) \<noteq> \<omega>" using i f
|
|
538 |
unfolding lebesgue.simple_integral_def setsum_\<omega> lebesgue.simple_function_def
|
|
539 |
by auto
|
|
540 |
ultimately have "f y = 0" by (auto split: split_if_asm)
|
|
541 |
then show False using `f y \<noteq> 0` by simp
|
|
542 |
qed
|
|
543 |
qed
|
|
544 |
qed
|
|
545 |
|
|
546 |
lemma (in measure_space) positive_integral_monotone_convergence:
|
|
547 |
fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pinfreal"
|
|
548 |
assumes i: "\<And>i. f i \<in> borel_measurable M" and mono: "\<And>x. mono (\<lambda>n. f n x)"
|
|
549 |
and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
|
|
550 |
shows "u \<in> borel_measurable M"
|
|
551 |
and "(\<lambda>i. positive_integral (f i)) ----> positive_integral u" (is ?ilim)
|
|
552 |
proof -
|
|
553 |
from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u]
|
|
554 |
show ?ilim using mono lim i by auto
|
|
555 |
have "(SUP i. f i) = u" using mono lim SUP_Lim_pinfreal
|
|
556 |
unfolding fun_eq_iff SUPR_fun_expand mono_def by auto
|
|
557 |
moreover have "(SUP i. f i) \<in> borel_measurable M"
|
|
558 |
using i by (rule borel_measurable_SUP)
|
|
559 |
ultimately show "u \<in> borel_measurable M" by simp
|
|
560 |
qed
|
|
561 |
|
|
562 |
lemma positive_integral_has_integral:
|
|
563 |
fixes f::"'a::ordered_euclidean_space => pinfreal"
|
|
564 |
assumes f:"f \<in> borel_measurable lebesgue"
|
|
565 |
and int_om:"lebesgue.positive_integral f \<noteq> \<omega>"
|
|
566 |
and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *)
|
|
567 |
shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.positive_integral f))) UNIV"
|
|
568 |
proof- let ?i = "lebesgue.positive_integral f"
|
|
569 |
from lebesgue.borel_measurable_implies_simple_function_sequence[OF f]
|
|
570 |
guess u .. note conjunctD2[OF this,rule_format] note u = conjunctD2[OF this(1)] this(2)
|
|
571 |
let ?u = "\<lambda>i x. real (u i x)" and ?f = "\<lambda>x. real (f x)"
|
|
572 |
have u_simple:"\<And>k. lebesgue.simple_integral (u k) = lebesgue.positive_integral (u k)"
|
|
573 |
apply(subst lebesgue.positive_integral_eq_simple_integral[THEN sym,OF u(1)]) ..
|
|
574 |
have int_u_le:"\<And>k. lebesgue.simple_integral (u k) \<le> lebesgue.positive_integral f"
|
|
575 |
unfolding u_simple apply(rule lebesgue.positive_integral_mono)
|
|
576 |
using isoton_Sup[OF u(3)] unfolding le_fun_def by auto
|
|
577 |
have u_int_om:"\<And>i. lebesgue.simple_integral (u i) \<noteq> \<omega>"
|
|
578 |
proof- case goal1 thus ?case using int_u_le[of i] int_om by auto qed
|
|
579 |
|
|
580 |
note u_int = simple_function_has_integral'[OF u(1) this]
|
|
581 |
have "(\<lambda>x. real (f x)) integrable_on UNIV \<and>
|
|
582 |
(\<lambda>k. Integration.integral UNIV (\<lambda>x. real (u k x))) ----> Integration.integral UNIV (\<lambda>x. real (f x))"
|
|
583 |
apply(rule monotone_convergence_increasing) apply(rule,rule,rule u_int)
|
|
584 |
proof safe case goal1 show ?case apply(rule real_of_pinfreal_mono) using u(2,3) by auto
|
|
585 |
next case goal2 show ?case using u(3) apply(subst lim_Real[THEN sym])
|
|
586 |
prefer 3 apply(subst Real_real') defer apply(subst Real_real')
|
|
587 |
using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] using f_om u by auto
|
|
588 |
next case goal3
|
|
589 |
show ?case apply(rule bounded_realI[where B="real (lebesgue.positive_integral f)"])
|
|
590 |
apply safe apply(subst abs_of_nonneg) apply(rule integral_nonneg,rule) apply(rule u_int)
