author  hoelzl 
Fri, 14 Jan 2011 15:59:49 +0100  
changeset 41546  2a12c23b7a34 
parent 41097  a1abfa4e2b44 
child 41654  32fe42892983 
permissions  rwrr 
40859  1 
(* 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 applyapply(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" 

123 
assumes measurable: "\<And>i. gmeasurable (A i)" 

124 
and finite: "\<And>n. gmeasure (UNION {.. n} A) \<le> B" 

125 
shows "gmeasurable (\<Union>i. A i)" 

126 
proof  

127 
have *: "\<And>n. \<Union>{A k k. k \<le> n} = (\<Union>i\<le>n. A i)" 

128 
"(\<Union>{A n n. n \<in> UNIV}) = (\<Union>i. A i)" by auto 

129 
show ?thesis 

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by (rule gmeasurable_countable_unions_strong[of A B, unfolded *, OF assms]) 

131 
qed 

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133 
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" 

159 
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" 

163 
proof (rule lebesgueI2) 

164 
fix n show "gmeasurable ((\<Union>i. A i) \<inter> cube n)" unfolding UN_extend_simps 

165 
using A 

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by (intro gmeasurable_countable_UNION[where B="gmeasure (cube n::'a set)"]) 

167 
(auto intro!: measure_subset gmeasure_setsum simp: UN_extend_simps simp del: gmeasure_cube_eq UN_simps) 

168 
qed } 

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{ fix A assume A: "A \<in> sets lebesgue" show "space lebesgue  A \<in> sets lebesgue" 

170 
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 

174 
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)})" 

182 
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)" applyapply(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)))" 

237 
proof (intro arg_cong[where f="SUPR UNIV"] ext) 

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fix n 

239 
have sums: "(\<lambda>i. gmeasure (A i \<inter> cube n)) sums gmeasure (\<Union>{A i \<inter> cube n i. i \<in> UNIV})" 

240 
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)" 

250 
unfolding atLeast0AtMost using A 

251 
proof (intro gmeasure_setsum[symmetric]) 

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fix s t :: nat assume "s \<noteq> t" then have "A t \<inter> A s = {}" 

253 
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 

255 
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 

258 
finally show "(\<Sum>k = 0..i. gmeasure (A k \<inter> cube n)) \<le> gmeasure (cube n :: 'b set)" . 

259 
qed 

260 
show "(\<Sum>\<^isub>\<infinity>i. Real (gmeasure (A i \<inter> cube n))) = Real (gmeasure ((\<Union>i. A i) \<inter> cube n))" 

261 
unfolding psuminf_def 

262 
apply (subst setsum_Real) 

263 
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)" 

267 
unfolding sums_def atLeast0LessThan by simp 

268 
qed (auto intro!: monoI setsum_nonneg setsum_mono2) 

269 
qed 

270 
qed 

271 
qed 

272 
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`] . 
279 
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))) 

282 
> integral UNIV (\<lambda>x. if x \<in> s then 1 else 0)" 

283 
proof(rule monotone_convergence_increasing) 

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have "lmeasure s \<le> Real m" using `lmeasure s = Real m` by simp 
285 
then have "\<forall>n. gmeasure (s \<inter> cube n) \<le> m" 

286 
unfolding lmeasure_def complete_lattice_class.SUP_le_iff 

287 
using `0 \<le> m` by (auto simp: measure_pos_le) 

288 
thus *:"bounded {integral UNIV (\<lambda>x. if x \<in> s \<inter> cube k then 1 else (0::real)) k. True}" 

289 
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 

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using s unfolding gmeasurable_def has_gmeasure_def by auto 
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have *:"\<And>n. n \<le> Suc n" by auto 
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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))" 

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unfolding Lim_sequentially 
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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 

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using ball_subset_cube[of N] by(auto simp: dist_norm) 
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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 

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apply(rule LIMSEQ_unique[OF _ **(2)]) unfolding measure_integral_univ[THEN sym,OF s] using * . 
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show ?thesis unfolding has_gmeasure * apply(rule integrable_integral) using ** by auto 
312 
qed 

313 

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lemma lmeasure_finite_gmeasurable: assumes "s \<in> sets lebesgue" "lmeasure s \<noteq> \<omega>" 
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shows "gmeasurable s" 
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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) 

