author  hoelzl 
Wed, 25 Apr 2012 15:09:18 +0200  
changeset 47757  5e6fe71e2390 
parent 47694  05663f75964c 
child 49777  6ac97ab9b295 
permissions  rwrr 
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(* Title: HOL/Probability/Lebesgue_Measure.thy 
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Author: Johannes Hölzl, TU München 

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Author: Robert Himmelmann, TU München 

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*) 

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header {* Lebsegue measure *} 
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theory Lebesgue_Measure 
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imports Finite_Product_Measure 
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begin 
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lemma borel_measurable_sets: 
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assumes "f \<in> measurable borel M" "A \<in> sets M" 

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shows "f ` A \<in> sets borel" 

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using measurable_sets[OF assms] by simp 

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lemma measurable_identity[intro,simp]: 

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"(\<lambda>x. x) \<in> measurable M M" 

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unfolding measurable_def by auto 

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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 (fastforce 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 (fastforce simp: cube_def eucl_le[where 'a='a]) 
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then show "n \<le> N" 
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by (fastforce 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 reals_Archimedean2[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 cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)" 
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unfolding cube_def subset_eq apply safe unfolding mem_interval apply auto done 
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subsection {* Lebesgue measure *} 
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definition lebesgue :: "'a::ordered_euclidean_space measure" where 
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"lebesgue = measure_of UNIV {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n} 

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(\<lambda>A. SUP n. ereal (integral (cube n) (indicator A)))" 

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lemma space_lebesgue[simp]: "space lebesgue = UNIV" 
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unfolding lebesgue_def by simp 

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lemma lebesgueI: "(\<And>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n) \<Longrightarrow> A \<in> sets lebesgue" 

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unfolding lebesgue_def by simp 

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lemma absolutely_integrable_on_indicator[simp]: 

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fixes A :: "'a::ordered_euclidean_space set" 

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shows "((indicator A :: _ \<Rightarrow> real) absolutely_integrable_on X) \<longleftrightarrow> 

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(indicator A :: _ \<Rightarrow> real) integrable_on X" 

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unfolding absolutely_integrable_on_def by simp 

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lemma LIMSEQ_indicator_UN: 

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"(\<lambda>k. indicator (\<Union> i<k. A i) x) > (indicator (\<Union>i. A i) x :: real)" 

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proof cases 

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assume "\<exists>i. x \<in> A i" then guess i .. note i = this 

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then have *: "\<And>n. (indicator (\<Union> i<n + Suc i. A i) x :: real) = 1" 

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"(indicator (\<Union> i. A i) x :: real) = 1" by (auto simp: indicator_def) 

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show ?thesis 

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apply (rule LIMSEQ_offset[of _ "Suc i"]) unfolding * by auto 

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qed (auto simp: indicator_def) 

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lemma indicator_add: 
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"A \<inter> B = {} \<Longrightarrow> (indicator A x::_::monoid_add) + indicator B x = indicator (A \<union> B) x" 

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unfolding indicator_def by auto 

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lemma sigma_algebra_lebesgue: 
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defines "leb \<equiv> {A. \<forall>n. (indicator A :: 'a::ordered_euclidean_space \<Rightarrow> real) integrable_on cube n}" 

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shows "sigma_algebra UNIV leb" 

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proof (safe intro!: sigma_algebra_iff2[THEN iffD2]) 

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fix A assume A: "A \<in> leb" 

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moreover have "indicator (UNIV  A) = (\<lambda>x. 1  indicator A x :: real)" 

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by (auto simp: fun_eq_iff indicator_def) 
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ultimately show "UNIV  A \<in> leb" 
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using A by (auto intro!: integrable_sub simp: cube_def leb_def) 

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next 
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fix n show "{} \<in> leb" 
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by (auto simp: cube_def indicator_def[abs_def] leb_def) 

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next 
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fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> leb" 
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have "\<forall>n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "\<forall>n. ?g integrable_on _") 

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proof (intro dominated_convergence[where g="?g"] ballI allI) 

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fix k n show "(indicator (\<Union>i<k. A i) :: _ \<Rightarrow> real) integrable_on cube n" 

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proof (induct k) 
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case (Suc k) 

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have *: "(\<Union> i<Suc k. A i) = (\<Union> i<k. A i) \<union> A k" 

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unfolding lessThan_Suc UN_insert by auto 

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have *: "(\<lambda>x. max (indicator (\<Union> i<k. A i) x) (indicator (A k) x) :: real) = 

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indicator (\<Union> i<Suc k. A i)" (is "(\<lambda>x. max (?f x) (?g x)) = _") 

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by (auto simp: fun_eq_iff * indicator_def) 

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show ?case 

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using absolutely_integrable_max[of ?f "cube n" ?g] A Suc 
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by (simp add: * leb_def subset_eq) 

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qed auto 
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qed (auto intro: LIMSEQ_indicator_UN simp: cube_def) 

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then show "(\<Union>i. A i) \<in> leb" by (auto simp: leb_def) 
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qed simp 
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lemma sets_lebesgue: "sets lebesgue = {A. \<forall>n. (indicator A :: _ \<Rightarrow> real) integrable_on cube n}" 
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unfolding lebesgue_def sigma_algebra.sets_measure_of_eq[OF sigma_algebra_lebesgue] .. 

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lemma lebesgueD: "A \<in> sets lebesgue \<Longrightarrow> (indicator A :: _ \<Rightarrow> real) integrable_on cube n" 

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unfolding sets_lebesgue by simp 

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lemma emeasure_lebesgue: 
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assumes "A \<in> sets lebesgue" 
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shows "emeasure lebesgue A = (SUP n. ereal (integral (cube n) (indicator A)))" 

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(is "_ = ?\<mu> A") 

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proof (rule emeasure_measure_of[OF lebesgue_def]) 

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have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff) 
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show "positive (sets lebesgue) ?\<mu>" 
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proof (unfold positive_def, intro conjI ballI) 

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show "?\<mu> {} = 0" by (simp add: integral_0 *) 

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fix A :: "'a set" assume "A \<in> sets lebesgue" then show "0 \<le> ?\<mu> A" 

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by (auto intro!: SUP_upper2 Integration.integral_nonneg simp: sets_lebesgue) 

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qed 
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next 
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show "countably_additive (sets lebesgue) ?\<mu>" 
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proof (intro countably_additive_def[THEN iffD2] allI impI) 
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fix A :: "nat \<Rightarrow> 'a set" assume rA: "range A \<subseteq> sets lebesgue" "disjoint_family A" 
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then have A[simp, intro]: "\<And>i n. (indicator (A i) :: _ \<Rightarrow> real) integrable_on cube n" 
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by (auto dest: lebesgueD) 

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let ?m = "\<lambda>n i. integral (cube n) (indicator (A i) :: _\<Rightarrow>real)" 
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let ?M = "\<lambda>n I. integral (cube n) (indicator (\<Union>i\<in>I. A i) :: _\<Rightarrow>real)" 

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have nn[simp, intro]: "\<And>i n. 0 \<le> ?m n i" by (auto intro!: Integration.integral_nonneg) 
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assume "(\<Union>i. A i) \<in> sets lebesgue" 
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then have UN_A[simp, intro]: "\<And>i n. (indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" 

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by (auto simp: sets_lebesgue) 
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show "(\<Sum>n. ?\<mu> (A n)) = ?\<mu> (\<Union>i. A i)" 

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proof (subst suminf_SUP_eq, safe intro!: incseq_SucI) 

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fix i n show "ereal (?m n i) \<le> ereal (?m (Suc n) i)" 
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using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le incseq_SucI) 
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next 
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fix i n show "0 \<le> ereal (?m n i)" 
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using rA unfolding lebesgue_def 
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by (auto intro!: SUP_upper2 integral_nonneg) 
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next 
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show "(SUP n. \<Sum>i. ereal (?m n i)) = (SUP n. ereal (?M n UNIV))" 
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proof (intro arg_cong[where f="SUPR UNIV"] ext sums_unique[symmetric] sums_ereal[THEN iffD2] sums_def[THEN iffD2]) 