|
|
591 |
unfolding integral_unique[OF u_int] defer apply(rule real_of_pinfreal_mono[OF _ int_u_le])
|
|
592 |
using u int_om by auto
|
|
593 |
qed note int = conjunctD2[OF this]
|
|
594 |
|
|
595 |
have "(\<lambda>i. lebesgue.simple_integral (u i)) ----> ?i" unfolding u_simple
|
|
596 |
apply(rule lebesgue.positive_integral_monotone_convergence(2))
|
|
597 |
apply(rule lebesgue.borel_measurable_simple_function[OF u(1)])
|
|
598 |
using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] by auto
|
|
599 |
hence "(\<lambda>i. real (lebesgue.simple_integral (u i))) ----> real ?i" apply-
|
|
600 |
apply(subst lim_Real[THEN sym]) prefer 3
|
|
601 |
apply(subst Real_real') defer apply(subst Real_real')
|
|
602 |
using u f_om int_om u_int_om by auto
|
|
603 |
note * = LIMSEQ_unique[OF this int(2)[unfolded integral_unique[OF u_int]]]
|
|
604 |
show ?thesis unfolding * by(rule integrable_integral[OF int(1)])
|
|
605 |
qed
|
|
606 |
|
|
607 |
lemma lebesgue_integral_has_integral:
|
|
608 |
fixes f::"'a::ordered_euclidean_space => real"
|
|
609 |
assumes f:"lebesgue.integrable f"
|
|
610 |
shows "(f has_integral (lebesgue.integral f)) UNIV"
|
|
611 |
proof- let ?n = "\<lambda>x. - min (f x) 0" and ?p = "\<lambda>x. max (f x) 0"
|
|
612 |
have *:"f = (\<lambda>x. ?p x - ?n x)" apply rule by auto
|
|
613 |
note f = lebesgue.integrableD[OF f]
|
|
614 |
show ?thesis unfolding lebesgue.integral_def apply(subst *)
|
|
615 |
proof(rule has_integral_sub) case goal1
|
|
616 |
have *:"\<forall>x. Real (f x) \<noteq> \<omega>" by auto
|
|
617 |
note lebesgue.borel_measurable_Real[OF f(1)]
|
|
618 |
from positive_integral_has_integral[OF this f(2) *]
|
|
619 |
show ?case unfolding real_Real_max .
|
|
620 |
next case goal2
|
|
621 |
have *:"\<forall>x. Real (- f x) \<noteq> \<omega>" by auto
|
|
622 |
note lebesgue.borel_measurable_uminus[OF f(1)]
|
|
623 |
note lebesgue.borel_measurable_Real[OF this]
|
|
624 |
from positive_integral_has_integral[OF this f(3) *]
|
|
625 |
show ?case unfolding real_Real_max minus_min_eq_max by auto
|
|
626 |
qed
|
|
627 |
qed
|
|
628 |
|
|
629 |
lemma continuous_on_imp_borel_measurable:
|
|
630 |
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
|
|
631 |
assumes "continuous_on UNIV f"
|
|
632 |
shows "f \<in> borel_measurable lebesgue"
|
|
633 |
apply(rule lebesgue.borel_measurableI)
|
|
634 |
using continuous_open_preimage[OF assms] unfolding vimage_def by auto
|
|
635 |
|
|
636 |
lemma (in measure_space) integral_monotone_convergence_pos':
|
|
637 |
assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
|
|
638 |
and pos: "\<And>x i. 0 \<le> f i x"
|
|
639 |
and lim: "\<And>x. (\<lambda>i. f i x) ----> u x"
|
|
640 |
and ilim: "(\<lambda>i. integral (f i)) ----> x"
|
|
641 |
shows "integrable u \<and> integral u = x"
|
|
642 |
using integral_monotone_convergence_pos[OF assms] by auto
|
|
643 |
|
|
644 |
definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where
|
|
645 |
"e2p x = (\<lambda>i\<in>{..<DIM('a)}. x$$i)"
|
|
646 |
|
|
647 |
definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where
|
|
648 |
"p2e x = (\<chi>\<chi> i. x i)"
|
|
649 |
|
|
650 |
lemma bij_euclidean_component:
|
|
651 |
"bij_betw (e2p::'a::ordered_euclidean_space \<Rightarrow> _) (UNIV :: 'a set)
|
|
652 |
({..<DIM('a)} \<rightarrow>\<^isub>E (UNIV :: real set))"
|
|
653 |
unfolding bij_betw_def e2p_def_raw
|
|
654 |
