author  hoelzl 
Tue, 18 Jan 2011 21:37:23 +0100  
changeset 41654  32fe42892983 
parent 41546  2a12c23b7a34 
child 41661  baf1964bc468 
permissions  rwrr 
40859  1 
(* Author: Robert Himmelmann, TU Muenchen *) 
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header {* Lebsegue measure *} 
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theory Lebesgue_Measure 

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imports Product_Measure Complete_Measure 
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begin 
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subsection {* Standard Cubes *} 

8 

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definition cube :: "nat \<Rightarrow> 'a::ordered_euclidean_space set" where 
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"cube n \<equiv> {\<chi>\<chi> i.  real n .. \<chi>\<chi> i. real n}" 

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lemma cube_closed[intro]: "closed (cube n)" 

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unfolding cube_def by auto 

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lemma cube_subset[intro]: "n \<le> N \<Longrightarrow> cube n \<subseteq> cube N" 

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by (fastsimp simp: eucl_le[where 'a='a] cube_def) 

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lemma cube_subset_iff: 
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"cube n \<subseteq> cube N \<longleftrightarrow> n \<le> N" 

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proof 

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assume subset: "cube n \<subseteq> (cube N::'a set)" 

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then have "((\<chi>\<chi> i. real n)::'a) \<in> cube N" 

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using DIM_positive[where 'a='a] 

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by (fastsimp simp: cube_def eucl_le[where 'a='a]) 

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then show "n \<le> N" 

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by (fastsimp simp: cube_def eucl_le[where 'a='a]) 

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next 

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assume "n \<le> N" then show "cube n \<subseteq> (cube N::'a set)" by (rule cube_subset) 

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qed 

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lemma ball_subset_cube:"ball (0::'a::ordered_euclidean_space) (real n) \<subseteq> cube n" 

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unfolding cube_def subset_eq mem_interval apply safe unfolding euclidean_lambda_beta' 

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proof fix x::'a and i assume as:"x\<in>ball 0 (real n)" "i<DIM('a)" 

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thus " real n \<le> x $$ i" "real n \<ge> x $$ i" 

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using component_le_norm[of x i] by(auto simp: dist_norm) 

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qed 

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lemma mem_big_cube: obtains n where "x \<in> cube n" 

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proof from real_arch_lt[of "norm x"] guess n .. 

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thus ?thesis applyapply(rule that[where n=n]) 

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apply(rule ball_subset_cube[unfolded subset_eq,rule_format]) 

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by (auto simp add:dist_norm) 

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qed 

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definition lebesgue :: "'a::ordered_euclidean_space algebra" where 
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"lebesgue = \<lparr> space = UNIV, sets = {A. \<forall>n. (indicator A :: 'a \<Rightarrow> real) integrable_on cube n} \<rparr>" 

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

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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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interpretation lebesgue: sigma_algebra lebesgue 
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proof (intro sigma_algebra_iff2[THEN iffD2] conjI allI ballI impI lebesgueI) 

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fix A n assume A: "A \<in> sets lebesgue" 

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

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by (auto simp: fun_eq_iff indicator_def) 

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then show "(indicator (space lebesgue  A) :: _ \<Rightarrow> real) integrable_on cube n" 

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using A by (auto intro!: integrable_sub dest: lebesgueD simp: cube_def) 

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next 

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fix n show "(indicator {} :: _\<Rightarrow>real) integrable_on cube n" 

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by (auto simp: cube_def indicator_def_raw) 

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next 

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fix A :: "nat \<Rightarrow> 'a set" and n ::nat assume "range A \<subseteq> sets lebesgue" 

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

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by (auto dest: lebesgueD) 

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show "(indicator (\<Union>i. A i) :: _ \<Rightarrow> real) integrable_on cube n" (is "?g integrable_on _") 

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

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fix k 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 by (simp add: *) 

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qed auto 

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qed (auto intro: LIMSEQ_indicator_UN simp: cube_def) 

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qed simp 

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definition "lmeasure A = (SUP n. Real (integral (cube n) (indicator A)))" 
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interpretation lebesgue: measure_space lebesgue lmeasure 
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proof 

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have *: "indicator {} = (\<lambda>x. 0 :: real)" by (simp add: fun_eq_iff) 

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show "lmeasure {} = 0" by (simp add: integral_0 * lmeasure_def) 

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next 
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show "countably_additive lebesgue lmeasure" 
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proof (intro countably_additive_def[THEN iffD2] allI impI) 

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fix A :: "nat \<Rightarrow> 'b 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 n i" = "integral (cube n) (indicator (A i) :: _\<Rightarrow>real)" 

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let "?M 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!: 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 dest: lebesgueD) 

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show "(\<Sum>\<^isub>\<infinity>n. lmeasure (A n)) = lmeasure (\<Union>i. A i)" unfolding lmeasure_def 

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proof (subst psuminf_SUP_eq) 

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fix n i show "Real (?m n i) \<le> Real (?m (Suc n) i)" 

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using cube_subset[of n "Suc n"] by (auto intro!: integral_subset_le) 

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next 

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show "(SUP n. \<Sum>\<^isub>\<infinity>i. Real (?m n i)) = (SUP n. Real (?M n UNIV))" 

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unfolding psuminf_def 

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proof (subst setsum_Real, (intro arg_cong[where f="SUPR UNIV"] ext ballI nn SUP_eq_LIMSEQ[THEN iffD2])+) 

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fix n :: nat show "mono (\<lambda>m. \<Sum>x<m. ?m n x)" 

134 
proof (intro mono_iff_le_Suc[THEN iffD2] allI) 

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fix m show "(\<Sum>x<m. ?m n x) \<le> (\<Sum>x<Suc m. ?m n x)" 

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using nn[of n m] by auto 

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qed 

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show "0 \<le> ?M n UNIV" 

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using UN_A by (auto intro!: integral_nonneg) 

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fix m show "0 \<le> (\<Sum>x<m. ?m n x)" by (auto intro!: setsum_nonneg) 

141 
next 

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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" 

146 
(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: integral_add[symmetric]) 

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qed auto } 

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ultimately show "(\<lambda>m. \<Sum>x<m. ?m n x) > ?M n UNIV" 

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by simp 

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qed 

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qed 

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qed 

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qed 
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lemma 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)" 

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(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]) 

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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_raw 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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let ?S = "range (\<lambda>(a, b). {a .. (b :: 'a\<Colon>ordered_euclidean_space)})" 

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have *:"?S \<subseteq> sets lebesgue" 

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proof (safe intro!: 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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have "s \<in> sigma_sets UNIV ?S" using assms 
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unfolding borel_eq_atLeastAtMost by (simp add: sigma_def) 

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thus ?thesis 

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using lebesgue.sigma_subset[of "\<lparr> space = UNIV, sets = ?S\<rparr>", simplified, OF *] 

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by (auto simp: sigma_def) 

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qed 
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lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set" 
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assumes "negligible s" shows "s \<in> sets lebesgue" 