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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 

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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]) 

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show ?thesis 
330 
unfolding lmeasure_iff_LIMSEQ[OF s `0 \<le> m`] unfolding * 

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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 

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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: 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

360 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal" 
38656  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 

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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" 

41023
9118eb4eb8dc
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hoelzl
parents:
40874
diff
changeset

424 
using has_gmeasure_interval[of a a] unfolding zero_pextreal_def 
40859  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: 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

478 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal" 
40859  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 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

489 
show ?case unfolding space_lebesgue real_of_pextreal_setsum'[OF *(1),THEN sym] 
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

490 
unfolding real_of_pextreal_setsum'[OF *(2),THEN sym] 
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

491 
unfolding real_of_pextreal_setsum space_lebesgue 
40859  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 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

499 
have *:"\<And>x. real (indicator (f ` {f y}) x::pextreal) = (if x \<in> f ` {f y} then 1 else 0)" by auto 
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

500 
show ?thesis unfolding real_of_pextreal_mult[THEN sym] 
40859  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': 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

513 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal" 
40859  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: 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

547 
fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pextreal" 
40859  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 

41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

555 
have "\<And>x. (SUP i. f i x) = u x" using mono lim SUP_Lim_pextreal 
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

556 
unfolding fun_eq_iff mono_def by auto 
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

557 
moreover have "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M" 
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

558 
using i by auto 
40859  559 
ultimately show "u \<in> borel_measurable M" by simp 
560 
qed 

561 

562 
lemma positive_integral_has_integral: 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

563 
fixes f::"'a::ordered_euclidean_space => pextreal" 
40859  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) 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

584 
proof safe case goal1 show ?case apply(rule real_of_pextreal_mono) using u(2,3) by auto 
40859  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) 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

591 
unfolding integral_unique[OF u_int] defer apply(rule real_of_pextreal_mono[OF _ int_u_le]) 
40859  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 

41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

629 
lemma lebesgue_positive_integral_eq_borel: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

630 
"f \<in> borel_measurable borel \<Longrightarrow> lebesgue.positive_integral f = borel.positive_integral f " 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

631 
by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

632 

2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

633 
lemma lebesgue_integral_eq_borel: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

634 
assumes "f \<in> borel_measurable borel" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

635 
shows "lebesgue.integrable f = borel.integrable f" (is ?P) 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

636 
and "lebesgue.integral f = borel.integral f" (is ?I) 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

637 
proof  
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

638 
have *: "sigma_algebra borel" by default 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

639 
have "sets borel \<subseteq> sets lebesgue" by auto 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

640 
from lebesgue.integral_subalgebra[OF assms this _ *] 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

641 
show ?P ?I by auto 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

642 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

643 

2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

644 
lemma borel_integral_has_integral: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

645 
fixes f::"'a::ordered_euclidean_space => real" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

646 
assumes f:"borel.integrable f" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

647 
shows "(f has_integral (borel.integral f)) UNIV" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

648 
proof  
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

649 
have borel: "f \<in> borel_measurable borel" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

650 
using f unfolding borel.integrable_def by auto 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

651 
from f show ?thesis 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

652 
using lebesgue_integral_has_integral[of f] 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

653 
unfolding lebesgue_integral_eq_borel[OF borel] by simp 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

654 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

655 

40859  656 
lemma continuous_on_imp_borel_measurable: 
657 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space" 

658 
assumes "continuous_on UNIV f" 

41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

659 
shows "f \<in> borel_measurable borel" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

660 
apply(rule borel.borel_measurableI) 
40859  661 
using continuous_open_preimage[OF assms] unfolding vimage_def by auto 
662 

663 
lemma (in measure_space) integral_monotone_convergence_pos': 

664 
assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)" 

665 
and pos: "\<And>x i. 0 \<le> f i x" 

666 
and lim: "\<And>x. (\<lambda>i. f i x) > u x" 

667 
and ilim: "(\<lambda>i. integral (f i)) > x" 

668 
shows "integrable u \<and> integral u = x" 

669 
using integral_monotone_convergence_pos[OF assms] by auto 

670 

671 
definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where 

672 
"e2p x = (\<lambda>i\<in>{..<DIM('a)}. x$$i)" 

673 

674 
definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where 

675 
"p2e x = (\<chi>\<chi> i. x i)" 