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fix n 
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have "\<And>m. (UNION {..<m} A) \<in> sets lebesgue" using rA by auto 

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from lebesgueD[OF this] 

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have "(\<lambda>m. ?M n {..< m}) > ?M n UNIV" 

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(is "(\<lambda>m. integral _ (?A m)) > ?I") 

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by (intro dominated_convergence(2)[where f="?A" and h="\<lambda>x. 1::real"]) 

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(auto intro: LIMSEQ_indicator_UN simp: cube_def) 

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moreover 

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{ fix m have *: "(\<Sum>x<m. ?m n x) = ?M n {..< m}" 

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proof (induct m) 

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case (Suc m) 

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have "(\<Union>i<m. A i) \<in> sets lebesgue" using rA by auto 

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then have "(indicator (\<Union>i<m. A i) :: _\<Rightarrow>real) integrable_on (cube n)" 

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by (auto dest!: lebesgueD) 

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moreover 

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have "(\<Union>i<m. A i) \<inter> A m = {}" 

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using rA(2)[unfolded disjoint_family_on_def, THEN bspec, of m] 

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by auto 

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then have "\<And>x. indicator (\<Union>i<Suc m. A i) x = 

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indicator (\<Union>i<m. A i) x + (indicator (A m) x :: real)" 

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by (auto simp: indicator_add lessThan_Suc ac_simps) 

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ultimately show ?case 

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using Suc A by (simp add: Integration.integral_add[symmetric]) 
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qed auto } 
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ultimately show "(\<lambda>m. \<Sum>x = 0..<m. ?m n x) > ?M n UNIV" 
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by (simp add: atLeast0LessThan) 
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qed 
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qed 

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qed 

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next 
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qed (auto, fact) 

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lemma has_integral_interval_cube: 
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fixes a b :: "'a::ordered_euclidean_space" 

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shows "(indicator {a .. b} has_integral 

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content ({\<chi>\<chi> i. max ( real n) (a $$ i) .. \<chi>\<chi> i. min (real n) (b $$ i)} :: 'a set)) (cube n)" 

203 
(is "(?I has_integral content ?R) (cube n)") 

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proof  
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let "{?N .. ?P}" = ?R 
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have [simp]: "(\<lambda>x. if x \<in> cube n then ?I x else 0) = indicator ?R" 

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by (auto simp: indicator_def cube_def fun_eq_iff eucl_le[where 'a='a]) 

208 
have "(?I has_integral content ?R) (cube n) \<longleftrightarrow> (indicator ?R has_integral content ?R) UNIV" 

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unfolding has_integral_restrict_univ[where s="cube n", symmetric] by simp 

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also have "\<dots> \<longleftrightarrow> ((\<lambda>x. 1) has_integral content ?R) ?R" 

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unfolding indicator_def [abs_def] has_integral_restrict_univ .. 
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finally show ?thesis 
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using has_integral_const[of "1::real" "?N" "?P"] by simp 

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qed 
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lemma lebesgueI_borel[intro, simp]: 
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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  
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have "s \<in> sigma_sets (space lebesgue) (range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)}))" 
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using assms by (simp add: borel_eq_atLeastAtMost) 

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also have "\<dots> \<subseteq> sets lebesgue" 

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proof (safe intro!: sigma_sets_subset lebesgueI) 

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fix n :: nat and a b :: 'a 
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let ?N = "\<chi>\<chi> i. max ( real n) (a $$ i)" 

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let ?P = "\<chi>\<chi> i. min (real n) (b $$ i)" 

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show "(indicator {a..b} :: 'a\<Rightarrow>real) integrable_on cube n" 

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unfolding integrable_on_def 

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using has_integral_interval_cube[of a b] by auto 

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qed 

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finally show ?thesis . 
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qed 
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lemma borel_measurable_lebesgueI: 
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"f \<in> borel_measurable borel \<Longrightarrow> f \<in> borel_measurable lebesgue" 
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unfolding measurable_def by simp 
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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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using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI) 
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lemma lmeasure_eq_0: 
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fixes S :: "'a::ordered_euclidean_space set" 
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assumes "negligible S" shows "emeasure lebesgue S = 0" 

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proof  
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have "\<And>n. integral (cube n) (indicator S :: 'a\<Rightarrow>real) = 0" 
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unfolding lebesgue_integral_def using assms 
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by (intro integral_unique some1_equality ex_ex1I) 
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(auto simp: cube_def negligible_def) 
47694  250 
then show ?thesis 
251 
using assms by (simp add: emeasure_lebesgue lebesgueI_negligible) 

40859  252 
qed 
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lemma lmeasure_iff_LIMSEQ: 

47694  255 
assumes A: "A \<in> sets lebesgue" and "0 \<le> m" 
256 
shows "emeasure lebesgue A = ereal m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) > m" 

257 
proof (subst emeasure_lebesgue[OF A], intro SUP_eq_LIMSEQ) 

41654  258 
show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))" 
259 
using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD) 

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qed 
38656  261 

41654  262 
lemma has_integral_indicator_UNIV: 
263 
fixes s A :: "'a::ordered_euclidean_space set" and x :: real 

264 
shows "((indicator (s \<inter> A) :: 'a\<Rightarrow>real) has_integral x) UNIV = ((indicator s :: _\<Rightarrow>real) has_integral x) A" 

265 
proof  

266 
have "(\<lambda>x. if x \<in> A then indicator s x else 0) = (indicator (s \<inter> A) :: _\<Rightarrow>real)" 

267 
by (auto simp: fun_eq_iff indicator_def) 

268 
then show ?thesis 

269 
unfolding has_integral_restrict_univ[where s=A, symmetric] by simp 

40859  270 
qed 
38656  271 

41654  272 
lemma 
273 
fixes s a :: "'a::ordered_euclidean_space set" 

274 
shows integral_indicator_UNIV: 

275 
"integral UNIV (indicator (s \<inter> A) :: 'a\<Rightarrow>real) = integral A (indicator s :: _\<Rightarrow>real)" 

276 
and integrable_indicator_UNIV: 

277 
"(indicator (s \<inter> A) :: 'a\<Rightarrow>real) integrable_on UNIV \<longleftrightarrow> (indicator s :: 'a\<Rightarrow>real) integrable_on A" 

278 
unfolding integral_def integrable_on_def has_integral_indicator_UNIV by auto 

279 

280 
lemma lmeasure_finite_has_integral: 

281 
fixes s :: "'a::ordered_euclidean_space set" 

47694  282 
assumes "s \<in> sets lebesgue" "emeasure lebesgue s = ereal m" "0 \<le> m" 
41654  283 
shows "(indicator s has_integral m) UNIV" 
284 
proof  

285 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

286 
have **: "(?I s) integrable_on UNIV \<and> (\<lambda>k. integral UNIV (?I (s \<inter> cube k))) > integral UNIV (?I s)" 

287 
proof (intro monotone_convergence_increasing allI ballI) 

288 
have LIMSEQ: "(\<lambda>n. integral (cube n) (?I s)) > m" 

289 
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1, 3)] . 