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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" assumes "negligible S" shows "lmeasure 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 integral_def using assms 

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by (intro some1_equality ex_ex1I has_integral_unique) 

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(auto simp: cube_def negligible_def intro: ) 

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then show ?thesis unfolding lmeasure_def by auto 

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qed 
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lemma lmeasure_iff_LIMSEQ: 

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assumes "A \<in> sets lebesgue" "0 \<le> m" 

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shows "lmeasure A = Real m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) > m" 
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unfolding lmeasure_def 

229 
proof (intro SUP_eq_LIMSEQ) 

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show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))" 

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using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD) 

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fix n show "0 \<le> integral (cube n) (indicator A::_=>real)" 

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using assms by (auto dest!: lebesgueD intro!: integral_nonneg) 

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qed fact 

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

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

239 
proof  

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

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by (auto simp: fun_eq_iff indicator_def) 

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then show ?thesis 

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

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qed 
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lemma 
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fixes s a :: "'a::ordered_euclidean_space set" 

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shows integral_indicator_UNIV: 

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

250 
and integrable_indicator_UNIV: 

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

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unfolding integral_def integrable_on_def has_integral_indicator_UNIV by auto 

253 

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lemma lmeasure_finite_has_integral: 

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

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assumes "s \<in> sets lebesgue" "lmeasure s = Real m" "0 \<le> m" 

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shows "(indicator s has_integral m) UNIV" 

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proof  

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

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

261 
proof (intro monotone_convergence_increasing allI ballI) 

262 
have LIMSEQ: "(\<lambda>n. integral (cube n) (?I s)) > m" 

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

264 
{ fix n have "integral (cube n) (?I s) \<le> m" 

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using cube_subset assms 

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

267 
(auto dest!: lebesgueD) } 

268 
moreover 

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

270 
using assms by (auto dest!: lebesgueD intro!: integral_nonneg) } 

271 
ultimately 

272 
show "bounded {integral UNIV (?I (s \<inter> cube k)) k. True}" 

273 
unfolding bounded_def 

274 
apply (rule_tac exI[of _ 0]) 

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apply (rule_tac exI[of _ m]) 

276 
by (auto simp: dist_real_def integral_indicator_UNIV) 

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

278 
unfolding integrable_indicator_UNIV using assms by (auto dest!: lebesgueD) 

279 
fix x show "?I (s \<inter> cube k) x \<le> ?I (s \<inter> cube (Suc k)) x" 

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

281 
next 

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fix x :: 'a 

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

284 
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x" 

285 
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } 

286 
note * = this 

287 
show "(\<lambda>k. ?I (s \<inter> cube k) x) > ?I s x" 

288 
by (rule LIMSEQ_offset[where k=k]) (auto simp: *) 

289 
qed 

290 
note ** = conjunctD2[OF this] 

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

292 
apply (intro LIMSEQ_unique[OF _ **(2)]) 

293 
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1,3)] integral_indicator_UNIV . 

294 
show ?thesis 

295 
unfolding m by (intro integrable_integral **) 

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qed 
297 

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lemma lmeasure_finite_integrable: assumes "s \<in> sets lebesgue" "lmeasure s \<noteq> \<omega>" 
299 
shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV" 

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proof (cases "lmeasure s") 
41654  301 
case (preal m) from lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this] 
302 
show ?thesis unfolding integrable_on_def by auto 

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qed (insert assms, auto) 
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lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV" 
306 
shows "s \<in> sets lebesgue" 

307 
proof (intro lebesgueI) 

308 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

309 
fix n show "(?I s) integrable_on cube n" unfolding cube_def 

310 
proof (intro integrable_on_subinterval) 

311 
show "(?I s) integrable_on UNIV" 

312 
unfolding integrable_on_def using assms by auto 

313 
qed auto 

38656  314 
qed 
315 

41654  316 
lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV" 
317 
shows "lmeasure s = Real m" 

318 
proof (intro lmeasure_iff_LIMSEQ[THEN iffD2]) 

319 
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" 

320 
show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] . 

321 
show "0 \<le> m" using assms by (rule has_integral_nonneg) auto 

322 
have "(\<lambda>n. integral UNIV (?I (s \<inter> cube n))) > integral UNIV (?I s)" 

323 
proof (intro dominated_convergence(2) ballI) 

324 
show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto 

325 
fix n show "?I (s \<inter> cube n) integrable_on UNIV" 

326 
unfolding integrable_indicator_UNIV using `s \<in> sets lebesgue` by (auto dest: lebesgueD) 

327 
fix x show "norm (?I (s \<inter> cube n) x) \<le> ?I s x" by (auto simp: indicator_def) 

328 
next 

329 
fix x :: 'a 

330 
from mem_big_cube obtain k where k: "x \<in> cube k" . 

331 
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x" 

332 
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } 

333 
note * = this 

334 
show "(\<lambda>k. ?I (s \<inter> cube k) x) > ?I s x" 

335 
by (rule LIMSEQ_offset[where k=k]) (auto simp: *) 

336 
qed 

337 
then show "(\<lambda>n. integral (cube n) (?I s)) > m" 

338 
unfolding integral_unique[OF assms] integral_indicator_UNIV by simp 

339 
qed 

340 

341 
lemma has_integral_iff_lmeasure: 

342 
"(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m)" 

40859  343 
proof 
41654  344 
assume "(indicator A has_integral m) UNIV" 
345 
with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this] 

346 
show "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m" 

347 
by (auto intro: has_integral_nonneg) 

40859  348 
next 
349 
assume "A \<in> sets lebesgue \<and> 0 \<le> m \<and> lmeasure A = Real m" 

41654  350 
then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto 
38656  351 
qed 
352 

41654  353 
lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" 
354 
shows "lmeasure s = Real (integral UNIV (indicator s))" 

355 
using assms unfolding integrable_on_def 

356 
proof safe 

357 
fix y :: real assume "(indicator s has_integral y) UNIV" 

358 
from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this] 

359 
show "lmeasure s = Real (integral UNIV (indicator s))" by simp 

40859  360 
qed 
38656  361 

362 
lemma lebesgue_simple_function_indicator: 

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

363 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal" 
38656  364 
assumes f:"lebesgue.simple_function f" 
365 
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f ` {y}) x))" 

366 
apply(rule,subst lebesgue.simple_function_indicator_representation[OF f]) by auto 

367 

41654  368 
lemma integral_eq_lmeasure: 
369 
"(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (lmeasure s)" 

370 
by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg) 

38656  371 

41654  372 
lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "lmeasure s \<noteq> \<omega>" 
373 
using lmeasure_eq_integral[OF assms] by auto 

38656  374 

40859  375 
lemma negligible_iff_lebesgue_null_sets: 
376 
"negligible A \<longleftrightarrow> A \<in> lebesgue.null_sets" 

377 
proof 

378 
assume "negligible A" 

379 
from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0] 

380 
show "A \<in> lebesgue.null_sets" by auto 

381 
next 

382 
assume A: "A \<in> lebesgue.null_sets" 