676 

41095  677 
lemma e2p_p2e[simp]: 
678 
"x \<in> extensional {..<DIM('a)} \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x" 

679 
by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def) 

40859  680 

41095  681 
lemma p2e_e2p[simp]: 
682 
"p2e (e2p x) = (x::'a::ordered_euclidean_space)" 

683 
by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def) 

40859  684 

41095  685 
lemma bij_inv_p2e_e2p: 
686 
shows "bij_inv ({..<DIM('a)} \<rightarrow>\<^isub>E UNIV) (UNIV :: 'a::ordered_euclidean_space set) 

687 
p2e e2p" (is "bij_inv ?P ?U _ _") 

688 
proof (rule bij_invI) 

689 
show "p2e \<in> ?P \<rightarrow> ?U" "e2p \<in> ?U \<rightarrow> ?P" by (auto simp: e2p_def) 

690 
qed auto 

40859  691 

692 
interpretation borel_product: product_sigma_finite "\<lambda>x. borel::real algebra" "\<lambda>x. lmeasure" 

693 
by default 

694 

695 
lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)" 

696 
unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto 

697 

41095  698 
lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)" 
699 
unfolding Pi_def by auto 

40859  700 

41095  701 
lemma measurable_e2p_on_generator: 
702 
"e2p \<in> measurable \<lparr> space = UNIV::'a set, sets = range lessThan \<rparr> 

703 
(product_algebra 

704 
(\<lambda>x. \<lparr> space = UNIV::real set, sets = range lessThan \<rparr>) 

705 
{..<DIM('a::ordered_euclidean_space)})" 

706 
(is "e2p \<in> measurable ?E ?P") 

707 
proof (unfold measurable_def, intro CollectI conjI ballI) 

708 
show "e2p \<in> space ?E \<rightarrow> space ?P" by (auto simp: e2p_def) 

709 
fix A assume "A \<in> sets ?P" 

710 
then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)" 

711 
and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)" 

712 
by (auto elim!: product_algebraE) 

713 
then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto 

714 
from this[THEN bchoice] guess xs .. 

715 
then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})" 

716 
using A by auto 

717 
have "e2p ` A = {..< (\<chi>\<chi> i. xs i) :: 'a}" 

718 
using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq 

719 
euclidean_eq[where 'a='a] eucl_less[where 'a='a]) 

720 
then show "e2p ` A \<inter> space ?E \<in> sets ?E" by simp 

40859  721 
qed 
722 

41095  723 
lemma measurable_p2e_on_generator: 
724 
"p2e \<in> measurable 

725 
(product_algebra 

726 
(\<lambda>x. \<lparr> space = UNIV::real set, sets = range lessThan \<rparr>) 

727 
{..<DIM('a::ordered_euclidean_space)}) 

728 
\<lparr> space = UNIV::'a set, sets = range lessThan \<rparr>" 

729 
(is "p2e \<in> measurable ?P ?E") 

730 
proof (unfold measurable_def, intro CollectI conjI ballI) 

731 
show "p2e \<in> space ?P \<rightarrow> space ?E" by simp 

732 
fix A assume "A \<in> sets ?E" 

733 
then obtain x where "A = {..<x}" by auto 

734 
then have "p2e ` A \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< x $$ i})" 

735 
using DIM_positive 

736 
by (auto simp: Pi_iff set_eq_iff p2e_def 

737 
euclidean_eq[where 'a='a] eucl_less[where 'a='a]) 

738 
then show "p2e ` A \<inter> space ?P \<in> sets ?P" by auto 

739 
qed 

740 

741 
lemma borel_vimage_algebra_eq: 

742 
defines "F \<equiv> product_algebra (\<lambda>x. \<lparr> space = (UNIV::real set), sets = range lessThan \<rparr>) {..<DIM('a::ordered_euclidean_space)}" 

743 
shows "sigma_algebra.vimage_algebra (borel::'a::ordered_euclidean_space algebra) (space (sigma F)) p2e = sigma F" 

744 
unfolding borel_eq_lessThan 

745 
proof (intro vimage_algebra_sigma) 

746 
let ?E = "\<lparr>space = (UNIV::'a set), sets = range lessThan\<rparr>" 