290 
{ fix n have "integral (cube n) (?I s) \<le> m" 

291 
using cube_subset assms 

292 
by (intro incseq_le[where L=m] LIMSEQ incseq_def[THEN iffD2] integral_subset_le allI impI) 

293 
(auto dest!: lebesgueD) } 

294 
moreover 

295 
{ fix n have "0 \<le> integral (cube n) (?I s)" 

47694  296 
using assms by (auto dest!: lebesgueD intro!: Integration.integral_nonneg) } 
41654  297 
ultimately 
298 
show "bounded {integral UNIV (?I (s \<inter> cube k)) k. True}" 

299 
unfolding bounded_def 

300 
apply (rule_tac exI[of _ 0]) 

301 
apply (rule_tac exI[of _ m]) 

302 
by (auto simp: dist_real_def integral_indicator_UNIV) 

303 
fix k show "?I (s \<inter> cube k) integrable_on UNIV" 

304 
unfolding integrable_indicator_UNIV using assms by (auto dest!: lebesgueD) 

305 
fix x show "?I (s \<inter> cube k) x \<le> ?I (s \<inter> cube (Suc k)) x" 

306 
using cube_subset[of k "Suc k"] by (auto simp: indicator_def) 

307 
next 

308 
fix x :: 'a 

309 
from mem_big_cube obtain k where k: "x \<in> cube k" . 

310 
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x" 

311 
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } 

312 
note * = this 

313 
show "(\<lambda>k. ?I (s \<inter> cube k) x) > ?I s x" 

314 
by (rule LIMSEQ_offset[where k=k]) (auto simp: *) 

315 
qed 

316 
note ** = conjunctD2[OF this] 

317 
have m: "m = integral UNIV (?I s)" 

318 
apply (intro LIMSEQ_unique[OF _ **(2)]) 

319 
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1,3)] integral_indicator_UNIV . 

320 
show ?thesis 

321 
unfolding m by (intro integrable_integral **) 

38656  322 
qed 
323 

47694  324 
lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "emeasure lebesgue s \<noteq> \<infinity>" 
41654  325 
shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV" 
47694  326 
proof (cases "emeasure lebesgue s") 
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327 
case (real m) 
47694  328 
with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this] emeasure_nonneg[of lebesgue s] 
41654  329 
show ?thesis unfolding integrable_on_def by auto 
47694  330 
qed (insert assms emeasure_nonneg[of lebesgue s], auto) 
38656  331 

41654  332 
lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV" 
333 
shows "s \<in> sets lebesgue" 

334 
proof (intro lebesgueI) 

335 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

336 
fix n show "(?I s) integrable_on cube n" unfolding cube_def 

337 
proof (intro integrable_on_subinterval) 

338 
show "(?I s) integrable_on UNIV" 

339 
unfolding integrable_on_def using assms by auto 

340 
qed auto 

38656  341 
qed 
342 

41654  343 
lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV" 
47694  344 
shows "emeasure lebesgue s = ereal m" 
41654  345 
proof (intro lmeasure_iff_LIMSEQ[THEN iffD2]) 
346 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

347 
show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] . 

348 
show "0 \<le> m" using assms by (rule has_integral_nonneg) auto 

349 
have "(\<lambda>n. integral UNIV (?I (s \<inter> cube n))) > integral UNIV (?I s)" 

350 
proof (intro dominated_convergence(2) ballI) 

351 
show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto 

352 
fix n show "?I (s \<inter> cube n) integrable_on UNIV" 

353 
unfolding integrable_indicator_UNIV using `s \<in> sets lebesgue` by (auto dest: lebesgueD) 

354 
fix x show "norm (?I (s \<inter> cube n) x) \<le> ?I s x" by (auto simp: indicator_def) 

355 
next 

356 
fix x :: 'a 

357 
from mem_big_cube obtain k where k: "x \<in> cube k" . 

358 
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x" 

359 
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } 

360 
note * = this 

361 
show "(\<lambda>k. ?I (s \<inter> cube k) x) > ?I s x" 

362 
by (rule LIMSEQ_offset[where k=k]) (auto simp: *) 

363 
qed 

364 
then show "(\<lambda>n. integral (cube n) (?I s)) > m" 

365 
unfolding integral_unique[OF assms] integral_indicator_UNIV by simp 

366 
qed 

367 

368 
lemma has_integral_iff_lmeasure: 

47694  369 
"(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> emeasure lebesgue A = ereal m)" 
40859  370 
proof 
41654  371 
assume "(indicator A has_integral m) UNIV" 
372 
with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this] 

47694  373 
show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> emeasure lebesgue A = ereal m" 
41654  374 
by (auto intro: has_integral_nonneg) 
40859  375 
next 
47694  376 
assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> emeasure lebesgue A = ereal m" 
41654  377 
then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto 
38656  378 
qed 
379 

41654  380 
lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" 
47694  381 
shows "emeasure lebesgue s = ereal (integral UNIV (indicator s))" 
41654  382 
using assms unfolding integrable_on_def 
383 
proof safe 

384 
fix y :: real assume "(indicator s has_integral y) UNIV" 

385 
from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this] 

47694  386 
show "emeasure lebesgue s = ereal (integral UNIV (indicator s))" by simp 
40859  387 
qed 
38656  388 

389 
lemma lebesgue_simple_function_indicator: 

43920  390 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal" 
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391 
assumes f:"simple_function lebesgue f" 
38656  392 
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f ` {y}) x))" 
47694  393 
by (rule, subst simple_function_indicator_representation[OF f]) auto 
38656  394 

41654  395 
lemma integral_eq_lmeasure: 
47694  396 
"(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (emeasure lebesgue s)" 
41654  397 
by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg) 
38656  398 

47694  399 
lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "emeasure lebesgue s \<noteq> \<infinity>" 
41654  400 
using lmeasure_eq_integral[OF assms] by auto 
38656  401 

40859  402 
lemma negligible_iff_lebesgue_null_sets: 
47694  403 
"negligible A \<longleftrightarrow> A \<in> null_sets lebesgue" 
40859  404 
proof 
405 
assume "negligible A" 

406 
from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0] 

47694  407 
show "A \<in> null_sets lebesgue" by auto 
40859  408 
next 
47694  409 
assume A: "A \<in> null_sets lebesgue" 
410 
then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] 

411 
by (auto simp: null_sets_def) 

41654  412 
show "negligible A" unfolding negligible_def 
413 
proof (intro allI) 

414 
fix a b :: 'a 

415 
have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on {a..b}" 

416 
by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *) 

417 
then have "integral {a..b} (indicator A) \<le> (integral UNIV (indicator A) :: real)" 

47694  418 
using * by (auto intro!: integral_subset_le) 
41654  419 
moreover have "(0::real) \<le> integral {a..b} (indicator A)" 
420 
using integrable by (auto intro!: integral_nonneg) 

421 
ultimately have "integral {a..b} (indicator A) = (0::real)" 

422 
using integral_unique[OF *] by auto 

423 
then show "(indicator A has_integral (0::real)) {a..b}" 

424 
using integrable_integral[OF integrable] by simp 

425 
qed 

426 
qed 

427 

428 
lemma integral_const[simp]: 

429 
fixes a b :: "'a::ordered_euclidean_space" 

430 
shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c" 

431 
by (rule integral_unique) (rule has_integral_const) 

432 

47694  433 
lemma lmeasure_UNIV[intro]: "emeasure lebesgue (UNIV::'a::ordered_euclidean_space set) = \<infinity>" 
434 
proof (simp add: emeasure_lebesgue, intro SUP_PInfty bexI) 

41981
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changeset

435 
fix n :: nat 
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diff
changeset

436 
have "indicator UNIV = (\<lambda>x::'a. 1 :: real)" by auto 
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diff
changeset

437 
moreover 
cdf7693bbe08
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diff
changeset

438 
{ have "real n \<le> (2 * real n) ^ DIM('a)" 
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changeset

439 
proof (cases n) 
cdf7693bbe08
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diff
changeset

440 
case 0 then show ?thesis by auto 
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changeset

441 
next 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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changeset

442 
case (Suc n') 
cdf7693bbe08
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diff
changeset

443 
have "real n \<le> (2 * real n)^1" by auto 
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changeset

444 
also have "(2 * real n)^1 \<le> (2 * real n) ^ DIM('a)" 
cdf7693bbe08
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changeset