41654  383 
then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] by auto 
384 
show "negligible A" unfolding negligible_def 

385 
proof (intro allI) 

386 
fix a b :: 'a 

387 
have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on {a..b}" 

388 
by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *) 

389 
then have "integral {a..b} (indicator A) \<le> (integral UNIV (indicator A) :: real)" 

390 
using * by (auto intro!: integral_subset_le has_integral_integrable) 

391 
moreover have "(0::real) \<le> integral {a..b} (indicator A)" 

392 
using integrable by (auto intro!: integral_nonneg) 

393 
ultimately have "integral {a..b} (indicator A) = (0::real)" 

394 
using integral_unique[OF *] by auto 

395 
then show "(indicator A has_integral (0::real)) {a..b}" 

396 
using integrable_integral[OF integrable] by simp 

397 
qed 

398 
qed 

399 

400 
lemma integral_const[simp]: 

401 
fixes a b :: "'a::ordered_euclidean_space" 

402 
shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c" 

403 
by (rule integral_unique) (rule has_integral_const) 

404 

405 
lemma lmeasure_UNIV[intro]: "lmeasure (UNIV::'a::ordered_euclidean_space set) = \<omega>" 

406 
unfolding lmeasure_def SUP_\<omega> 

407 
proof (intro allI impI) 

408 
fix x assume "x < \<omega>" 

409 
then obtain r where r: "x = Real r" "0 \<le> r" by (cases x) auto 

410 
then obtain n where n: "r < of_nat n" using ex_less_of_nat by auto 

411 
show "\<exists>i\<in>UNIV. x < Real (integral (cube i) (indicator UNIV::'a\<Rightarrow>real))" 

412 
proof (intro bexI[of _ n]) 

413 
have [simp]: "indicator UNIV = (\<lambda>x. 1)" by (auto simp: fun_eq_iff) 

414 
{ fix m :: nat assume "0 < m" then have "real n \<le> (\<Prod>x<m. 2 * real n)" 

415 
proof (induct m) 

416 
case (Suc m) 

417 
show ?case 

418 
proof cases 

419 
assume "m = 0" then show ?thesis by (simp add: lessThan_Suc) 

420 
next 

421 
assume "m \<noteq> 0" then have "real n \<le> (\<Prod>x<m. 2 * real n)" using Suc by auto 

422 
then show ?thesis 

423 
by (auto simp: lessThan_Suc field_simps mult_le_cancel_left1) 

424 
qed 

425 
qed auto } note this[OF DIM_positive[where 'a='a], simp] 

426 
then have [simp]: "0 \<le> (\<Prod>x<DIM('a). 2 * real n)" using real_of_nat_ge_zero by arith 

427 
have "x < Real (of_nat n)" using n r by auto 

428 
also have "Real (of_nat n) \<le> Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" 

429 
by (auto simp: real_eq_of_nat[symmetric] cube_def content_closed_interval_cases) 

430 
finally show "x < Real (integral (cube n) (indicator UNIV::'a\<Rightarrow>real))" . 

431 
qed auto 

40859  432 
qed 
433 

434 
lemma 

435 
fixes a b ::"'a::ordered_euclidean_space" 

436 
shows lmeasure_atLeastAtMost[simp]: "lmeasure {a..b} = Real (content {a..b})" 

41654  437 
proof  
438 
have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV" 

439 
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def_raw) 

440 
from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV 

441 
by (simp add: indicator_def_raw) 

40859  442 
qed 
443 

444 
lemma atLeastAtMost_singleton_euclidean[simp]: 

445 
fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}" 

446 
by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a]) 

447 

448 
lemma content_singleton[simp]: "content {a} = 0" 

449 
proof  

450 
have "content {a .. a} = 0" 

451 
by (subst content_closed_interval) auto 

452 
then show ?thesis by simp 

453 
qed 

454 

455 
lemma lmeasure_singleton[simp]: 

456 
fixes a :: "'a::ordered_euclidean_space" shows "lmeasure {a} = 0" 

41654  457 
using lmeasure_atLeastAtMost[of a a] by simp 
40859  458 

459 
declare content_real[simp] 

460 

461 
lemma 

462 
fixes a b :: real 

463 
shows lmeasure_real_greaterThanAtMost[simp]: 

464 
"lmeasure {a <.. b} = Real (if a \<le> b then b  a else 0)" 

465 
proof cases 

466 
assume "a < b" 

41654  467 
then have "lmeasure {a <.. b} = lmeasure {a .. b}  lmeasure {a}" 
468 
by (subst lebesgue.measure_Diff[symmetric]) 

469 
(auto intro!: arg_cong[where f=lmeasure]) 

40859  470 
then show ?thesis by auto 
471 
qed auto 

472 

473 
lemma 

474 
fixes a b :: real 

475 
shows lmeasure_real_atLeastLessThan[simp]: 

41654  476 
"lmeasure {a ..< b} = Real (if a \<le> b then b  a else 0)" 
40859  477 
proof cases 
478 
assume "a < b" 

41654  479 
then have "lmeasure {a ..< b} = lmeasure {a .. b}  lmeasure {b}" 
480 
by (subst lebesgue.measure_Diff[symmetric]) 

481 
(auto intro!: arg_cong[where f=lmeasure]) 

482 
then show ?thesis by auto 

483 
qed auto 

484 

485 
lemma 

486 
fixes a b :: real 

487 
shows lmeasure_real_greaterThanLessThan[simp]: 

488 
"lmeasure {a <..< b} = Real (if a \<le> b then b  a else 0)" 

489 
proof cases 

490 
assume "a < b" 

491 
then have "lmeasure {a <..< b} = lmeasure {a <.. b}  lmeasure {b}" 

492 
by (subst lebesgue.measure_Diff[symmetric]) 

493 
(auto intro!: arg_cong[where f=lmeasure]) 

40859  494 
then show ?thesis by auto 
495 
qed auto 

496 

497 
interpretation borel: measure_space borel lmeasure 

498 
proof 

499 
show "countably_additive borel lmeasure" 

500 
using lebesgue.ca unfolding countably_additive_def 

501 
apply safe apply (erule_tac x=A in allE) by auto 

502 
qed auto 

503 

504 
interpretation borel: sigma_finite_measure borel lmeasure 

505 
proof (default, intro conjI exI[of _ "\<lambda>n. cube n"]) 

506 
show "range cube \<subseteq> sets borel" by (auto intro: borel_closed) 

507 
{ fix x have "\<exists>n. x\<in>cube n" using mem_big_cube by auto } 

508 
thus "(\<Union>i. cube i) = space borel" by auto 

41654  509 
show "\<forall>i. lmeasure (cube i) \<noteq> \<omega>" unfolding cube_def by auto 
40859  510 
qed 
511 

512 
interpretation lebesgue: sigma_finite_measure lebesgue lmeasure 

513 
proof 

514 
from borel.sigma_finite guess A .. 