747 
show "bij_inv (space (sigma F)) (space (sigma ?E)) p2e e2p" 

748 
using bij_inv_p2e_e2p unfolding F_def by simp 

749 
show "sets F \<subseteq> Pow (space F)" "sets ?E \<subseteq> Pow (space ?E)" unfolding F_def 

750 
by (intro product_algebra_sets_into_space) auto 

751 
show "p2e \<in> measurable F ?E" 

752 
"e2p \<in> measurable ?E F" 

753 
unfolding F_def using measurable_p2e_on_generator measurable_e2p_on_generator by auto 

754 
qed 

755 

756 
lemma product_borel_eq_vimage: 

757 
"sigma (product_algebra (\<lambda>x. borel) {..<DIM('a::ordered_euclidean_space)}) = 

758 
sigma_algebra.vimage_algebra borel (extensional {..<DIM('a)}) 

759 
(p2e:: _ \<Rightarrow> 'a::ordered_euclidean_space)" 

760 
unfolding borel_vimage_algebra_eq[simplified] 

761 
unfolding borel_eq_lessThan 

762 
apply(subst sigma_product_algebra_sigma_eq[where S="\<lambda>i. \<lambda>n. lessThan (real n)"]) 

763 
unfolding lessThan_iff 

764 
proof fix i assume i:"i<DIM('a)" 

765 
show "(\<lambda>n. {..<real n}) \<up> space \<lparr>space = UNIV, sets = range lessThan\<rparr>" 

766 
by(auto intro!:real_arch_lt isotoneI) 

767 
qed auto 

768 

40859  769 
lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R") 
41095  770 
apply(rule image_Int[THEN sym]) 
771 
using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)] 

40859  772 
unfolding bij_betw_def by auto 
773 

774 
lemma Int_stable_cuboids: fixes x::"'a::ordered_euclidean_space" 

775 
shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). e2p ` {a..b})\<rparr>" 

776 
unfolding Int_stable_def algebra.select_convs 

777 
proof safe fix a b x y::'a 

778 
have *:"e2p ` {a..b} \<inter> e2p ` {x..y} = 

779 
(\<lambda>(a, b). e2p ` {a..b}) (\<chi>\<chi> i. max (a $$ i) (x $$ i), \<chi>\<chi> i. min (b $$ i) (y $$ i)::'a)" 

780 
unfolding e2p_Int inter_interval by auto 

781 
show "e2p ` {a..b} \<inter> e2p ` {x..y} \<in> range (\<lambda>(a, b). e2p ` {a..b::'a})" unfolding * 

782 
apply(rule range_eqI) .. 

783 
qed 

784 

785 
lemma Int_stable_cuboids': fixes x::"'a::ordered_euclidean_space" 

786 
shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>" 

787 
unfolding Int_stable_def algebra.select_convs 

788 
apply safe unfolding inter_interval by 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(12) 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" 

41095  811 
shows "e2p ` A = p2e ` A \<inter> extensional {..<DIM('a)}" 
40859  812 
proof(rule set_eqI,rule) 
813 
fix x assume "x \<in> e2p ` A" then guess y unfolding image_iff .. note y=this 

41095  814 
show "x \<in> p2e ` A \<inter> extensional {..<DIM('a)}" 
40859  815 
apply safe apply(rule vimageI[OF _ y(1)]) unfolding y p2e_e2p by auto 
41095  816 
next fix x assume "x \<in> p2e ` A \<inter> extensional {..<DIM('a)}" 
40859  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 
applyapply(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) 

41095  882 
using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)] using A(2) unfolding bij_betw_def by auto 
40859  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 

41095  889 
ultimately show "x \<in> p2e ` (\<Union>i. A i) \<inter> extensional {..<DIM('a)}" by auto 
890 
next fix x assume x:"x \<in> p2e ` (\<Union>i. A i) \<inter> extensional {..<DIM('a)}" 

40859  891 
hence "p2e x \<in> (\<Union>i. A i)" by auto 
892 
hence "\<exists>n. x \<in> e2p ` A n" apply safe apply(rule_tac x=i in exI) 

893 
unfolding image_iff apply(rule_tac x="p2e x" in bexI) 