445 
using Suc DIM_positive[where 'a='a] by (intro power_increasing) (auto simp: real_of_nat_Suc) 
cdf7693bbe08
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changeset

446 
finally show ?thesis . 
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changeset

447 
qed } 
43920  448 
ultimately show "ereal (real n) \<le> ereal (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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changeset

449 
using integral_const DIM_positive[where 'a='a] 
cdf7693bbe08
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changeset

450 
by (auto simp: cube_def content_closed_interval_cases setprod_constant) 
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changeset

451 
qed simp 
40859  452 

453 
lemma 

454 
fixes a b ::"'a::ordered_euclidean_space" 

47694  455 
shows lmeasure_atLeastAtMost[simp]: "emeasure lebesgue {a..b} = ereal (content {a..b})" 
41654  456 
proof  
457 
have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV" 

46905  458 
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def [abs_def]) 
41654  459 
from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV 
46905  460 
by (simp add: indicator_def [abs_def]) 
40859  461 
qed 
462 

463 
lemma atLeastAtMost_singleton_euclidean[simp]: 

464 
fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}" 

465 
by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a]) 

466 

467 
lemma content_singleton[simp]: "content {a} = 0" 

468 
proof  

469 
have "content {a .. a} = 0" 

470 
by (subst content_closed_interval) auto 

471 
then show ?thesis by simp 

472 
qed 

473 

474 
lemma lmeasure_singleton[simp]: 

47694  475 
fixes a :: "'a::ordered_euclidean_space" shows "emeasure lebesgue {a} = 0" 
41654  476 
using lmeasure_atLeastAtMost[of a a] by simp 
40859  477 

478 
declare content_real[simp] 

479 

480 
lemma 

481 
fixes a b :: real 

482 
shows lmeasure_real_greaterThanAtMost[simp]: 

47694  483 
"emeasure lebesgue {a <.. b} = ereal (if a \<le> b then b  a else 0)" 
40859  484 
proof cases 
485 
assume "a < b" 

47694  486 
then have "emeasure lebesgue {a <.. b} = emeasure lebesgue {a .. b}  emeasure lebesgue {a}" 
487 
by (subst emeasure_Diff[symmetric]) 

488 
(auto intro!: arg_cong[where f="emeasure lebesgue"]) 

40859  489 
then show ?thesis by auto 
490 
qed auto 

491 

492 
lemma 

493 
fixes a b :: real 

494 
shows lmeasure_real_atLeastLessThan[simp]: 

47694  495 
"emeasure lebesgue {a ..< b} = ereal (if a \<le> b then b  a else 0)" 
40859  496 
proof cases 
497 
assume "a < b" 

47694  498 
then have "emeasure lebesgue {a ..< b} = emeasure lebesgue {a .. b}  emeasure lebesgue {b}" 
499 
by (subst emeasure_Diff[symmetric]) 

500 
(auto intro!: arg_cong[where f="emeasure lebesgue"]) 

41654  501 
then show ?thesis by auto 
502 
qed auto 

503 

504 
lemma 

505 
fixes a b :: real 

506 
shows lmeasure_real_greaterThanLessThan[simp]: 

47694  507 
"emeasure lebesgue {a <..< b} = ereal (if a \<le> b then b  a else 0)" 
41654  508 
proof cases 
509 
assume "a < b" 

47694  510 
then have "emeasure lebesgue {a <..< b} = emeasure lebesgue {a <.. b}  emeasure lebesgue {b}" 
511 
by (subst emeasure_Diff[symmetric]) 

512 
(auto intro!: arg_cong[where f="emeasure lebesgue"]) 

40859  513 
then show ?thesis by auto 
514 
qed auto 

515 

41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

516 
subsection {* LebesgueBorel measure *} 
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

517 

47694  518 
definition "lborel = measure_of UNIV (sets borel) (emeasure lebesgue)" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

519 

3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

520 
lemma 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

521 
shows space_lborel[simp]: "space lborel = UNIV" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

522 
and sets_lborel[simp]: "sets lborel = sets borel" 
47694  523 
and measurable_lborel1[simp]: "measurable lborel = measurable borel" 
524 
and measurable_lborel2[simp]: "measurable A lborel = measurable A borel" 

525 
using sigma_sets_eq[of borel] 

526 
by (auto simp add: lborel_def measurable_def[abs_def]) 

40859  527 

47694  528 
lemma emeasure_lborel[simp]: "A \<in> sets borel \<Longrightarrow> emeasure lborel A = emeasure lebesgue A" 
529 
by (rule emeasure_measure_of[OF lborel_def]) 

530 
(auto simp: positive_def emeasure_nonneg countably_additive_def intro!: suminf_emeasure) 

40859  531 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

532 
interpretation lborel: sigma_finite_measure lborel 
47694  533 
proof (default, intro conjI exI[of _ "\<lambda>n. cube n"]) 
534 
show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed) 

535 
{ fix x :: 'a have "\<exists>n. x\<in>cube n" using mem_big_cube by auto } 

536 
then show "(\<Union>i. cube i) = (space lborel :: 'a set)" using mem_big_cube by auto 

537 
show "\<forall>i. emeasure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def) 

538 
qed 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

539 

3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

540 
interpretation lebesgue: sigma_finite_measure lebesgue 
40859  541 
proof 
47694  542 
from lborel.sigma_finite guess A :: "nat \<Rightarrow> 'a set" .. 
543 
then show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. emeasure lebesgue (A i) \<noteq> \<infinity>)" 

544 
by (intro exI[of _ A]) (auto simp: subset_eq) 

40859  545 
qed 
546 

41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

547 
subsection {* Lebesgue integrable implies Gauge integrable *} 
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

548 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

549 
lemma has_integral_cmult_real: 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

550 
fixes c :: real 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

551 
assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

552 
shows "((\<lambda>x. c * f x) has_integral c * x) A" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

553 
proof cases 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

554 
assume "c \<noteq> 0" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

555 
from has_integral_cmul[OF assms[OF this], of c] show ?thesis 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

556 
unfolding real_scaleR_def . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

557 
qed simp 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

558 

40859  559 
lemma simple_function_has_integral: 
43920  560 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

561 
assumes f:"simple_function lebesgue f" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

562 
and f':"range f \<subseteq> {0..<\<infinity>}" 
47694  563 
and om:"\<And>x. x \<in> range f \<Longrightarrow> emeasure lebesgue (f ` {x} \<inter> UNIV) = \<infinity> \<Longrightarrow> x = 0" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

564 
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

565 
unfolding simple_integral_def space_lebesgue 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

566 
proof (subst lebesgue_simple_function_indicator) 
47694  567 
let ?M = "\<lambda>x. emeasure lebesgue (f ` {x} \<inter> UNIV)" 
46731  568 
let ?F = "\<lambda>x. indicator (f ` {x})" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

569 
{ fix x y assume "y \<in> range f" 
43920  570 
from subsetD[OF f' this] have "y * ?F y x = ereal (real y * ?F y x)" 
571 
by (cases rule: ereal2_cases[of y "?F y x"]) 

572 
(auto simp: indicator_def one_ereal_def split: split_if_asm) } 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

573 
moreover 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

574 
{ fix x assume x: "x\<in>range f" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

575 
have "x * ?M x = real x * real (?M x)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

576 
proof cases 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

577 
assume "x \<noteq> 0" with om[OF x] have "?M x \<noteq> \<infinity>" by auto 
47694  578 
with subsetD[OF f' x] f[THEN simple_functionD(2)] show ?thesis 
43920  579 
by (cases rule: ereal2_cases[of x "?M x"]) auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

580 
qed simp } 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

581 
ultimately 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

582 
have "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV \<longleftrightarrow> 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

583 
((\<lambda>x. \<Sum>y\<in>range f. real y * ?F y x) has_integral (\<Sum>x\<in>range f. real x * real (?M x))) UNIV" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