proof let ?e = "\<lambda>x.\<lambda>i\<in>{..<DIM('a::ordered_euclidean_space)}. (x::'a)$$i"
|
|
655 |
show "inj ?e" unfolding inj_on_def restrict_def apply(subst euclidean_eq) apply safe
|
|
656 |
apply(drule_tac x=i in fun_cong) by auto
|
|
657 |
{ fix x::"nat \<Rightarrow> real" assume x:"\<forall>i. i \<notin> {..<DIM('a)} \<longrightarrow> x i = undefined"
|
|
658 |
hence "x = ?e (\<chi>\<chi> i. x i)" apply-apply(rule,case_tac "xa<DIM('a)") by auto
|
|
659 |
hence "x \<in> range ?e" by fastsimp
|
|
660 |
} thus "range ?e = ({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)}"
|
|
661 |
unfolding extensional_def using DIM_positive by auto
|
|
662 |
qed
|
|
663 |
|
|
664 |
lemma bij_p2e:
|
|
665 |
"bij_betw (p2e::_ \<Rightarrow> 'a::ordered_euclidean_space) ({..<DIM('a)} \<rightarrow>\<^isub>E (UNIV :: real set))
|
|
666 |
(UNIV :: 'a set)" (is "bij_betw ?p ?U _")
|
|
667 |
unfolding bij_betw_def
|
|
668 |
proof show "inj_on ?p ?U" unfolding inj_on_def p2e_def
|
|
669 |
apply(subst euclidean_eq) apply(safe,rule) unfolding extensional_def
|
|
670 |
apply(case_tac "xa<DIM('a)") by auto
|
|
671 |
{ fix x::'a have "x \<in> ?p ` extensional {..<DIM('a)}"
|
|
672 |
unfolding image_iff apply(rule_tac x="\<lambda>i. if i<DIM('a) then x$$i else undefined" in bexI)
|
|
673 |
apply(subst euclidean_eq,safe) unfolding p2e_def extensional_def by auto
|
|
674 |
} thus "?p ` ?U = UNIV" by auto
|
|
675 |
qed
|
|
676 |
|
|
677 |
lemma e2p_p2e[simp]: fixes z::"'a::ordered_euclidean_space"
|
|
678 |
assumes "x \<in> extensional {..<DIM('a)}"
|
|
679 |
shows "e2p (p2e x::'a) = x"
|
|
680 |
proof fix i::nat
|
|
681 |
show "e2p (p2e x::'a) i = x i" unfolding e2p_def p2e_def restrict_def
|
|
682 |
using assms unfolding extensional_def by auto
|
|
683 |
qed
|
|
684 |
|
|
685 |
lemma p2e_e2p[simp]: fixes x::"'a::ordered_euclidean_space"
|
|
686 |
shows "p2e (e2p x) = x"
|
|
687 |
apply(subst euclidean_eq) unfolding e2p_def p2e_def restrict_def by auto
|
|
688 |
|
|
689 |
interpretation borel_product: product_sigma_finite "\<lambda>x. borel::real algebra" "\<lambda>x. lmeasure"
|
|
690 |
by default
|
|
691 |
|
|
692 |
lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)"
|
|
693 |
unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto
|
|
694 |
|
|
695 |
lemma borel_vimage_algebra_eq:
|
|
696 |
"sigma_algebra.vimage_algebra
|
|
697 |
(borel :: ('a::ordered_euclidean_space) algebra) ({..<DIM('a)} \<rightarrow>\<^isub>E UNIV) p2e =
|
|
698 |
sigma (product_algebra (\<lambda>x. \<lparr> space = UNIV::real set, sets = range (\<lambda>a. {..<a}) \<rparr>) {..<DIM('a)} )"
|
|
699 |
proof- note bor = borel_eq_lessThan
|
|
700 |
def F \<equiv> "product_algebra (\<lambda>x. \<lparr> space = UNIV::real set, sets = range (\<lambda>a. {..<a}) \<rparr>) {..<DIM('a)}"
|
|
701 |
def E \<equiv> "\<lparr>space = (UNIV::'a set), sets = range lessThan\<rparr>"
|
|
702 |
have *:"(({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)}) = space F" unfolding F_def by auto
|
|
703 |
show ?thesis unfolding F_def[symmetric] * bor
|
|
704 |
proof(rule vimage_algebra_sigma,unfold E_def[symmetric])
|
|
705 |
show "sets E \<subseteq> Pow (space E)" "p2e \<in> space F \<rightarrow> space E" unfolding E_def by auto
|
|
706 |
next fix A assume "A \<in> sets F"
|
|
707 |
hence A:"A \<in> (Pi\<^isub>E {..<DIM('a)}) ` ({..<DIM('a)} \<rightarrow> range lessThan)"
|
|
708 |
unfolding F_def product_algebra_def algebra.simps .