515 
moreover then have "range A \<subseteq> sets lebesgue" using lebesgueI_borel by blast 

516 
ultimately show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets lebesgue \<and> (\<Union>i. A i) = space lebesgue \<and> (\<forall>i. lmeasure (A i) \<noteq> \<omega>)" 

517 
by auto 

518 
qed 

519 

520 
lemma simple_function_has_integral: 

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

521 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal" 
40859  522 
assumes f:"lebesgue.simple_function f" 
523 
and f':"\<forall>x. f x \<noteq> \<omega>" 

524 
and om:"\<forall>x\<in>range f. lmeasure (f ` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0" 

525 
shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV" 

526 
unfolding lebesgue.simple_integral_def 

527 
apply(subst lebesgue_simple_function_indicator[OF f]) 

41654  528 
proof  
529 
case goal1 

40859  530 
have *:"\<And>x. \<forall>y\<in>range f. y * indicator (f ` {y}) x \<noteq> \<omega>" 
531 
"\<forall>x\<in>range f. x * lmeasure (f ` {x} \<inter> UNIV) \<noteq> \<omega>" 

532 
using f' om unfolding indicator_def by auto 

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

533 
show ?case unfolding space_lebesgue real_of_pextreal_setsum'[OF *(1),THEN sym] 
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

534 
unfolding real_of_pextreal_setsum'[OF *(2),THEN sym] 
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40874
diff
changeset

535 
unfolding real_of_pextreal_setsum space_lebesgue 
40859  536 
apply(rule has_integral_setsum) 
537 
proof safe show "finite (range f)" using f by (auto dest: lebesgue.simple_functionD) 

538 
fix y::'a show "((\<lambda>x. real (f y * indicator (f ` {f y}) x)) has_integral 

539 
real (f y * lmeasure (f ` {f y} \<inter> UNIV))) UNIV" 

540 
proof(cases "f y = 0") case False 

41654  541 
have mea:"(indicator (f ` {f y}) ::_\<Rightarrow>real) integrable_on UNIV" 
542 
apply(rule lmeasure_finite_integrable) 

40859  543 
using assms unfolding lebesgue.simple_function_def using False by auto 
41654  544 
have *:"\<And>x. real (indicator (f ` {f y}) x::pextreal) = (indicator (f ` {f y}) x)" 
545 
by (auto simp: indicator_def) 

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

546 
show ?thesis unfolding real_of_pextreal_mult[THEN sym] 
40859  547 
apply(rule has_integral_cmul[where 'b=real, unfolded real_scaleR_def]) 
41654  548 
unfolding Int_UNIV_right lmeasure_eq_integral[OF mea,THEN sym] 
549 
unfolding integral_eq_lmeasure[OF mea, symmetric] * 

550 
apply(rule integrable_integral) using mea . 

40859  551 
qed auto 
41654  552 
qed 
553 
qed 

40859  554 

555 
lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s" 

556 
unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI) 

557 
using assms by auto 

558 

559 
lemma simple_function_has_integral': 

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

560 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> pextreal" 
40859  561 
assumes f:"lebesgue.simple_function f" 
562 
and i: "lebesgue.simple_integral f \<noteq> \<omega>" 

563 
shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.simple_integral f))) UNIV" 

564 
proof let ?f = "\<lambda>x. if f x = \<omega> then 0 else f x" 

565 
{ fix x have "real (f x) = real (?f x)" by (cases "f x") auto } note * = this 

566 
have **:"{x. f x \<noteq> ?f x} = f ` {\<omega>}" by auto 

567 
have **:"lmeasure {x\<in>space lebesgue. f x \<noteq> ?f x} = 0" 

568 
using lebesgue.simple_integral_omega[OF assms] by(auto simp add:**) 

569 
show ?thesis apply(subst lebesgue.simple_integral_cong'[OF f _ **]) 

570 
apply(rule lebesgue.simple_function_compose1[OF f]) 

571 
unfolding * defer apply(rule simple_function_has_integral) 

572 
proof 

573 
show "lebesgue.simple_function ?f" 

574 
using lebesgue.simple_function_compose1[OF f] . 

575 
show "\<forall>x. ?f x \<noteq> \<omega>" by auto 

576 
show "\<forall>x\<in>range ?f. lmeasure (?f ` {x} \<inter> UNIV) = \<omega> \<longrightarrow> x = 0" 

577 
proof (safe, simp, safe, rule ccontr) 

578 
fix y assume "f y \<noteq> \<omega>" "f y \<noteq> 0" 

579 
hence "(\<lambda>x. if f x = \<omega> then 0 else f x) ` {if f y = \<omega> then 0 else f y} = f ` {f y}" 

580 
by (auto split: split_if_asm) 

581 
moreover assume "lmeasure ((\<lambda>x. if f x = \<omega> then 0 else f x) ` {if f y = \<omega> then 0 else f y}) = \<omega>" 

582 
ultimately have "lmeasure (f ` {f y}) = \<omega>" by simp 

583 
moreover 

584 
have "f y * lmeasure (f ` {f y}) \<noteq> \<omega>" using i f 

585 
unfolding lebesgue.simple_integral_def setsum_\<omega> lebesgue.simple_function_def 

586 
by auto 

587 
ultimately have "f y = 0" by (auto split: split_if_asm) 

588 
then show False using `f y \<noteq> 0` by simp 

589 
qed 

590 
qed 

591 
qed 

592 

593 
lemma (in measure_space) positive_integral_monotone_convergence: 

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

594 
fixes f::"nat \<Rightarrow> 'a \<Rightarrow> pextreal" 
40859  595 
assumes i: "\<And>i. f i \<in> borel_measurable M" and mono: "\<And>x. mono (\<lambda>n. f n x)" 
596 
and lim: "\<And>x. (\<lambda>i. f i x) > u x" 

597 
shows "u \<in> borel_measurable M" 

598 
and "(\<lambda>i. positive_integral (f i)) > positive_integral u" (is ?ilim) 

599 
proof  

600 
from positive_integral_isoton[unfolded isoton_fun_expand isoton_iff_Lim_mono, of f u] 

601 
show ?ilim using mono lim i by auto 

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

602 
have "\<And>x. (SUP i. f i x) = u x" using mono lim SUP_Lim_pextreal 
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

603 
unfolding fun_eq_iff mono_def by auto 
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

604 
moreover have "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M" 
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset

605 
using i by auto 
40859  606 
ultimately show "u \<in> borel_measurable M" by simp 
607 
qed 

608 

609 
lemma positive_integral_has_integral: 

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

610 
fixes f::"'a::ordered_euclidean_space => pextreal" 
40859  611 
assumes f:"f \<in> borel_measurable lebesgue" 
612 
and int_om:"lebesgue.positive_integral f \<noteq> \<omega>" 

613 
and f_om:"\<forall>x. f x \<noteq> \<omega>" (* TODO: remove this *) 

614 
shows "((\<lambda>x. real (f x)) has_integral (real (lebesgue.positive_integral f))) UNIV" 

615 
proof let ?i = "lebesgue.positive_integral f" 

616 
from lebesgue.borel_measurable_implies_simple_function_sequence[OF f] 

617 
guess u .. note conjunctD2[OF this,rule_format] note u = conjunctD2[OF this(1)] this(2) 

618 
let ?u = "\<lambda>i x. real (u i x)" and ?f = "\<lambda>x. real (f x)" 

619 
have u_simple:"\<And>k. lebesgue.simple_integral (u k) = lebesgue.positive_integral (u k)" 

620 
apply(subst lebesgue.positive_integral_eq_simple_integral[THEN sym,OF u(1)]) .. 