584 
by simp 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

585 
also have \<dots> 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

586 
proof (intro has_integral_setsum has_integral_cmult_real lmeasure_finite_has_integral 
47694  587 
real_of_ereal_pos emeasure_nonneg ballI) 
588 
show *: "finite (range f)" "\<And>y. f ` {y} \<in> sets lebesgue" 

589 
using simple_functionD[OF f] by auto 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

590 
fix y assume "real y \<noteq> 0" "y \<in> range f" 
47694  591 
with * om[OF this(2)] show "emeasure lebesgue (f ` {y}) = ereal (real (?M y))" 
43920  592 
by (auto simp: ereal_real) 
41654  593 
qed 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

594 
finally show "((\<lambda>x. real (\<Sum>y\<in>range f. y * ?F y x)) has_integral real (\<Sum>x\<in>range f. x * ?M x)) UNIV" . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

595 
qed fact 
40859  596 

597 
lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s" 

598 
unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI) 

599 
using assms by auto 

600 

601 
lemma simple_function_has_integral': 

43920  602 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> ereal" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

603 
assumes f: "simple_function lebesgue f" "\<And>x. 0 \<le> f x" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

604 
and i: "integral\<^isup>S lebesgue f \<noteq> \<infinity>" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

605 
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>S lebesgue f))) UNIV" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

606 
proof  
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

607 
let ?f = "\<lambda>x. if x \<in> f ` {\<infinity>} then 0 else f x" 
47694  608 
note f(1)[THEN simple_functionD(2)] 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

609 
then have [simp, intro]: "\<And>X. f ` X \<in> sets lebesgue" by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

610 
have f': "simple_function lebesgue ?f" 
47694  611 
using f by (intro simple_function_If_set) auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

612 
have rng: "range ?f \<subseteq> {0..<\<infinity>}" using f by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

613 
have "AE x in lebesgue. f x = ?f x" 
47694  614 
using simple_integral_PInf[OF f i] 
615 
by (intro AE_I[where N="f ` {\<infinity>} \<inter> space lebesgue"]) auto 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

616 
from f(1) f' this have eq: "integral\<^isup>S lebesgue f = integral\<^isup>S lebesgue ?f" 
47694  617 
by (rule simple_integral_cong_AE) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

618 
have real_eq: "\<And>x. real (f x) = real (?f x)" by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

619 

cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

620 
show ?thesis 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

621 
unfolding eq real_eq 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

622 
proof (rule simple_function_has_integral[OF f' rng]) 
47694  623 
fix x assume x: "x \<in> range ?f" and inf: "emeasure lebesgue (?f ` {x} \<inter> UNIV) = \<infinity>" 
624 
have "x * emeasure lebesgue (?f ` {x} \<inter> UNIV) = (\<integral>\<^isup>S y. x * indicator (?f ` {x}) y \<partial>lebesgue)" 

625 
using f'[THEN simple_functionD(2)] 

626 
by (simp add: simple_integral_cmult_indicator) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

627 
also have "\<dots> \<le> integral\<^isup>S lebesgue f" 
47694  628 
using f'[THEN simple_functionD(2)] f 
629 
by (intro simple_integral_mono simple_function_mult simple_function_indicator) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

630 
(auto split: split_indicator) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

631 
finally show "x = 0" unfolding inf using i subsetD[OF rng x] by (auto split: split_if_asm) 
40859  632 
qed 
633 
qed 

634 

635 
lemma positive_integral_has_integral: 

43920  636 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

637 
assumes f: "f \<in> borel_measurable lebesgue" "range f \<subseteq> {0..<\<infinity>}" "integral\<^isup>P lebesgue f \<noteq> \<infinity>" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

638 
shows "((\<lambda>x. real (f x)) has_integral (real (integral\<^isup>P lebesgue f))) UNIV" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

639 
proof  
47694  640 
from borel_measurable_implies_simple_function_sequence'[OF f(1)] 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

641 
guess u . note u = this 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

642 
have SUP_eq: "\<And>x. (SUP i. u i x) = f x" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

643 
using u(4) f(2)[THEN subsetD] by (auto split: split_max) 
46731  644 
let ?u = "\<lambda>i x. real (u i x)" 
47694  645 
note u_eq = positive_integral_eq_simple_integral[OF u(1,5), symmetric] 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

646 
{ fix i 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

647 
note u_eq 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

648 
also have "integral\<^isup>P lebesgue (u i) \<le> (\<integral>\<^isup>+x. max 0 (f x) \<partial>lebesgue)" 
47694  649 
by (intro positive_integral_mono) (auto intro: SUP_upper simp: u(4)[symmetric]) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

650 
finally have "integral\<^isup>S lebesgue (u i) \<noteq> \<infinity>" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

651 
unfolding positive_integral_max_0 using f by auto } 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

652 
note u_fin = this 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

653 
then have u_int: "\<And>i. (?u i has_integral real (integral\<^isup>S lebesgue (u i))) UNIV" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

654 
by (rule simple_function_has_integral'[OF u(1,5)]) 
43920  655 
have "\<forall>x. \<exists>r\<ge>0. f x = ereal r" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

656 
proof 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

657 
fix x from f(2) have "0 \<le> f x" "f x \<noteq> \<infinity>" by (auto simp: subset_eq) 
43920  658 
then show "\<exists>r\<ge>0. f x = ereal r" by (cases "f x") auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

659 
qed 
43920  660 
from choice[OF this] obtain f' where f': "f = (\<lambda>x. ereal (f' x))" "\<And>x. 0 \<le> f' x" by auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

661 

43920  662 
have "\<forall>i. \<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = ereal (r x)" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

663 
proof 
43920  664 
fix i show "\<exists>r. \<forall>x. 0 \<le> r x \<and> u i x = ereal (r x)" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

665 
proof (intro choice allI) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

666 
fix i x have "u i x \<noteq> \<infinity>" using u(3)[of i] by (auto simp: image_iff) metis 
43920  667 
then show "\<exists>r\<ge>0. u i x = ereal r" using u(5)[of i x] by (cases "u i x") auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

668 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

669 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

670 
from choice[OF this] obtain u' where 
43920  671 
u': "u = (\<lambda>i x. ereal (u' i x))" "\<And>i x. 0 \<le> u' i x" by (auto simp: fun_eq_iff) 
40859  672 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

673 
have convergent: "f' integrable_on UNIV \<and> 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

674 
(\<lambda>k. integral UNIV (u' k)) > integral UNIV f'" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

675 
proof (intro monotone_convergence_increasing allI ballI) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

676 
show int: "\<And>k. (u' k) integrable_on UNIV" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

677 
using u_int unfolding integrable_on_def u' by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

678 
show "\<And>k x. u' k x \<le> u' (Suc k) x" using u(2,3,5) 
43920  679 
by (auto simp: incseq_Suc_iff le_fun_def image_iff u' intro!: real_of_ereal_positive_mono) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

680 
show "\<And>x. (\<lambda>k. u' k x) > f' x" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

681 
using SUP_eq u(2) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

682 
by (intro SUP_eq_LIMSEQ[THEN iffD1]) (auto simp: u' f' incseq_mono incseq_Suc_iff le_fun_def) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

683 
show "bounded {integral UNIV (u' k)k. True}" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

684 
proof (safe intro!: bounded_realI) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

685 
fix k 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

686 
have "\<bar>integral UNIV (u' k)\<bar> = integral UNIV (u' k)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

687 
by (intro abs_of_nonneg integral_nonneg int ballI u') 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

688 
also have "\<dots> = real (integral\<^isup>S lebesgue (u k))" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

689 
using u_int[THEN integral_unique] by (simp add: u') 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

690 
also have "\<dots> = real (integral\<^isup>P lebesgue (u k))" 
47694  691 
using positive_integral_eq_simple_integral[OF u(1,5)] by simp 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

