author | hoelzl |
Wed, 10 Oct 2012 12:12:35 +0200 | |
changeset 49801 | f3471f09bb86 |
parent 49784 | 5e5b2da42a69 |
child 50003 | 8c213922ed49 |
permissions | -rw-r--r-- |
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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_indicator': |
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"A \<in> sets borel \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. indicator A (f x)) \<in> borel_measurable M" |
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using measurable_comp[OF _ borel_measurable_indicator, of f M borel A] by (auto simp add: comp_def) |
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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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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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||
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lemma borel_cube[intro]: "cube n \<in> sets borel" |
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unfolding cube_def by auto |
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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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|
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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 apply-apply(rule that[where n=n]) |
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apply(rule ball_subset_cube[unfolded subset_eq,rule_format]) |
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by (auto simp add:dist_norm) |
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qed |
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lemma 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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prefer p2e before e2p; use measure_unique_Int_stable_vimage;
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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 |
|
198 |
qed |
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qed (auto, fact) |
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41654 | 201 |
lemma has_integral_interval_cube: |
202 |
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 |
208 |
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 [abs_def] has_integral_restrict_univ .. |
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finally show ?thesis |
215 |
using has_integral_const[of "1::real" "?N" "?P"] by simp |
|
40859 | 216 |
qed |
38656 | 217 |
|
41654 | 218 |
lemma lebesgueI_borel[intro, simp]: |
219 |
fixes s::"'a::ordered_euclidean_space set" |
|
40859 | 220 |
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)}))" |
223 |
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) |
|
41654 | 226 |
fix n :: nat and a b :: 'a |
227 |
let ?N = "\<chi>\<chi> i. max (- real n) (a $$ i)" |
|
228 |
let ?P = "\<chi>\<chi> i. min (real n) (b $$ i)" |
|
229 |
show "(indicator {a..b} :: 'a\<Rightarrow>real) integrable_on cube n" |
|
230 |
unfolding integrable_on_def |
|
231 |
using has_integral_interval_cube[of a b] by auto |
|
232 |
qed |
|
47694 | 233 |
finally show ?thesis . |
38656 | 234 |
qed |
235 |
||
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equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
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|
236 |
lemma borel_measurable_lebesgueI: |
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equate positive Lebesgue integral and MV-Analysis' Gauge integral
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237 |
"f \<in> borel_measurable borel \<Longrightarrow> f \<in> borel_measurable lebesgue" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
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parents:
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|
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unfolding measurable_def by simp |
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equate positive Lebesgue integral and MV-Analysis' Gauge integral
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239 |
|
40859 | 240 |
lemma lebesgueI_negligible[dest]: fixes s::"'a::ordered_euclidean_space set" |
241 |
assumes "negligible s" shows "s \<in> sets lebesgue" |
|
41654 | 242 |
using assms by (force simp: cube_def integrable_on_def negligible_def intro!: lebesgueI) |
38656 | 243 |
|
41654 | 244 |
lemma lmeasure_eq_0: |
47694 | 245 |
fixes S :: "'a::ordered_euclidean_space set" |