|
|
709 |
then guess B unfolding image_iff .. note B=this
|
|
710 |
hence "\<forall>x<DIM('a). B x \<in> range lessThan" by auto
|
|
711 |
hence "\<forall>x. \<exists>xa. x<DIM('a) \<longrightarrow> B x = {..<xa}" unfolding image_iff by auto
|
|
712 |
from choice[OF this] guess b .. note b=this
|
|
713 |
hence b':"\<forall>i<DIM('a). Sup (B i) = b i" using Sup_lessThan by auto
|
|
714 |
|
|
715 |
show "A \<in> (\<lambda>X. p2e -` X \<inter> space F) ` sets E" unfolding image_iff B
|
|
716 |
proof(rule_tac x="{..< \<chi>\<chi> i. Sup (B i)}" in bexI)
|
|
717 |
show "Pi\<^isub>E {..<DIM('a)} B = p2e -` {..<(\<chi>\<chi> i. Sup (B i))::'a} \<inter> space F"
|
|
718 |
unfolding F_def E_def product_algebra_def algebra.simps
|
|
719 |
proof(rule,unfold subset_eq,rule_tac[!] ballI)
|
|
720 |
fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} B"
|
|
721 |
hence *:"\<forall>i<DIM('a). x i < b i" "\<forall>i\<ge>DIM('a). x i = undefined"
|
|
722 |
unfolding Pi_def extensional_def using b by auto
|
|
723 |
have "(p2e x::'a) < (\<chi>\<chi> i. Sup (B i))" unfolding less_prod_def eucl_less[of "p2e x"]
|
|
724 |
apply safe unfolding euclidean_lambda_beta b'[rule_format] p2e_def using * by auto
|
|
725 |
moreover have "x \<in> extensional {..<DIM('a)}"
|
|
726 |
using *(2) unfolding extensional_def by auto
|
|
727 |
ultimately show "x \<in> p2e -` {..<(\<chi>\<chi> i. Sup (B i)) ::'a} \<inter>
|
|
728 |
(({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})" by auto
|
|
729 |
next fix x assume as:"x \<in> p2e -` {..<(\<chi>\<chi> i. Sup (B i))::'a} \<inter>
|
|
730 |
(({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
|
|
731 |
hence "p2e x < ((\<chi>\<chi> i. Sup (B i))::'a)" by auto
|
|
732 |
hence "\<forall>i<DIM('a). x i \<in> B i" apply-apply(subst(asm) eucl_less)
|
|
733 |
unfolding p2e_def using b b' by auto
|
|
734 |
thus "x \<in> Pi\<^isub>E {..<DIM('a)} B" using as unfolding Pi_def extensional_def by auto
|
|
735 |
qed
|
|
736 |
show "{..<(\<chi>\<chi> i. Sup (B i))::'a} \<in> sets E" unfolding E_def algebra.simps by auto
|
|
737 |
qed
|
|
738 |
next fix A assume "A \<in> sets E"
|
|
739 |
then guess a unfolding E_def algebra.simps image_iff .. note a = this(2)
|
|
740 |
def B \<equiv> "\<lambda>i. {..<a $$ i}"
|
|
741 |
show "p2e -` A \<inter> space F \<in> sets F" unfolding F_def
|
|
742 |
unfolding product_algebra_def algebra.simps image_iff
|
|
743 |
apply(rule_tac x=B in bexI) apply rule unfolding subset_eq apply(rule_tac[1-2] ballI)
|
|
744 |
proof- show "B \<in> {..<DIM('a)} \<rightarrow> range lessThan" unfolding B_def by auto
|
|
745 |
fix x assume as:"x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
|
|
746 |
hence "p2e x \<in> A" by auto
|
|
747 |
hence "\<forall>i<DIM('a). x i \<in> B i" unfolding B_def a lessThan_iff
|
|
748 |
apply-apply(subst (asm) eucl_less) unfolding p2e_def by auto
|
|
749 |
thus "x \<in> Pi\<^isub>E {..<DIM('a)} B" using as unfolding Pi_def extensional_def by auto
|
|
750 |
next fix x assume x:"x \<in> Pi\<^isub>E {..<DIM('a)} B"
|
|
751 |
moreover have "p2e x \<in> A" unfolding a lessThan_iff p2e_def apply(subst eucl_less)