621 
have int_u_le:"\<And>k. lebesgue.simple_integral (u k) \<le> lebesgue.positive_integral f" 

622 
unfolding u_simple apply(rule lebesgue.positive_integral_mono) 

623 
using isoton_Sup[OF u(3)] unfolding le_fun_def by auto 

624 
have u_int_om:"\<And>i. lebesgue.simple_integral (u i) \<noteq> \<omega>" 

625 
proof case goal1 thus ?case using int_u_le[of i] int_om by auto qed 

626 

627 
note u_int = simple_function_has_integral'[OF u(1) this] 

628 
have "(\<lambda>x. real (f x)) integrable_on UNIV \<and> 

629 
(\<lambda>k. Integration.integral UNIV (\<lambda>x. real (u k x))) > Integration.integral UNIV (\<lambda>x. real (f x))" 

630 
apply(rule monotone_convergence_increasing) apply(rule,rule,rule u_int) 

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

631 
proof safe case goal1 show ?case apply(rule real_of_pextreal_mono) using u(2,3) by auto 
40859  632 
next case goal2 show ?case using u(3) apply(subst lim_Real[THEN sym]) 
633 
prefer 3 apply(subst Real_real') defer apply(subst Real_real') 

634 
using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] using f_om u by auto 

635 
next case goal3 

636 
show ?case apply(rule bounded_realI[where B="real (lebesgue.positive_integral f)"]) 

637 
apply safe apply(subst abs_of_nonneg) apply(rule integral_nonneg,rule) apply(rule u_int) 

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

638 
unfolding integral_unique[OF u_int] defer apply(rule real_of_pextreal_mono[OF _ int_u_le]) 
40859  639 
using u int_om by auto 
640 
qed note int = conjunctD2[OF this] 

641 

642 
have "(\<lambda>i. lebesgue.simple_integral (u i)) > ?i" unfolding u_simple 

643 
apply(rule lebesgue.positive_integral_monotone_convergence(2)) 

644 
apply(rule lebesgue.borel_measurable_simple_function[OF u(1)]) 

645 
using isotone_Lim[OF u(3)[unfolded isoton_fun_expand, THEN spec]] by auto 

646 
hence "(\<lambda>i. real (lebesgue.simple_integral (u i))) > real ?i" apply 

647 
apply(subst lim_Real[THEN sym]) prefer 3 

648 
apply(subst Real_real') defer apply(subst Real_real') 

649 
using u f_om int_om u_int_om by auto 

650 
note * = LIMSEQ_unique[OF this int(2)[unfolded integral_unique[OF u_int]]] 

651 
show ?thesis unfolding * by(rule integrable_integral[OF int(1)]) 

652 
qed 

653 

654 
lemma lebesgue_integral_has_integral: 

655 
fixes f::"'a::ordered_euclidean_space => real" 

656 
assumes f:"lebesgue.integrable f" 

657 
shows "(f has_integral (lebesgue.integral f)) UNIV" 

658 
proof let ?n = "\<lambda>x.  min (f x) 0" and ?p = "\<lambda>x. max (f x) 0" 

659 
have *:"f = (\<lambda>x. ?p x  ?n x)" apply rule by auto 

660 
note f = lebesgue.integrableD[OF f] 

661 
show ?thesis unfolding lebesgue.integral_def apply(subst *) 

662 
proof(rule has_integral_sub) case goal1 

663 
have *:"\<forall>x. Real (f x) \<noteq> \<omega>" by auto 

664 
note lebesgue.borel_measurable_Real[OF f(1)] 

665 
from positive_integral_has_integral[OF this f(2) *] 

666 
show ?case unfolding real_Real_max . 

667 
next case goal2 

668 
have *:"\<forall>x. Real ( f x) \<noteq> \<omega>" by auto 

669 
note lebesgue.borel_measurable_uminus[OF f(1)] 

670 
note lebesgue.borel_measurable_Real[OF this] 

671 
from positive_integral_has_integral[OF this f(3) *] 

672 
show ?case unfolding real_Real_max minus_min_eq_max by auto 

673 
qed 

674 
qed 

675 

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

676 
lemma lebesgue_positive_integral_eq_borel: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

677 
"f \<in> borel_measurable borel \<Longrightarrow> lebesgue.positive_integral f = borel.positive_integral f " 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

678 
by (auto intro!: lebesgue.positive_integral_subalgebra[symmetric]) default 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

679 

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

680 
lemma lebesgue_integral_eq_borel: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

681 
assumes "f \<in> borel_measurable borel" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

682 
shows "lebesgue.integrable f = borel.integrable f" (is ?P) 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

683 
and "lebesgue.integral f = borel.integral f" (is ?I) 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

684 
proof  
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

685 
have *: "sigma_algebra borel" by default 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

686 
have "sets borel \<subseteq> sets lebesgue" by auto 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

687 
from lebesgue.integral_subalgebra[OF assms this _ *] 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

688 
show ?P ?I by auto 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

689 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

690 

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

691 
lemma borel_integral_has_integral: 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

692 
fixes f::"'a::ordered_euclidean_space => real" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

693 
assumes f:"borel.integrable f" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

694 
shows "(f has_integral (borel.integral f)) UNIV" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

695 
proof  
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

696 
have borel: "f \<in> borel_measurable borel" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

697 
using f unfolding borel.integrable_def by auto 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

698 
from f show ?thesis 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

699 
using lebesgue_integral_has_integral[of f] 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

700 
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

701 
qed 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

702 

40859  703 
lemma continuous_on_imp_borel_measurable: 
704 
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space" 

705 
assumes "continuous_on UNIV f" 

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

706 
shows "f \<in> borel_measurable borel" 
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset

707 
apply(rule borel.borel_measurableI) 
40859  708 
using continuous_open_preimage[OF assms] unfolding vimage_def by auto 
709 

710 
lemma (in measure_space) integral_monotone_convergence_pos': 

711 
assumes i: "\<And>i. integrable (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)" 

712 
and pos: "\<And>x i. 0 \<le> f i x" 

713 
and lim: "\<And>x. (\<lambda>i. f i x) > u x" 

714 
and ilim: "(\<lambda>i. integral (f i)) > x" 

715 
shows "integrable u \<and> integral u = x" 

716 
using integral_monotone_convergence_pos[OF assms] by auto 

717 

718 
definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where 

719 
"e2p x = (\<lambda>i\<in>{..<DIM('a)}. x$$i)" 