246 |
assumes "negligible S" shows "emeasure lebesgue S = 0" |
|
40859 | 247 |
proof - |
41654 | 248 |
have "\<And>n. integral (cube n) (indicator S :: 'a\<Rightarrow>real) = 0" |
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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parents:
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|
249 |
unfolding lebesgue_integral_def using assms |
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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|
250 |
by (intro integral_unique some1_equality ex_ex1I) |
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251 |
(auto simp: cube_def negligible_def) |
47694 | 252 |
then show ?thesis |
253 |
using assms by (simp add: emeasure_lebesgue lebesgueI_negligible) |
|
40859 | 254 |
qed |
255 |
||
256 |
lemma lmeasure_iff_LIMSEQ: |
|
47694 | 257 |
assumes A: "A \<in> sets lebesgue" and "0 \<le> m" |
258 |
shows "emeasure lebesgue A = ereal m \<longleftrightarrow> (\<lambda>n. integral (cube n) (indicator A :: _ \<Rightarrow> real)) ----> m" |
|
259 |
proof (subst emeasure_lebesgue[OF A], intro SUP_eq_LIMSEQ) |
|
41654 | 260 |
show "mono (\<lambda>n. integral (cube n) (indicator A::_=>real))" |
261 |
using cube_subset assms by (intro monoI integral_subset_le) (auto dest!: lebesgueD) |
|
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262 |
qed |
38656 | 263 |
|
41654 | 264 |
lemma has_integral_indicator_UNIV: |
265 |
fixes s A :: "'a::ordered_euclidean_space set" and x :: real |
|
266 |
shows "((indicator (s \<inter> A) :: 'a\<Rightarrow>real) has_integral x) UNIV = ((indicator s :: _\<Rightarrow>real) has_integral x) A" |
|
267 |
proof - |
|
268 |
have "(\<lambda>x. if x \<in> A then indicator s x else 0) = (indicator (s \<inter> A) :: _\<Rightarrow>real)" |
|
269 |
by (auto simp: fun_eq_iff indicator_def) |
|
270 |
then show ?thesis |
|
271 |
unfolding has_integral_restrict_univ[where s=A, symmetric] by simp |
|
40859 | 272 |
qed |
38656 | 273 |
|
41654 | 274 |
lemma |
275 |
fixes s a :: "'a::ordered_euclidean_space set" |
|
276 |
shows integral_indicator_UNIV: |
|
277 |
"integral UNIV (indicator (s \<inter> A) :: 'a\<Rightarrow>real) = integral A (indicator s :: _\<Rightarrow>real)" |
|
278 |
and integrable_indicator_UNIV: |
|
279 |
"(indicator (s \<inter> A) :: 'a\<Rightarrow>real) integrable_on UNIV \<longleftrightarrow> (indicator s :: 'a\<Rightarrow>real) integrable_on A" |
|
280 |
unfolding integral_def integrable_on_def has_integral_indicator_UNIV by auto |
|
281 |
||
282 |
lemma lmeasure_finite_has_integral: |
|
283 |
fixes s :: "'a::ordered_euclidean_space set" |
|
49777 | 284 |
assumes "s \<in> sets lebesgue" "emeasure lebesgue s = ereal m" |
41654 | 285 |
shows "(indicator s has_integral m) UNIV" |
286 |
proof - |
|
287 |
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" |
|
49777 | 288 |
have "0 \<le> m" |
289 |
using emeasure_nonneg[of lebesgue s] `emeasure lebesgue s = ereal m` by simp |
|
41654 | 290 |
have **: "(?I s) integrable_on UNIV \<and> (\<lambda>k. integral UNIV (?I (s \<inter> cube k))) ----> integral UNIV (?I s)" |
291 |
proof (intro monotone_convergence_increasing allI ballI) |
|
292 |
have LIMSEQ: "(\<lambda>n. integral (cube n) (?I s)) ----> m" |
|
49777 | 293 |
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1) `0 \<le> m`] . |
41654 | 294 |
{ fix n have "integral (cube n) (?I s) \<le> m" |
295 |
using cube_subset assms |
|
296 |
by (intro incseq_le[where L=m] LIMSEQ incseq_def[THEN iffD2] integral_subset_le allI impI) |
|
297 |
(auto dest!: lebesgueD) } |
|
298 |
moreover |
|
299 |
{ fix n have "0 \<le> integral (cube n) (?I s)" |
|
47694 | 300 |
using assms by (auto dest!: lebesgueD intro!: Integration.integral_nonneg) } |
41654 | 301 |
ultimately |
302 |
show "bounded {integral UNIV (?I (s \<inter> cube k)) |k. True}" |
|
303 |
unfolding bounded_def |
|
304 |
apply (rule_tac exI[of _ 0]) |
|
305 |
apply (rule_tac exI[of _ m]) |
|
306 |
by (auto simp: dist_real_def integral_indicator_UNIV) |
|
307 |
fix k show "?I (s \<inter> cube k) integrable_on UNIV" |
|
308 |
unfolding integrable_indicator_UNIV using assms by (auto dest!: lebesgueD) |
|
309 |
fix x show "?I (s \<inter> cube k) x \<le> ?I (s \<inter> cube (Suc k)) x" |
|
310 |
using cube_subset[of k "Suc k"] by (auto simp: indicator_def) |
|
311 |
next |
|
312 |
fix x :: 'a |
|
313 |
from mem_big_cube obtain k where k: "x \<in> cube k" . |
|