692 
also have "\<dots> \<le> real (integral\<^isup>P lebesgue f)" using f 
47694  693 
by (auto intro!: real_of_ereal_positive_mono positive_integral_positive 
694 
positive_integral_mono SUP_upper simp: SUP_eq[symmetric]) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

695 
finally show "\<bar>integral UNIV (u' k)\<bar> \<le> real (integral\<^isup>P lebesgue f)" . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

696 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

697 
qed 
40859  698 

43920  699 
have "integral\<^isup>P lebesgue f = ereal (integral UNIV f')" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

700 
proof (rule tendsto_unique[OF trivial_limit_sequentially]) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

701 
have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) > (SUP i. integral\<^isup>P lebesgue (u i))" 
47694  702 
unfolding u_eq by (intro LIMSEQ_ereal_SUPR incseq_positive_integral u) 
703 
also note positive_integral_monotone_convergence_SUP 

704 
[OF u(2) borel_measurable_simple_function[OF u(1)] u(5), symmetric] 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

705 
finally show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) > integral\<^isup>P lebesgue f" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

706 
unfolding SUP_eq . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

707 

cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

708 
{ fix k 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

709 
have "0 \<le> integral\<^isup>S lebesgue (u k)" 
47694  710 
using u by (auto intro!: simple_integral_positive) 
43920  711 
then have "integral\<^isup>S lebesgue (u k) = ereal (real (integral\<^isup>S lebesgue (u k)))" 
712 
using u_fin by (auto simp: ereal_real) } 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

713 
note * = this 
43920  714 
show "(\<lambda>k. integral\<^isup>S lebesgue (u k)) > ereal (integral UNIV f')" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

715 
using convergent using u_int[THEN integral_unique, symmetric] 
47694  716 
by (subst *) (simp add: u') 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

717 
qed 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

718 
then show ?thesis using convergent by (simp add: f' integrable_integral) 
40859  719 
qed 
720 

721 
lemma lebesgue_integral_has_integral: 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

722 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

723 
assumes f: "integrable lebesgue f" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

724 
shows "(f has_integral (integral\<^isup>L lebesgue f)) UNIV" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

725 
proof  
43920  726 
let ?n = "\<lambda>x. real (ereal (max 0 ( f x)))" and ?p = "\<lambda>x. real (ereal (max 0 (f x)))" 
727 
have *: "f = (\<lambda>x. ?p x  ?n x)" by (auto simp del: ereal_max) 

47694  728 
{ fix f :: "'a \<Rightarrow> real" have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. ereal (max 0 (f x)) \<partial>lebesgue)" 
729 
by (intro positive_integral_cong_pos) (auto split: split_max) } 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

730 
note eq = this 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

731 
show ?thesis 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

732 
unfolding lebesgue_integral_def 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

733 
apply (subst *) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

734 
apply (rule has_integral_sub) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

735 
unfolding eq[of f] eq[of "\<lambda>x.  f x"] 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

736 
apply (safe intro!: positive_integral_has_integral) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

737 
using integrableD[OF f] 
43920  738 
by (auto simp: zero_ereal_def[symmetric] positive_integral_max_0 split: split_max 
47694  739 
intro!: measurable_If) 
40859  740 
qed 
741 

47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

742 
lemma lebesgue_simple_integral_eq_borel: 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

743 
assumes f: "f \<in> borel_measurable borel" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

744 
shows "integral\<^isup>S lebesgue f = integral\<^isup>S lborel f" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

745 
using f[THEN measurable_sets] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

746 
by (auto intro!: setsum_cong arg_cong2[where f="op *"] emeasure_lborel[symmetric] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

747 
simp: simple_integral_def) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

748 

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

749 
lemma lebesgue_positive_integral_eq_borel: 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

750 
assumes f: "f \<in> borel_measurable borel" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

751 
shows "integral\<^isup>P lebesgue f = integral\<^isup>P lborel f" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

752 
proof  
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

753 
from f have "integral\<^isup>P lebesgue (\<lambda>x. max 0 (f x)) = integral\<^isup>P lborel (\<lambda>x. max 0 (f x))" 
47694  754 
by (auto intro!: positive_integral_subalgebra[symmetric]) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

755 
then show ?thesis unfolding positive_integral_max_0 . 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

756 
qed 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

757 

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

758 
lemma lebesgue_integral_eq_borel: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

759 
assumes "f \<in> borel_measurable borel" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

760 
shows "integrable lebesgue f \<longleftrightarrow> integrable lborel f" (is ?P) 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

761 
and "integral\<^isup>L lebesgue f = integral\<^isup>L lborel f" (is ?I) 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

762 
proof  
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

763 
have "sets lborel \<subseteq> sets lebesgue" by auto 
47694  764 
from integral_subalgebra[of f lborel, OF _ this _ _] assms 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

765 
show ?P ?I by auto 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

766 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

767 

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

768 
lemma borel_integral_has_integral: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

769 
fixes f::"'a::ordered_euclidean_space => real" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

770 
assumes f:"integrable lborel f" 
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

771 
shows "(f has_integral (integral\<^isup>L lborel f)) UNIV" 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

772 
proof  
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

773 
have borel: "f \<in> borel_measurable borel" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

774 
using f unfolding integrable_def by auto 
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

775 
from f show ?thesis 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

776 
using lebesgue_integral_has_integral[of f] 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

777 
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

778 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

779 

41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

780 
subsection {* Equivalence between product spaces and euclidean spaces *} 
40859  781 

782 
definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where 

783 
"e2p x = (\<lambda>i\<in>{..<DIM('a)}. x$$i)" 

784 

785 
definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where 

786 
"p2e x = (\<chi>\<chi> i. x i)" 

787 

41095  788 
lemma e2p_p2e[simp]: 
789 
"x \<in> extensional {..<DIM('a)} \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x" 

790 
by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def) 

40859  791 

41095  792 
lemma p2e_e2p[simp]: 
793 
"p2e (e2p x) = (x::'a::ordered_euclidean_space)" 

794 
by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def) 

40859  795 

47694  796 
interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure" 
40859  797 
by default 
798 

47694  799 
interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure" "{..<n}" for n :: nat 
800 
by default auto 

801 

802 
lemma bchoice_iff: "(\<forall>x\<in>A. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>A. P x (f x))" 

803 
by metis 

40859  804 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

805 
lemma sets_product_borel: 
47694  806 
assumes I: "finite I" 
807 
shows "sets (\<Pi>\<^isub>M i\<in>I. lborel) = sigma_sets (\<Pi>\<^isub>E i\<in>I. UNIV) { \<Pi>\<^isub>E i\<in>I. {..< x i :: real}  x. True}" (is "_ = ?G") 

808 
proof (subst sigma_prod_algebra_sigma_eq[where S="\<lambda>_ i::nat. {..<real i}" and E="\<lambda>_. range lessThan", OF I]) 

809 
show "sigma_sets (space (Pi\<^isub>M I (\<lambda>i. lborel))) {Pi\<^isub>E I F F. \<forall>i\<in>I. F i \<in> range lessThan} = ?G" 

810 
by (intro arg_cong2[where f=sigma_sets]) (auto simp: space_PiM image_iff bchoice_iff) 

811 
qed (auto simp: borel_eq_lessThan incseq_def reals_Archimedean2 image_iff intro: real_natceiling_ge) 

40859  812 

41661  813 
lemma measurable_e2p: 
47694  814 
"e2p \<in> measurable (borel::'a::ordered_euclidean_space measure) (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure))" 
815 
proof (rule measurable_sigma_sets[OF sets_product_borel]) 

816 
fix A :: "(nat \<Rightarrow> real) set" assume "A \<in> {\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..<x i} x. True} " 

817 
then obtain x where "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..<x i})" by auto 

818 
then have "e2p ` A = {..< (\<chi>\<chi> i. x i) :: 'a}" 