720 

721 
definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where 

722 
"p2e x = (\<chi>\<chi> i. x i)" 

723 

41095  724 
lemma e2p_p2e[simp]: 
725 
"x \<in> extensional {..<DIM('a)} \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x" 

726 
by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def) 

40859  727 

41095  728 
lemma p2e_e2p[simp]: 
729 
"p2e (e2p x) = (x::'a::ordered_euclidean_space)" 

730 
by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def) 

40859  731 

41095  732 
lemma bij_inv_p2e_e2p: 
733 
shows "bij_inv ({..<DIM('a)} \<rightarrow>\<^isub>E UNIV) (UNIV :: 'a::ordered_euclidean_space set) 

734 
p2e e2p" (is "bij_inv ?P ?U _ _") 

735 
proof (rule bij_invI) 

736 
show "p2e \<in> ?P \<rightarrow> ?U" "e2p \<in> ?U \<rightarrow> ?P" by (auto simp: e2p_def) 

737 
qed auto 

40859  738 

739 
interpretation borel_product: product_sigma_finite "\<lambda>x. borel::real algebra" "\<lambda>x. lmeasure" 

740 
by default 

741 

742 
lemma cube_subset_Suc[intro]: "cube n \<subseteq> cube (Suc n)" 

743 
unfolding cube_def_raw subset_eq apply safe unfolding mem_interval by auto 

744 

41095  745 
lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)" 
746 
unfolding Pi_def by auto 

40859  747 

41095  748 
lemma measurable_e2p_on_generator: 
749 
"e2p \<in> measurable \<lparr> space = UNIV::'a set, sets = range lessThan \<rparr> 

750 
(product_algebra 

751 
(\<lambda>x. \<lparr> space = UNIV::real set, sets = range lessThan \<rparr>) 

752 
{..<DIM('a::ordered_euclidean_space)})" 

753 
(is "e2p \<in> measurable ?E ?P") 

754 
proof (unfold measurable_def, intro CollectI conjI ballI) 

755 
show "e2p \<in> space ?E \<rightarrow> space ?P" by (auto simp: e2p_def) 

756 
fix A assume "A \<in> sets ?P" 

757 
then obtain E where A: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. E i)" 

758 
and "E \<in> {..<DIM('a)} \<rightarrow> (range lessThan)" 

759 
by (auto elim!: product_algebraE) 

760 
then have "\<forall>i\<in>{..<DIM('a)}. \<exists>xs. E i = {..< xs}" by auto 

761 
from this[THEN bchoice] guess xs .. 

762 
then have A_eq: "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< xs i})" 

763 
using A by auto 

764 
have "e2p ` A = {..< (\<chi>\<chi> i. xs i) :: 'a}" 

765 
using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def A_eq 

766 
euclidean_eq[where 'a='a] eucl_less[where 'a='a]) 

767 
then show "e2p ` A \<inter> space ?E \<in> sets ?E" by simp 

40859  768 
qed 
769 

41095  770 
lemma measurable_p2e_on_generator: 
771 
"p2e \<in> measurable 

772 
(product_algebra 

773 
(\<lambda>x. \<lparr> space = UNIV::real set, sets = range lessThan \<rparr>) 

774 
{..<DIM('a::ordered_euclidean_space)}) 

775 
\<lparr> space = UNIV::'a set, sets = range lessThan \<rparr>" 

776 
(is "p2e \<in> measurable ?P ?E") 

777 
proof (unfold measurable_def, intro CollectI conjI ballI) 

778 
show "p2e \<in> space ?P \<rightarrow> space ?E" by simp 

779 
fix A assume "A \<in> sets ?E" 

780 
then obtain x where "A = {..<x}" by auto 

781 
then have "p2e ` A \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..< x $$ i})" 

782 
using DIM_positive 

783 
by (auto simp: Pi_iff set_eq_iff p2e_def 

784 
euclidean_eq[where 'a='a] eucl_less[where 'a='a]) 

785 
then show "p2e ` A \<inter> space ?P \<in> sets ?P" by auto 

786 
qed 

787 

788 
lemma borel_vimage_algebra_eq: 

789 
defines "F \<equiv> product_algebra (\<lambda>x. \<lparr> space = (UNIV::real set), sets = range lessThan \<rparr>) {..<DIM('a::ordered_euclidean_space)}" 

790 
shows "sigma_algebra.vimage_algebra (borel::'a::ordered_euclidean_space algebra) (space (sigma F)) p2e = sigma F" 

791 
unfolding borel_eq_lessThan 

792 
proof (intro vimage_algebra_sigma) 

793 
let ?E = "\<lparr>space = (UNIV::'a set), sets = range lessThan\<rparr>" 

794 
show "bij_inv (space (sigma F)) (space (sigma ?E)) p2e e2p" 

795 
using bij_inv_p2e_e2p unfolding F_def by simp 

796 
show "sets F \<subseteq> Pow (space F)" "sets ?E \<subseteq> Pow (space ?E)" unfolding F_def 

797 
by (intro product_algebra_sets_into_space) auto 

798 
show "p2e \<in> measurable F ?E" 

799 
"e2p \<in> measurable ?E F" 

800 
unfolding F_def using measurable_p2e_on_generator measurable_e2p_on_generator by auto 

801 
qed 

802 

803 
lemma product_borel_eq_vimage: 

804 
"sigma (product_algebra (\<lambda>x. borel) {..<DIM('a::ordered_euclidean_space)}) = 

805 
sigma_algebra.vimage_algebra borel (extensional {..<DIM('a)}) 

806 
(p2e:: _ \<Rightarrow> 'a::ordered_euclidean_space)" 

807 
unfolding borel_vimage_algebra_eq[simplified] 

808 
unfolding borel_eq_lessThan 

809 
apply(subst sigma_product_algebra_sigma_eq[where S="\<lambda>i. \<lambda>n. lessThan (real n)"]) 

810 
unfolding lessThan_iff 

811 
proof fix i assume i:"i<DIM('a)" 

812 
show "(\<lambda>n. {..<real n}) \<up> space \<lparr>space = UNIV, sets = range lessThan\<rparr>" 

813 
by(auto intro!:real_arch_lt isotoneI) 

814 
qed auto 

815 

40859  816 
lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R") 
41095  817 
apply(rule image_Int[THEN sym]) 
818 
using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)] 

40859  819 
unfolding bij_betw_def by auto 
820 

821 
lemma Int_stable_cuboids: fixes x::"'a::ordered_euclidean_space" 

822 
shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). e2p ` {a..b})\<rparr>" 

823 
unfolding Int_stable_def algebra.select_convs 

824 
proof safe fix a b x y::'a 

825 
have *:"e2p ` {a..b} \<inter> e2p ` {x..y} = 

826 
(\<lambda>(a, b). e2p ` {a..b}) (\<chi>\<chi> i. max (a $$ i) (x $$ i), \<chi>\<chi> i. min (b $$ i) (y $$ i)::'a)" 

827 
unfolding e2p_Int inter_interval by auto 

828 
show "e2p ` {a..b} \<inter> e2p ` {x..y} \<in> range (\<lambda>(a, b). e2p ` {a..b::'a})" unfolding * 

829 
apply(rule range_eqI) .. 