314 |
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x" |
|
315 |
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } |
|
316 |
note * = this |
|
317 |
show "(\<lambda>k. ?I (s \<inter> cube k) x) ----> ?I s x" |
|
318 |
by (rule LIMSEQ_offset[where k=k]) (auto simp: *) |
|
319 |
qed |
|
320 |
note ** = conjunctD2[OF this] |
|
321 |
have m: "m = integral UNIV (?I s)" |
|
322 |
apply (intro LIMSEQ_unique[OF _ **(2)]) |
|
49777 | 323 |
using assms(2) unfolding lmeasure_iff_LIMSEQ[OF assms(1) `0 \<le> m`] integral_indicator_UNIV . |
41654 | 324 |
show ?thesis |
325 |
unfolding m by (intro integrable_integral **) |
|
38656 | 326 |
qed |
327 |
||
47694 | 328 |
lemma lmeasure_finite_integrable: assumes s: "s \<in> sets lebesgue" and "emeasure lebesgue s \<noteq> \<infinity>" |
41654 | 329 |
shows "(indicator s :: _ \<Rightarrow> real) integrable_on UNIV" |
47694 | 330 |
proof (cases "emeasure lebesgue s") |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
331 |
case (real m) |
47694 | 332 |
with lmeasure_finite_has_integral[OF `s \<in> sets lebesgue` this] emeasure_nonneg[of lebesgue s] |
41654 | 333 |
show ?thesis unfolding integrable_on_def by auto |
47694 | 334 |
qed (insert assms emeasure_nonneg[of lebesgue s], auto) |
38656 | 335 |
|
41654 | 336 |
lemma has_integral_lebesgue: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV" |
337 |
shows "s \<in> sets lebesgue" |
|
338 |
proof (intro lebesgueI) |
|
339 |
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" |
|
340 |
fix n show "(?I s) integrable_on cube n" unfolding cube_def |
|
341 |
proof (intro integrable_on_subinterval) |
|
342 |
show "(?I s) integrable_on UNIV" |
|
343 |
unfolding integrable_on_def using assms by auto |
|
344 |
qed auto |
|
38656 | 345 |
qed |
346 |
||
41654 | 347 |
lemma has_integral_lmeasure: assumes "((indicator s :: _\<Rightarrow>real) has_integral m) UNIV" |
47694 | 348 |
shows "emeasure lebesgue s = ereal m" |
41654 | 349 |
proof (intro lmeasure_iff_LIMSEQ[THEN iffD2]) |
350 |
let ?I = "indicator :: 'a set \<Rightarrow> 'a \<Rightarrow> real" |
|
351 |
show "s \<in> sets lebesgue" using has_integral_lebesgue[OF assms] . |
|
352 |
show "0 \<le> m" using assms by (rule has_integral_nonneg) auto |
|
353 |
have "(\<lambda>n. integral UNIV (?I (s \<inter> cube n))) ----> integral UNIV (?I s)" |
|
354 |
proof (intro dominated_convergence(2) ballI) |
|
355 |
show "(?I s) integrable_on UNIV" unfolding integrable_on_def using assms by auto |
|
356 |
fix n show "?I (s \<inter> cube n) integrable_on UNIV" |
|
357 |
unfolding integrable_indicator_UNIV using `s \<in> sets lebesgue` by (auto dest: lebesgueD) |
|
358 |
fix x show "norm (?I (s \<inter> cube n) x) \<le> ?I s x" by (auto simp: indicator_def) |
|
359 |
next |
|
360 |
fix x :: 'a |
|
361 |
from mem_big_cube obtain k where k: "x \<in> cube k" . |
|
362 |
{ fix n have "?I (s \<inter> cube (n + k)) x = ?I s x" |
|
363 |
using k cube_subset[of k "n + k"] by (auto simp: indicator_def) } |
|
364 |
note * = this |
|
365 |
show "(\<lambda>k. ?I (s \<inter> cube k) x) ----> ?I s x" |
|
366 |
by (rule LIMSEQ_offset[where k=k]) (auto simp: *) |
|
367 |
qed |
|
368 |
then show "(\<lambda>n. integral (cube n) (?I s)) ----> m" |
|
369 |
unfolding integral_unique[OF assms] integral_indicator_UNIV by simp |
|
370 |
qed |
|
371 |
||
372 |
lemma has_integral_iff_lmeasure: |
|
49777 | 373 |
"(indicator A has_integral m) UNIV \<longleftrightarrow> (A \<in> sets lebesgue \<and> emeasure lebesgue A = ereal m)" |
40859 | 374 |
proof |
41654 | 375 |
assume "(indicator A has_integral m) UNIV" |
376 |
with has_integral_lmeasure[OF this] has_integral_lebesgue[OF this] |
|
49777 | 377 |
show "A \<in> sets lebesgue \<and> emeasure lebesgue A = ereal m" |
41654 | 378 |
by (auto intro: has_integral_nonneg) |
40859 | 379 |
next |
49777 | 380 |
assume "A \<in> sets lebesgue \<and> emeasure lebesgue A = ereal m" |
41654 | 381 |
then show "(indicator A has_integral m) UNIV" by (intro lmeasure_finite_has_integral) auto |
38656 | 382 |
qed |
383 |
||
41654 | 384 |
lemma lmeasure_eq_integral: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" |
47694 | 385 |
shows "emeasure lebesgue s = ereal (integral UNIV (indicator s))" |
41654 | 386 |
using assms unfolding integrable_on_def |
387 |
proof safe |
|
388 |
fix y :: real assume "(indicator s has_integral y) UNIV" |
|
389 |
from this[unfolded has_integral_iff_lmeasure] integral_unique[OF this] |
|