894 
apply(subst e2p_p2e) using x by auto 

895 
thus "x \<in> (\<Union>n. e2p ` A n)" by auto 

896 
qed 

897 
qed 

898 
qed auto 

899 
qed 

900 

901 
lemma e2p_p2e'[simp]: fixes x::"'a::ordered_euclidean_space" 

902 
assumes "A \<subseteq> extensional {..<DIM('a)}" 

903 
shows "e2p ` (p2e ` A ::'a set) = A" 

904 
apply(rule set_eqI) unfolding image_iff Bex_def apply safe defer 

905 
apply(rule_tac x="p2e x" in exI,safe) using assms by auto 

906 

907 
lemma range_p2e:"range (p2e::_\<Rightarrow>'a::ordered_euclidean_space) = UNIV" 

908 
apply safe defer unfolding image_iff apply(rule_tac x="\<lambda>i. x $$ i" in bexI) 

909 
unfolding p2e_def by auto 

910 

911 
lemma p2e_inv_extensional:"(A::'a::ordered_euclidean_space set) 

912 
= p2e ` (p2e ` A \<inter> extensional {..<DIM('a)})" 

913 
unfolding p2e_def_raw apply safe unfolding image_iff 

914 
proof fix x assume "x\<in>A" 

915 
let ?y = "\<lambda>i. if i<DIM('a) then x$$i else undefined" 

916 
have *:"Chi ?y = x" apply(subst euclidean_eq) by auto 

917 
show "\<exists>xa\<in>Chi ` A \<inter> extensional {..<DIM('a)}. x = Chi xa" apply(rule_tac x="?y" in bexI) 

918 
apply(subst euclidean_eq) unfolding extensional_def using `x\<in>A` by(auto simp: *) 

919 
qed 

920 

921 
lemma borel_fubini_positiv_integral: 

41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

922 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pextreal" 
40859  923 
assumes f: "f \<in> borel_measurable borel" 
924 
shows "borel.positive_integral f = 

925 
borel_product.product_positive_integral {..<DIM('a)} (f \<circ> p2e)" 

41095  926 
proof def U \<equiv> "extensional {..<DIM('a)} :: (nat \<Rightarrow> real) set" 
40859  927 
interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto 
41095  928 
have *:"sigma_algebra.vimage_algebra borel U (p2e:: _ \<Rightarrow> 'a) 
40859  929 
= sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)})" 
41095  930 
unfolding U_def product_borel_eq_vimage[symmetric] .. 
931 
show ?thesis 

932 
unfolding borel.positive_integral_vimage[unfolded space_borel, OF bij_inv_p2e_e2p[THEN bij_inv_bij_betw(1)]] 

40859  933 
apply(subst fprod.positive_integral_cong_measure[THEN sym, of "\<lambda>A. lmeasure (p2e ` A)"]) 
934 
unfolding U_def[symmetric] *[THEN sym] o_def 

935 
proof fix A assume A:"A \<in> sets (sigma_algebra.vimage_algebra borel U (p2e ::_ \<Rightarrow> 'a))" 

936 
hence *:"A \<subseteq> extensional {..<DIM('a)}" unfolding U_def by auto 

41095  937 
from A guess B unfolding borel.in_vimage_algebra U_def .. 
938 
then have "(p2e ` A::'a set) \<in> sets borel" 

939 
by (simp add: p2e_inv_extensional[of B, symmetric]) 

40859  940 
from lmeasure_measure_eq_borel_prod[OF this] show "lmeasure (p2e ` A::'a set) = 
941 
finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} A" 

942 
unfolding e2p_p2e'[OF *] . 

943 
qed auto 

944 
qed 

945 

946 
lemma borel_fubini: 

947 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 

948 
assumes f: "f \<in> borel_measurable borel" 

949 
shows "borel.integral f = borel_product.product_integral {..<DIM('a)} (f \<circ> p2e)" 

950 
proof interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto 

951 
have 1:"(\<lambda>x. Real (f x)) \<in> borel_measurable borel" using f by auto 

952 
have 2:"(\<lambda>x. Real ( f x)) \<in> borel_measurable borel" using f by auto 

953 
show ?thesis unfolding fprod.integral_def borel.integral_def 

954 
unfolding borel_fubini_positiv_integral[OF 1] borel_fubini_positiv_integral[OF 2] 

955 
unfolding o_def .. 

38656  956 
qed 
957 

958 
end 