830 
qed 

831 

832 
lemma Int_stable_cuboids': fixes x::"'a::ordered_euclidean_space" 

833 
shows "Int_stable \<lparr>space = UNIV, sets = range (\<lambda>(a, b::'a). {a..b})\<rparr>" 

834 
unfolding Int_stable_def algebra.select_convs 

835 
apply safe unfolding inter_interval by auto 

836 

837 
lemma inj_on_disjoint_family_on: assumes "disjoint_family_on A S" "inj f" 

838 
shows "disjoint_family_on (\<lambda>x. f ` A x) S" 

839 
unfolding disjoint_family_on_def 

840 
proof(rule,rule,rule) 

841 
fix x1 x2 assume x:"x1 \<in> S" "x2 \<in> S" "x1 \<noteq> x2" 

842 
show "f ` A x1 \<inter> f ` A x2 = {}" 

843 
proof(rule ccontr) case goal1 

844 
then obtain z where z:"z \<in> f ` A x1 \<inter> f ` A x2" by auto 

845 
then obtain z1 z2 where z12:"z1 \<in> A x1" "z2 \<in> A x2" "f z1 = z" "f z2 = z" by auto 

846 
hence "z1 = z2" using assms(2) unfolding inj_on_def by blast 

847 
hence "x1 = x2" using z12(12) using assms[unfolded disjoint_family_on_def] using x by auto 

848 
thus False using x(3) by auto 

849 
qed 

850 
qed 

851 

852 
declare restrict_extensional[intro] 

853 

854 
lemma e2p_extensional[intro]:"e2p (y::'a::ordered_euclidean_space) \<in> extensional {..<DIM('a)}" 

855 
unfolding e2p_def by auto 

856 

857 
lemma e2p_image_vimage: fixes A::"'a::ordered_euclidean_space set" 

41095  858 
shows "e2p ` A = p2e ` A \<inter> extensional {..<DIM('a)}" 
40859  859 
proof(rule set_eqI,rule) 
860 
fix x assume "x \<in> e2p ` A" then guess y unfolding image_iff .. note y=this 

41095  861 
show "x \<in> p2e ` A \<inter> extensional {..<DIM('a)}" 
40859  862 
apply safe apply(rule vimageI[OF _ y(1)]) unfolding y p2e_e2p by auto 
41095  863 
next fix x assume "x \<in> p2e ` A \<inter> extensional {..<DIM('a)}" 
40859  864 
thus "x \<in> e2p ` A" unfolding image_iff apply(rule_tac x="p2e x" in bexI) apply(subst e2p_p2e) by auto 
865 
qed 

866 

867 
lemma lmeasure_measure_eq_borel_prod: 

868 
fixes A :: "('a::ordered_euclidean_space) set" 

869 
assumes "A \<in> sets borel" 

870 
shows "lmeasure A = borel_product.product_measure {..<DIM('a)} (e2p ` A :: (nat \<Rightarrow> real) set)" 

871 
proof (rule measure_unique_Int_stable[where X=A and A=cube]) 

872 
interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto 

873 
show "Int_stable \<lparr> space = UNIV :: 'a set, sets = range (\<lambda>(a,b). {a..b}) \<rparr>" 

874 
(is "Int_stable ?E" ) using Int_stable_cuboids' . 

875 
show "borel = sigma ?E" using borel_eq_atLeastAtMost . 

41654  876 
show "\<And>i. lmeasure (cube i) \<noteq> \<omega>" unfolding cube_def by auto 
40859  877 
show "\<And>X. X \<in> sets ?E \<Longrightarrow> 
878 
lmeasure X = borel_product.product_measure {..<DIM('a)} (e2p ` X :: (nat \<Rightarrow> real) set)" 

879 
proof case goal1 then obtain a b where X:"X = {a..b}" by auto 

880 
{ presume *:"X \<noteq> {} \<Longrightarrow> ?case" 

881 
show ?case apply(cases,rule *,assumption) by auto } 

882 
def XX \<equiv> "\<lambda>i. {a $$ i .. b $$ i}" assume "X \<noteq> {}" note X' = this[unfolded X interval_ne_empty] 

883 
have *:"Pi\<^isub>E {..<DIM('a)} XX = e2p ` X" apply(rule set_eqI) 

884 
proof fix x assume "x \<in> Pi\<^isub>E {..<DIM('a)} XX" 

885 
thus "x \<in> e2p ` X" unfolding image_iff apply(rule_tac x="\<chi>\<chi> i. x i" in bexI) 

886 
unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by rule auto 

887 
next fix x assume "x \<in> e2p ` X" then guess y unfolding image_iff .. note y = this 

888 
show "x \<in> Pi\<^isub>E {..<DIM('a)} XX" unfolding y using y(1) 

889 
unfolding Pi_def extensional_def e2p_def restrict_def X mem_interval XX_def by auto 

890 
qed 

891 
have "lmeasure X = (\<Prod>x<DIM('a). Real (b $$ x  a $$ x))" using X' apply unfolding X 

892 
unfolding lmeasure_atLeastAtMost content_closed_interval apply(subst Real_setprod) by auto 

893 
also have "... = (\<Prod>i<DIM('a). lmeasure (XX i))" apply(rule setprod_cong2) 

894 
unfolding XX_def lmeasure_atLeastAtMost apply(subst content_real) using X' by auto 

895 
also have "... = borel_product.product_measure {..<DIM('a)} (e2p ` X)" unfolding *[THEN sym] 

896 
apply(rule fprod.measure_times[THEN sym]) unfolding XX_def by auto 

897 
finally show ?case . 

898 
qed 

899 

900 
show "range cube \<subseteq> sets \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>" 

901 
unfolding cube_def_raw by auto 

902 
have "\<And>x. \<exists>xa. x \<in> cube xa" apply(rule_tac x=x in mem_big_cube) by fastsimp 

903 
thus "cube \<up> space \<lparr>space = UNIV, sets = range (\<lambda>(a, b). {a..b})\<rparr>" 

904 
applyapply(rule isotoneI) apply(rule cube_subset_Suc) by auto 

905 
show "A \<in> sets borel " by fact 

906 
show "measure_space borel lmeasure" by default 

907 
show "measure_space borel 

908 
(\<lambda>a::'a set. finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` a))" 

909 
apply default unfolding countably_additive_def 

910 
proof safe fix A::"nat \<Rightarrow> 'a set" assume A:"range A \<subseteq> sets borel" "disjoint_family A" 

911 
"(\<Union>i. A i) \<in> sets borel" 

912 
note fprod.ca[unfolded countably_additive_def,rule_format] 

913 
note ca = this[of "\<lambda> n. e2p ` (A n)"] 