|
|
752 |
using x unfolding Pi_def extensional_def B_def by auto
|
|
753 |
ultimately show "x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})" by auto
|
|
754 |
qed
|
|
755 |
qed
|
|
756 |
qed
|
|
757 |
|
|
758 |
lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
|
|
759 |
apply(rule image_Int[THEN sym]) using bij_euclidean_component
|
|
760 |
unfolding bij_betw_def by auto
|
|
761 |
|
|
762 |
lemma Int_stable_cuboids: fixes x::"'a::ordered_euclidean_space"
|
|
763 |
shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). e2p ` {a..b})\<rparr>"
|
|
764 |
unfolding Int_stable_def algebra.select_convs
|
|
765 |
proof safe fix a b x y::'a
|
|
766 |
have *:"e2p ` {a..b} \<inter> e2p ` {x..y} =
|
|
767 |
(\<lambda>(a, b). e2p ` {a..b}) (\<chi>\<chi> i. max (a $$ i) (x $$ i), \<chi>\<chi> i. min (b $$ i) (y $$ i)::'a)"
|
|
768 |
unfolding e2p_Int inter_interval by auto
|
|
769 |
show "e2p ` {a..b} \<inter> e2p ` {x..y} \<in> range (\<lambda>(a, b). e2p ` {a..b::'a})" unfolding *
|
|
770 |
apply(rule range_eqI) ..
|
|
771 |
qed
|
|
772 |
|
|
773 |
lemma Int_stable_cuboids': fixes x::"'a::ordered_euclidean_space"
|
|
774 |
shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>"
|
|
775 |
unfolding Int_stable_def algebra.select_convs
|
|
776 |
apply safe unfolding inter_interval by auto
|
|
777 |
|
|
778 |
lemma product_borel_eq_vimage:
|
|
779 |
"sigma (product_algebra (\<lambda>x. borel) {..<DIM('a::ordered_euclidean_space)}) =
|
|
780 |
sigma_algebra.vimage_algebra borel (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})
|
|
781 |
(p2e:: _ \<Rightarrow> 'a::ordered_euclidean_space)"
|
|
782 |
unfolding borel_vimage_algebra_eq unfolding borel_eq_lessThan
|
|
783 |
apply(subst sigma_product_algebra_sigma_eq[where S="\<lambda>i. \<lambda>n. lessThan (real n)"])
|
|
784 |
unfolding lessThan_iff
|
|
785 |
proof- fix i assume i:"i<DIM('a)"
|
|
786 |
show "(\<lambda>n. {..<real n}) \<up> space \<lparr>space = UNIV, sets = range lessThan\<rparr>"
|
|
787 |
by(auto intro!:real_arch_lt isotoneI)
|
|
788 |
qed auto
|
|
789 |
|
|
790 |
lemma inj_on_disjoint_family_on: assumes "disjoint_family_on A S" "inj f"
|
|
791 |
shows "disjoint_family_on (\<lambda>x. f ` A x) S"
|
|
792 |
unfolding disjoint_family_on_def
|
|
793 |
proof(rule,rule,rule)
|
|
794 |
fix x1 x2 assume x:"x1 \<in> S" "x2 \<in> S" "x1 \<noteq> x2"
|
|
795 |
show "f ` A x1 \<inter> f ` A x2 = {}"
|
|
796 |
proof(rule ccontr) case goal1
|
|
797 |
then obtain z where z:"z \<in> f ` A x1 \<inter> f ` A x2" by auto
|
|
798 |
then obtain z1 z2 where z12:"z1 \<in> A x1" "z2 \<in> A x2" "f z1 = z" "f z2 = z" by auto
|
|
799 |
hence "z1 = z2" using assms(2) unfolding inj_on_def by blast
|
|
800 |
hence "x1 = x2" using z12(1-2) using assms[unfolded disjoint_family_on_def] using x by auto
|
|
801 |
thus False using x(3) by auto
|
|
802 |
qed
|
|
803 |
qed
|
|
804 |
|
|
805 |
declare restrict_extensional[intro]
|
|
806 |
|
|
807 |
lemma e2p_extensional[intro]:"e2p (y::'a::ordered_euclidean_space) \<in> extensional {..<DIM('a)}"
|
|
808 |