819 
using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def 

820 
euclidean_eq[where 'a='a] eucl_less[where 'a='a]) 

821 
then show "e2p ` A \<inter> space (borel::'a measure) \<in> sets borel" by simp 

822 
qed (auto simp: e2p_def) 

41661  823 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

824 
lemma measurable_p2e: 
47694  825 
"p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure)) 
826 
(borel :: 'a::ordered_euclidean_space measure)" 

41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

827 
(is "p2e \<in> measurable ?P _") 
47694  828 
proof (safe intro!: borel_measurable_iff_halfspace_le[THEN iffD2]) 
829 
fix x i 

830 
let ?A = "{w \<in> space ?P. (p2e w :: 'a) $$ i \<le> x}" 

831 
assume "i < DIM('a)" 

832 
then have "?A = (\<Pi>\<^isub>E j\<in>{..<DIM('a)}. if i = j then {.. x} else UNIV)" 

833 
using DIM_positive by (auto simp: space_PiM p2e_def split: split_if_asm) 

834 
then show "?A \<in> sets ?P" 

835 
by auto 

836 
qed 

837 

838 
lemma Int_stable_atLeastAtMost: 

839 
fixes x::"'a::ordered_euclidean_space" 

840 
shows "Int_stable (range (\<lambda>(a, b::'a). {a..b}))" 

841 
by (auto simp: inter_interval Int_stable_def) 

41095  842 

47694  843 
lemma lborel_eqI: 
844 
fixes M :: "'a::ordered_euclidean_space measure" 

845 
assumes emeasure_eq: "\<And>a b. emeasure M {a .. b} = content {a .. b}" 

846 
assumes sets_eq: "sets M = sets borel" 

847 
shows "lborel = M" 

848 
proof (rule measure_eqI_generator_eq[OF Int_stable_atLeastAtMost]) 

849 
let ?P = "\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel" 

850 
let ?E = "range (\<lambda>(a, b). {a..b} :: 'a set)" 

851 
show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E" 

852 
by (simp_all add: borel_eq_atLeastAtMost sets_eq) 

47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

853 

47694  854 
show "range cube \<subseteq> ?E" unfolding cube_def [abs_def] by auto 
855 
show "incseq cube" using cube_subset_Suc by (auto intro!: incseq_SucI) 

856 
{ fix x :: 'a have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastforce } 

857 
then show "(\<Union>i. cube i :: 'a set) = UNIV" by auto 

858 

859 
{ fix i show "emeasure lborel (cube i) \<noteq> \<infinity>" unfolding cube_def by auto } 

860 
{ fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X" 

861 
by (auto simp: emeasure_eq) } 

862 
qed 

40859  863 

41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

864 
lemma lborel_eq_lborel_space: 
47694  865 
"(lborel :: 'a measure) = distr (\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel) lborel p2e" 
866 
(is "?B = ?D") 

867 
proof (rule lborel_eqI) 

868 
show "sets ?D = sets borel" by simp 

869 
let ?P = "(\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel)" 

870 
fix a b :: 'a 

871 
have *: "p2e ` {a .. b} \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {a $$ i .. b $$ i})" 

872 
by (auto simp: Pi_iff eucl_le[where 'a='a] p2e_def space_PiM) 

873 
have "emeasure ?P (p2e ` {a..b} \<inter> space ?P) = content {a..b}" 

874 
proof cases 

875 
assume "{a..b} \<noteq> {}" 

876 
then have "a \<le> b" 

877 
by (simp add: interval_ne_empty eucl_le[where 'a='a]) 

878 
then have "emeasure lborel {a..b} = (\<Prod>x<DIM('a). emeasure lborel {a $$ x .. b $$ x})" 

879 
by (auto simp: content_closed_interval eucl_le[where 'a='a] 

880 
intro!: setprod_ereal[symmetric]) 

881 
also have "\<dots> = emeasure ?P (p2e ` {a..b} \<inter> space ?P)" 

882 
unfolding * by (subst lborel_space.measure_times) auto 

883 
finally show ?thesis by simp 

884 
qed simp 

885 
then show "emeasure ?D {a .. b} = content {a .. b}" 

886 
by (simp add: emeasure_distr measurable_p2e) 

41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

887 
qed 
40859  888 

889 
lemma borel_fubini_positiv_integral: 

43920  890 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal" 
40859  891 
assumes f: "f \<in> borel_measurable borel" 
47694  892 
shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel)" 
893 
by (subst lborel_eq_lborel_space) (simp add: positive_integral_distr measurable_p2e f) 

40859  894 

41704
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

895 
lemma borel_fubini_integrable: 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

896 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

897 
shows "integrable lborel f \<longleftrightarrow> 
47694  898 
integrable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel) (\<lambda>x. f (p2e x))" 
41704
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

899 
(is "_ \<longleftrightarrow> integrable ?B ?f") 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

900 
proof 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

901 
assume "integrable lborel f" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

902 
moreover then have f: "f \<in> borel_measurable borel" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

903 
by auto 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

904 
moreover with measurable_p2e 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

905 
have "f \<circ> p2e \<in> borel_measurable ?B" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

906 
by (rule measurable_comp) 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

907 
ultimately show "integrable ?B ?f" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

908 
by (simp add: comp_def borel_fubini_positiv_integral integrable_def) 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

909 
next 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

910 
assume "integrable ?B ?f" 
47694  911 
moreover 
912 
then have "?f \<circ> e2p \<in> borel_measurable (borel::'a measure)" 

913 
by (auto intro!: measurable_e2p) 

41704
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

914 
then have "f \<in> borel_measurable borel" 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

915 
by (simp cong: measurable_cong) 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

916 
ultimately show "integrable lborel f" 
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

917 
by (simp add: borel_fubini_positiv_integral integrable_def) 
41704
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

918 
qed 
8c539202f854
add borel_fubini_integrable; remove unused bijectivity rules for measureable functions
hoelzl
parents:
41689
diff
changeset

919 

40859  920 
lemma borel_fubini: 
921 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 

922 
assumes f: "f \<in> borel_measurable borel" 

47694  923 
shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>((\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel))" 
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset

924 
using f by (simp add: borel_fubini_positiv_integral lebesgue_integral_def) 
38656  925 

47694  926 
lemma borel_measurable_indicator': 
927 
"A \<in> sets borel \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. indicator A (f x)) \<in> borel_measurable M" 

928 
using measurable_comp[OF _ borel_measurable_indicator, of f M borel A] by (auto simp add: comp_def) 

42164  929 

930 
lemma lebesgue_real_affine: 

47694  931 
fixes c :: real assumes "c \<noteq> 0" 
932 
shows "lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. \<bar>c\<bar>)" (is "_ = ?D") 

933 
proof (rule lborel_eqI) 

934 
fix a b show "emeasure ?D {a..b} = content {a .. b}" 

935 
proof cases 

936 
assume "0 < c" 

937 
then have "(\<lambda>x. t + c * x) ` {a..b} = {(a  t) / c .. (b  t) / c}" 

938 
by (auto simp: field_simps) 

939 
with `0 < c` show ?thesis 

940 
by (cases "a \<le> b") 

941 
(auto simp: field_simps emeasure_density positive_integral_distr positive_integral_cmult 

942 
borel_measurable_indicator' emeasure_distr) 

943 
next 

944 
assume "\<not> 0 < c" with `c \<noteq> 0` have "c < 0" by auto 

945 
then have *: "(\<lambda>x. t + c * x) ` {a..b} = {(b  t) / c .. (a  t) / c}" 

946 
by (auto simp: field_simps) 

947 
with `c < 0` show ?thesis 

948 
by (cases "a \<le> b") 

949 
(auto simp: field_simps emeasure_density positive_integral_distr 

950 
positive_integral_cmult borel_measurable_indicator' emeasure_distr) 

42164  951 
qed 
47694  952 
qed simp 
42164  953 

47757
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

954 
lemma borel_cube[intro]: "cube n \<in> sets borel" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