914 
show "(\<Sum>\<^isub>\<infinity>n. finite_product_sigma_finite.measure 

915 
(\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` A n)) = 

916 
finite_product_sigma_finite.measure (\<lambda>x. borel) 

917 
(\<lambda>x. lmeasure) {..<DIM('a)} (e2p ` (\<Union>i. A i))" unfolding image_UN 

918 
proof(rule ca) show "range (\<lambda>n. e2p ` A n) \<subseteq> sets 

919 
(sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))" 

920 
unfolding product_borel_eq_vimage 

921 
proof case goal1 

922 
then guess y unfolding image_iff .. note y=this(2) 

923 
show ?case unfolding borel.in_vimage_algebra y apply 

924 
apply(rule_tac x="A y" in bexI,rule e2p_image_vimage) 

925 
using A(1) by auto 

926 
qed 

927 

928 
show "disjoint_family (\<lambda>n. e2p ` A n)" apply(rule inj_on_disjoint_family_on) 

41095  929 
using bij_inv_p2e_e2p[THEN bij_inv_bij_betw(2)] using A(2) unfolding bij_betw_def by auto 
40859  930 
show "(\<Union>n. e2p ` A n) \<in> sets (sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)}))" 
931 
unfolding product_borel_eq_vimage borel.in_vimage_algebra 

932 
proof(rule bexI[OF _ A(3)],rule set_eqI,rule) 

933 
fix x assume x:"x \<in> (\<Union>n. e2p ` A n)" hence "p2e x \<in> (\<Union>i. A i)" by auto 

934 
moreover have "x \<in> extensional {..<DIM('a)}" 

935 
using x unfolding extensional_def e2p_def_raw by auto 

41095  936 
ultimately show "x \<in> p2e ` (\<Union>i. A i) \<inter> extensional {..<DIM('a)}" by auto 
937 
next fix x assume x:"x \<in> p2e ` (\<Union>i. A i) \<inter> extensional {..<DIM('a)}" 

40859  938 
hence "p2e x \<in> (\<Union>i. A i)" by auto 
939 
hence "\<exists>n. x \<in> e2p ` A n" apply safe apply(rule_tac x=i in exI) 

940 
unfolding image_iff apply(rule_tac x="p2e x" in bexI) 

941 
apply(subst e2p_p2e) using x by auto 

942 
thus "x \<in> (\<Union>n. e2p ` A n)" by auto 

943 
qed 

944 
qed 

945 
qed auto 

946 
qed 

947 

948 
lemma e2p_p2e'[simp]: fixes x::"'a::ordered_euclidean_space" 

949 
assumes "A \<subseteq> extensional {..<DIM('a)}" 

950 
shows "e2p ` (p2e ` A ::'a set) = A" 

951 
apply(rule set_eqI) unfolding image_iff Bex_def apply safe defer 

952 
apply(rule_tac x="p2e x" in exI,safe) using assms by auto 

953 

954 
lemma range_p2e:"range (p2e::_\<Rightarrow>'a::ordered_euclidean_space) = UNIV" 

955 
apply safe defer unfolding image_iff apply(rule_tac x="\<lambda>i. x $$ i" in bexI) 

956 
unfolding p2e_def by auto 

957 

958 
lemma p2e_inv_extensional:"(A::'a::ordered_euclidean_space set) 

959 
= p2e ` (p2e ` A \<inter> extensional {..<DIM('a)})" 

960 
unfolding p2e_def_raw apply safe unfolding image_iff 

961 
proof fix x assume "x\<in>A" 

962 
let ?y = "\<lambda>i. if i<DIM('a) then x$$i else undefined" 

963 
have *:"Chi ?y = x" apply(subst euclidean_eq) by auto 

964 
show "\<exists>xa\<in>Chi ` A \<inter> extensional {..<DIM('a)}. x = Chi xa" apply(rule_tac x="?y" in bexI) 

965 
apply(subst euclidean_eq) unfolding extensional_def using `x\<in>A` by(auto simp: *) 

966 
qed 

967 

968 
lemma borel_fubini_positiv_integral: 

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

969 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> pextreal" 
40859  970 
assumes f: "f \<in> borel_measurable borel" 
971 
shows "borel.positive_integral f = 

972 
borel_product.product_positive_integral {..<DIM('a)} (f \<circ> p2e)" 

41095  973 
proof def U \<equiv> "extensional {..<DIM('a)} :: (nat \<Rightarrow> real) set" 
40859  974 
interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto 
41095  975 
have *:"sigma_algebra.vimage_algebra borel U (p2e:: _ \<Rightarrow> 'a) 
40859  976 
= sigma (product_algebra (\<lambda>x. borel) {..<DIM('a)})" 
41095  977 
unfolding U_def product_borel_eq_vimage[symmetric] .. 
978 
show ?thesis 

979 
unfolding borel.positive_integral_vimage[unfolded space_borel, OF bij_inv_p2e_e2p[THEN bij_inv_bij_betw(1)]] 

40859  980 
apply(subst fprod.positive_integral_cong_measure[THEN sym, of "\<lambda>A. lmeasure (p2e ` A)"]) 
981 
unfolding U_def[symmetric] *[THEN sym] o_def 

982 
proof fix A assume A:"A \<in> sets (sigma_algebra.vimage_algebra borel U (p2e ::_ \<Rightarrow> 'a))" 

983 
hence *:"A \<subseteq> extensional {..<DIM('a)}" unfolding U_def by auto 

41095  984 
from A guess B unfolding borel.in_vimage_algebra U_def .. 
985 
then have "(p2e ` A::'a set) \<in> sets borel" 

986 
by (simp add: p2e_inv_extensional[of B, symmetric]) 

40859  987 
from lmeasure_measure_eq_borel_prod[OF this] show "lmeasure (p2e ` A::'a set) = 
988 
finite_product_sigma_finite.measure (\<lambda>x. borel) (\<lambda>x. lmeasure) {..<DIM('a)} A" 

989 
unfolding e2p_p2e'[OF *] . 

990 
qed auto 

991 
qed 

992 

993 
lemma borel_fubini: 

994 
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" 

995 
assumes f: "f \<in> borel_measurable borel" 

996 
shows "borel.integral f = borel_product.product_integral {..<DIM('a)} (f \<circ> p2e)" 

997 
proof interpret fprod: finite_product_sigma_finite "\<lambda>x. borel" "\<lambda>x. lmeasure" "{..<DIM('a)}" by default auto 

998 
have 1:"(\<lambda>x. Real (f x)) \<in> borel_measurable borel" using f by auto 

999 
have 2:"(\<lambda>x. Real ( f x)) \<in> borel_measurable borel" using f by auto 

1000 
show ?thesis unfolding fprod.integral_def borel.integral_def 

1001 
unfolding borel_fubini_positiv_integral[OF 1] borel_fubini_positiv_integral[OF 2] 

1002 
unfolding o_def .. 

38656  1003 
qed 
1004 

1005 
end 