unfolding e2p_def by auto
|
|
809 |
|
|
810 |
lemma e2p_image_vimage: fixes A::"'a::ordered_euclidean_space set"
|
|
811 |
shows "e2p ` A = p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
|
|
812 |
proof(rule set_eqI,rule)
|
|
813 |
fix x assume "x \<in> e2p ` A" then guess y unfolding image_iff .. note y=this
|
|
814 |
show "x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
|
|
815 |
apply safe apply(rule vimageI[OF _ y(1)]) unfolding y p2e_e2p by auto
|
|
816 |
next fix x assume "x \<in> p2e -` A \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)})"
|
|
817 |
thus "x \<in> e2p ` A" unfolding image_iff apply(rule_tac x="p2e x" in bexI) apply(subst e2p_p2e) by auto
|
|
818 |
qed
|
|
819 |
|
|
820 |
lemma lmeasure_measure_eq_borel_prod:
|
|
821 |
fixes A :: "('a::ordered_euclidean_space) set"
|
|
822 |
assumes "A \<in> sets borel"
|
|
823 |
shows "lmeasure A = borel_product.product_measure {..<DIM('a)} (e2p ` A :: (nat \<Rightarrow> real) set)"
|
|
824 |
proof (rule measure_unique_Int_stable[where X=A and A=cube])
|
|
825 |
interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
|
|
826 |
show "Int_stable \<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>"
|
|
827 |
(is "Int_stable ?E" ) using Int_stable_cuboids' .
|
|
828 |
show "borel = sigma ?E" using borel_eq_atLeastAtMost .
|
|
829 |
show "\<And>i. lmeasure (cube i) \<noteq> \<omega>" unfolding lmeasure_cube by auto
|
|
830 |
show "\<And>X. X \<in> sets ?E \<Longrightarrow>
|
|
831 |
lmeasure X = borel_product.product_measure {..<DIM('a)} (e2p ` X :: (nat \<Rightarrow> real) set)"
|
|
832 |
proof- case goal1 then obtain a b where X:"X = {a..b}" by auto
|
|
833 |
{ presume *:"X \<noteq> {} \<Longrightarrow> ?case"
|
|
834 |
show ?case apply(cases,rule *,assumption) by auto }
|
|
835 |
def XX \<equiv> "\<lambda>i. {a $$ i .. b $$ i}" assume "X \<noteq> {}" note X' = this[unfolded X interval_ne_empty]
|
|
836 |
have *:"Pi\<^isub>E {..<DIM('a)} XX = e2p ` X" apply(rule set_eqI)
|
|
837 |
proof fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} XX"
|
|
838 |
thus "x \<in> e2p ` X" unfolding image_iff apply(rule_tac x="\<chi>\<chi> i. x i" in bexI)
|
|
839 |
unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by rule auto
|
|
840 |
next fix x assume "x \<in> e2p ` X" then guess y unfolding image_iff .. note y = this
|
|
841 |
show "x \<in> Pi\<^isub>E {..<DIM('a)} XX" unfolding y using y(1)
|
|
842 |
unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by auto
|
|
843 |
qed
|
|
844 |
have "lmeasure X = (\<Prod>x<DIM('a). Real (b $$ x - a $$ x))" using X' apply- unfolding X
|
|
845 |
unfolding lmeasure_atLeastAtMost content_closed_interval apply(subst Real_setprod) by auto
|
|
846 |
also have "... = (\<Prod>i<DIM('a). lmeasure (XX i))" apply(rule setprod_cong2)
|
|
847 |
unfolding XX_def lmeasure_atLeastAtMost apply(subst content_real) using X' by auto
|
|
848 |
also have "... = borel_product.product_measure {..<DIM('a)} (e2p ` X)" unfolding *[THEN sym]
|
|
849 |
apply(rule fprod.measure_times[THEN sym]) unfolding XX_def by auto
|
|
850 |
finally show ?case .