955 
unfolding cube_def by auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

956 

5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

957 
lemma integrable_on_cmult_iff: 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

958 
fixes c :: real assumes "c \<noteq> 0" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

959 
shows "(\<lambda>x. c * f x) integrable_on s \<longleftrightarrow> f integrable_on s" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

960 
using integrable_cmul[of "\<lambda>x. c * f x" s "1 / c"] integrable_cmul[of f s c] `c \<noteq> 0` 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

961 
by auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

962 

5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

963 
lemma positive_integral_borel_has_integral: 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

964 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

965 
assumes f_borel: "f \<in> borel_measurable borel" and nonneg: "\<And>x. 0 \<le> f x" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

966 
assumes I: "(f has_integral I) UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

967 
shows "(\<integral>\<^isup>+x. f x \<partial>lborel) = I" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

968 
proof  
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

969 
from f_borel have "(\<lambda>x. ereal (f x)) \<in> borel_measurable borel" by auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

970 
from borel_measurable_implies_simple_function_sequence'[OF this] guess F . note F = this 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

971 

5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

972 
have lebesgue_eq: "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. ereal (f x) \<partial>lborel)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

973 
using f_borel by (intro lebesgue_positive_integral_eq_borel) auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

974 
also have "\<dots> = (SUP i. integral\<^isup>S lborel (F i))" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

975 
using F 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

976 
by (subst positive_integral_monotone_convergence_simple) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

977 
(simp_all add: positive_integral_max_0 simple_function_def) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

978 
also have "\<dots> \<le> ereal I" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

979 
proof (rule SUP_least) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

980 
fix i :: nat 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

981 

5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

982 
{ fix z 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

983 
from F(4)[of z] have "F i z \<le> ereal (f z)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

984 
by (metis SUP_upper UNIV_I ereal_max_0 min_max.sup_absorb2 nonneg) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

985 
with F(5)[of i z] have "real (F i z) \<le> f z" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

986 
by (cases "F i z") simp_all } 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

987 
note F_bound = this 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

988 

5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

989 
{ fix x :: ereal assume x: "x \<noteq> 0" "x \<in> range (F i)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

990 
with F(3,5)[of i] have [simp]: "real x \<noteq> 0" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

991 
by (metis image_iff order_eq_iff real_of_ereal_le_0) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

992 
let ?s = "(\<lambda>n z. real x * indicator (F i ` {x} \<inter> cube n) z) :: nat \<Rightarrow> 'a \<Rightarrow> real" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

993 
have "(\<lambda>z::'a. real x * indicator (F i ` {x}) z) integrable_on UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

994 
proof (rule dominated_convergence(1)) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

995 
fix n :: nat 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

996 
have "(\<lambda>z. indicator (F i ` {x} \<inter> cube n) z :: real) integrable_on cube n" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

997 
using x F(1)[of i] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

998 
by (intro lebesgueD) (auto simp: simple_function_def) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

999 
then have cube: "?s n integrable_on cube n" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1000 
by (simp add: integrable_on_cmult_iff) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1001 
show "?s n integrable_on UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1002 
by (rule integrable_on_superset[OF _ _ cube]) auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1003 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1004 
show "f integrable_on UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1005 
unfolding integrable_on_def using I by auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1006 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1007 
fix n from F_bound show "\<forall>x\<in>UNIV. norm (?s n x) \<le> f x" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1008 
using nonneg F(5) by (auto split: split_indicator) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1009 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1010 
show "\<forall>z\<in>UNIV. (\<lambda>n. ?s n z) > real x * indicator (F i ` {x}) z" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1011 
proof 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1012 
fix z :: 'a 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1013 
from mem_big_cube[of z] guess j . 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1014 
then have x: "eventually (\<lambda>n. ?s n z = real x * indicator (F i ` {x}) z) sequentially" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1015 
by (auto intro!: eventually_sequentiallyI[where c=j] dest!: cube_subset split: split_indicator) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1016 
then show "(\<lambda>n. ?s n z) > real x * indicator (F i ` {x}) z" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1017 
by (rule Lim_eventually) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1018 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1019 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1020 
then have "(indicator (F i ` {x}) :: 'a \<Rightarrow> real) integrable_on UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1021 
by (simp add: integrable_on_cmult_iff) } 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1022 
note F_finite = lmeasure_finite[OF this] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1023 

5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1024 
have "((\<lambda>x. real (F i x)) has_integral real (integral\<^isup>S lebesgue (F i))) UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1025 
proof (rule simple_function_has_integral[of "F i"]) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1026 
show "simple_function lebesgue (F i)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1027 
using F(1) by (simp add: simple_function_def) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1028 
show "range (F i) \<subseteq> {0..<\<infinity>}" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1029 
using F(3,5)[of i] by (auto simp: image_iff) metis 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1030 
next 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1031 
fix x assume "x \<in> range (F i)" "emeasure lebesgue (F i ` {x} \<inter> UNIV) = \<infinity>" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1032 
with F_finite[of x] show "x = 0" by auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1033 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1034 
from this I have "real (integral\<^isup>S lebesgue (F i)) \<le> I" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1035 
by (rule has_integral_le) (intro ballI F_bound) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1036 
moreover 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1037 
{ fix x assume x: "x \<in> range (F i)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1038 
with F(3,5)[of i] have "x = 0 \<or> (0 < x \<and> x \<noteq> \<infinity>)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1039 
by (auto simp: image_iff le_less) metis 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1040 
with F_finite[OF _ x] x have "x * emeasure lebesgue (F i ` {x} \<inter> UNIV) \<noteq> \<infinity>" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1041 
by auto } 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1042 
then have "integral\<^isup>S lebesgue (F i) \<noteq> \<infinity>" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1043 
unfolding simple_integral_def setsum_Pinfty space_lebesgue by blast 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1044 
moreover have "0 \<le> integral\<^isup>S lebesgue (F i)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1045 
using F(1,5) by (intro simple_integral_positive) (auto simp: simple_function_def) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1046 
moreover have "integral\<^isup>S lebesgue (F i) = integral\<^isup>S lborel (F i)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1047 
using F(1)[of i, THEN borel_measurable_simple_function] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1048 
by (rule lebesgue_simple_integral_eq_borel) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1049 
ultimately show "integral\<^isup>S lborel (F i) \<le> ereal I" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1050 
by (cases "integral\<^isup>S lborel (F i)") auto 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1051 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1052 
also have "\<dots> < \<infinity>" by simp 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1053 
finally have finite: "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) \<noteq> \<infinity>" by simp 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1054 
have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable lebesgue" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1055 
using f_borel by (auto intro: borel_measurable_lebesgueI) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1056 
from positive_integral_has_integral[OF borel _ finite] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1057 
have "(f has_integral real (\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue)) UNIV" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1058 
using nonneg by (simp add: subset_eq) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1059 
with I have "I = real (\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue)" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1060 
by (rule has_integral_unique) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1061 
with finite positive_integral_positive[of _ "\<lambda>x. ereal (f x)"] show ?thesis 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1062 
by (cases "\<integral>\<^isup>+ x. ereal (f x) \<partial>lborel") (auto simp: lebesgue_eq) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1063 
qed 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1064 

5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1065 
lemma has_integral_iff_positive_integral: 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1066 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1067 
assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1068 
shows "(f has_integral I) UNIV \<longleftrightarrow> integral\<^isup>P lborel f = I" 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1069 
using f positive_integral_borel_has_integral[of f I] positive_integral_has_integral[of f] 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1070 
by (auto simp: subset_eq borel_measurable_lebesgueI lebesgue_positive_integral_eq_borel) 
5e6fe71e2390
equate positive Lebesgue integral and MVAnalysis' Gauge integral
hoelzl
parents:
47694
diff
changeset

1071 

38656  1072 
end 