|
|
851 |
qed
|
|
852 |
|
|
853 |
show "range cube \<subseteq> sets \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
|
|
854 |
unfolding cube_def_raw by auto
|
|
855 |
have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastsimp
|
|
856 |
thus "cube \<up> space \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>"
|
|
857 |
apply-apply(rule isotoneI) apply(rule cube_subset_Suc) by auto
|
|
858 |
show "A \<in> sets borel " by fact
|
|
859 |
show "measure_space borel lmeasure" by default
|
|
860 |
show "measure_space borel
|
|
861 |
(\<lambda>a::'a set. finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` a))"
|
|
862 |
apply default unfolding countably_additive_def
|
|
863 |
proof safe fix A::"nat \<Rightarrow> 'a set" assume A:"range A \<subseteq> sets borel" "disjoint_family A"
|
|
864 |
"(\<Union>i. A i) \<in> sets borel"
|
|
865 |
note fprod.ca[unfolded countably_additive_def,rule_format]
|
|
866 |
note ca = this[of "\<lambda> n. e2p ` (A n)"]
|
|
867 |
show "(\<Sum>\<^isub>\<infinity>n. finite_product_sigma_finite.measure
|
|
868 |
(\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` A n)) =
|
|
869 |
finite_product_sigma_finite.measure (\<lambda>x. borel)
|
|
870 |
(\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` (\<Union>i. A i))" unfolding image_UN
|
|
871 |
proof(rule ca) show "range (\<lambda>n. e2p ` A n) \<subseteq> sets
|
|
872 |
(sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))"
|
|
873 |
unfolding product_borel_eq_vimage
|
|
874 |
proof case goal1
|
|
875 |
then guess y unfolding image_iff .. note y=this(2)
|
|
876 |
show ?case unfolding borel.in_vimage_algebra y apply-
|
|
877 |
apply(rule_tac x="A y" in bexI,rule e2p_image_vimage)
|
|
878 |
using A(1) by auto
|
|
879 |
qed
|
|
880 |
|
|
881 |
show "disjoint_family (\<lambda>n. e2p ` A n)" apply(rule inj_on_disjoint_family_on)
|
|
882 |
using bij_euclidean_component using A(2) unfolding bij_betw_def by auto
|
|
883 |
show "(\<Union>n. e2p ` A n) \<in> sets (sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))"
|
|
884 |
unfolding product_borel_eq_vimage borel.in_vimage_algebra
|
|
885 |
proof(rule bexI[OF _ A(3)],rule set_eqI,rule)
|
|
886 |
fix x assume x:"x \<in> (\<Union>n. e2p ` A n)" hence "p2e x \<in> (\<Union>i. A i)" by auto
|
|
887 |
moreover have "x \<in> extensional {..<DIM('a)}"
|
|
888 |
using x unfolding extensional_def e2p_def_raw by auto
|
|
889 |
ultimately show "x \<in> p2e -` (\<Union>i. A i) \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter>
|
|
890 |
extensional {..<DIM('a)})" by auto
|
|
891 |
next fix x assume x:"x \<in> p2e -` (\<Union>i. A i) \<inter> (({..<DIM('a)} \<rightarrow> UNIV) \<inter>
|
|
892 |
extensional {..<DIM('a)})"
|
|
893 |
hence "p2e x \<in> (\<Union>i. A i)" by auto
|
|
894 |
hence "\<exists>n. x \<in> e2p ` A n" apply safe apply(rule_tac x=i in exI)
|
|
895 |
unfolding image_iff apply(rule_tac x="p2e x" in bexI)
|
|
896 |
apply(subst e2p_p2e) using x by auto
|
|
897 |
thus "x \<in> (\<Union>n. e2p ` A n)" by auto
|
|
898 |
qed
|
|
899 |
qed
|
|
900 |
qed auto
|
|
901 |
qed
|
|
902 |
|
|
903 |
lemma e2p_p2e'[simp]: fixes x::"'a::ordered_euclidean_space"
|
|
904 |
assumes "A \<subseteq> extensional {..<DIM('a)}"
|
|
905 |
shows "e2p ` (p2e ` A ::'a set) = A"
|
|
906 |
apply(rule set_eqI) unfolding image_iff Bex_def apply safe defer
|
|
907 |
apply(rule_tac x="p2e x" in exI,safe) using assms by auto
|
|
908 |
|
|
909 |
lemma range_p2e:"range (p2e::_\<Rightarrow>'a::ordered_euclidean_space) = UNIV"
|
|
910 |
apply safe defer unfolding image_iff apply(rule_tac x="\<lambda>i. x $$ i" in bexI)
|
|
911 |
unfolding p2e_def by auto
|
|
912 |
|
|
913 |
lemma p2e_inv_extensional:"(A::'a::ordered_euclidean_space set)
|
|
914 |
= p2e ` (p2e -` A \<inter> extensional {..<DIM('a)})"
|
|
915 |
unfolding p2e_def_raw apply safe unfolding image_iff
|
|
916 |
proof- fix x assume "x\<in>A"
|
|
917 |
let ?y = "\<lambda>i. if i<DIM('a) then x$$i else undefined"
|
|
918 |
have *:"Chi ?y = x" apply(subst euclidean_eq) by auto
|
|
919 |
show "\<exists>xa\<in>Chi -` A \<inter> extensional {..<DIM('a)}. x = Chi xa" apply(rule_tac x="?y" in bexI)
|
|
920 |
apply(subst euclidean_eq) unfolding extensional_def using `x\<in>A` by(auto simp: *)
|
|
921 |
qed
|
|
922 |
|
|
923 |
lemma borel_fubini_positiv_integral:
|
|
924 |
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pinfreal"
|
|
925 |
assumes f: "f \<in> borel_measurable borel"
|
|
926 |
shows "borel.positive_integral f =
|
|
927 |
borel_product.product_positive_integral {..<DIM('a)} (f \<circ> p2e)"
|
|
928 |
proof- def U \<equiv> "(({..<DIM('a)} \<rightarrow> UNIV) \<inter> extensional {..<DIM('a)}):: (nat \<Rightarrow> real) set"
|
|
929 |
interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
|
|
930 |
have "\<And>x. \<exists>i::nat. x < real i" by (metis real_arch_lt)
|
|
931 |
hence "(\<lambda>n::nat. {..<real n}) \<up> UNIV" apply-apply(rule isotoneI) by auto
|
|
932 |
hence *:"sigma_algebra.vimage_algebra borel U (p2e:: _ \<Rightarrow> 'a)
|
|
933 |
= sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)})"
|
|
934 |
unfolding U_def apply-apply(subst borel_vimage_algebra_eq)
|
|
935 |
apply(subst sigma_product_algebra_sigma_eq[where S="\<lambda>x. \<lambda>n. {..<(\<chi>\<chi> i. real n)}", THEN sym])
|
|
936 |
unfolding borel_eq_lessThan[THEN sym] by auto
|
|
937 |
show ?thesis unfolding borel.positive_integral_vimage[unfolded space_borel,OF bij_p2e]
|
|
938 |
apply(subst fprod.positive_integral_cong_measure[THEN sym, of "\<lambda>A. lmeasure (p2e ` A)"])
|
|
939 |
unfolding U_def[symmetric] *[THEN sym] o_def
|
|
940 |
proof- fix A assume A:"A \<in> sets (sigma_algebra.vimage_algebra borel U (p2e ::_ \<Rightarrow> 'a))"
|
|
941 |
hence *:"A \<subseteq> extensional {..<DIM('a)}" unfolding U_def by auto
|
|
942 |
from A guess B unfolding borel.in_vimage_algebra U_def .. note B=this
|
|
943 |
have "(p2e ` A::'a set) \<in> sets borel" unfolding B apply(subst Int_left_commute)
|
|
944 |
apply(subst Int_absorb1) unfolding p2e_inv_extensional[of B,THEN sym] using B(1) by auto
|
|
945 |
from lmeasure_measure_eq_borel_prod[OF this] show "lmeasure (p2e ` A::'a set) =
|
|
946 |
finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} A"
|
|
947 |
unfolding e2p_p2e'[OF *] .
|
|
948 |
qed auto
|
|
949 |
qed
|
|
950 |
|
|
951 |
lemma borel_fubini:
|
|
952 |
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real"
|
|
953 |
assumes f: "f \<in> borel_measurable borel"
|
|
954 |
shows "borel.integral f = borel_product.product_integral {..<DIM('a)} (f \<circ> p2e)"
|
|
955 |
proof- interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto
|
|
956 |
have 1:"(\<lambda>x. Real (f x)) \<in> borel_measurable borel" using f by auto
|
|
957 |
have 2:"(\<lambda>x. Real (- f x)) \<in> borel_measurable borel" using f by auto
|
|
958 |
show ?thesis unfolding fprod.integral_def borel.integral_def
|
|
959 |
unfolding borel_fubini_positiv_integral[OF 1] borel_fubini_positiv_integral[OF 2]
|
|
960 |
unfolding o_def ..
|
38656
|
961 |
qed
|
|
962 |
|
|
963 |
end
|