47694 | 390 |
show "emeasure lebesgue s = ereal (integral UNIV (indicator s))" by simp |
40859 | 391 |
qed |
38656 | 392 |
|
393 |
lemma lebesgue_simple_function_indicator: |
|
43920 | 394 |
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
|
395 |
assumes f:"simple_function lebesgue f" |
38656 | 396 |
shows "f = (\<lambda>x. (\<Sum>y \<in> f ` UNIV. y * indicator (f -` {y}) x))" |
47694 | 397 |
by (rule, subst simple_function_indicator_representation[OF f]) auto |
38656 | 398 |
|
41654 | 399 |
lemma integral_eq_lmeasure: |
47694 | 400 |
"(indicator s::_\<Rightarrow>real) integrable_on UNIV \<Longrightarrow> integral UNIV (indicator s) = real (emeasure lebesgue s)" |
41654 | 401 |
by (subst lmeasure_eq_integral) (auto intro!: integral_nonneg) |
38656 | 402 |
|
47694 | 403 |
lemma lmeasure_finite: assumes "(indicator s::_\<Rightarrow>real) integrable_on UNIV" shows "emeasure lebesgue s \<noteq> \<infinity>" |
41654 | 404 |
using lmeasure_eq_integral[OF assms] by auto |
38656 | 405 |
|
40859 | 406 |
lemma negligible_iff_lebesgue_null_sets: |
47694 | 407 |
"negligible A \<longleftrightarrow> A \<in> null_sets lebesgue" |
40859 | 408 |
proof |
409 |
assume "negligible A" |
|
410 |
from this[THEN lebesgueI_negligible] this[THEN lmeasure_eq_0] |
|
47694 | 411 |
show "A \<in> null_sets lebesgue" by auto |
40859 | 412 |
next |
47694 | 413 |
assume A: "A \<in> null_sets lebesgue" |
414 |
then have *:"((indicator A) has_integral (0::real)) UNIV" using lmeasure_finite_has_integral[of A] |
|
415 |
by (auto simp: null_sets_def) |
|
41654 | 416 |
show "negligible A" unfolding negligible_def |
417 |
proof (intro allI) |
|
418 |
fix a b :: 'a |
|
419 |
have integrable: "(indicator A :: _\<Rightarrow>real) integrable_on {a..b}" |
|
420 |
by (intro integrable_on_subinterval has_integral_integrable) (auto intro: *) |
|
421 |
then have "integral {a..b} (indicator A) \<le> (integral UNIV (indicator A) :: real)" |
|
47694 | 422 |
using * by (auto intro!: integral_subset_le) |
41654 | 423 |
moreover have "(0::real) \<le> integral {a..b} (indicator A)" |
424 |
using integrable by (auto intro!: integral_nonneg) |
|
425 |
ultimately have "integral {a..b} (indicator A) = (0::real)" |
|
426 |
using integral_unique[OF *] by auto |
|
427 |
then show "(indicator A has_integral (0::real)) {a..b}" |
|
428 |
using integrable_integral[OF integrable] by simp |
|
429 |
qed |
|
430 |
qed |
|
431 |
||
432 |
lemma integral_const[simp]: |
|
433 |
fixes a b :: "'a::ordered_euclidean_space" |
|
434 |
shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c" |
|
435 |
by (rule integral_unique) (rule has_integral_const) |
|
436 |
||
47694 | 437 |
lemma lmeasure_UNIV[intro]: "emeasure lebesgue (UNIV::'a::ordered_euclidean_space set) = \<infinity>" |
438 |
proof (simp add: emeasure_lebesgue, intro SUP_PInfty bexI) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
439 |
fix n :: nat |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
440 |
have "indicator UNIV = (\<lambda>x::'a. 1 :: real)" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
441 |
moreover |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
442 |
{ have "real n \<le> (2 * real n) ^ DIM('a)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
443 |
proof (cases n) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
444 |
case 0 then show ?thesis by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
445 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
446 |
case (Suc n') |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
447 |
have "real n \<le> (2 * real n)^1" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
448 |
also have "(2 * real n)^1 \<le> (2 * real n) ^ DIM('a)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
449 |
using Suc DIM_positive[where 'a='a] by (intro power_increasing) (auto simp: real_of_nat_Suc) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
450 |
finally show ?thesis . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
451 |
qed } |
43920 | 452 |
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
hoelzl
parents:
41831
diff
changeset
|
453 |
using integral_const DIM_positive[where 'a='a] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
454 |
by (auto simp: cube_def content_closed_interval_cases setprod_constant) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
455 |
qed simp |
40859 | 456 |
|
49777 | 457 |
lemma lmeasure_complete: "A \<subseteq> B \<Longrightarrow> B \<in> null_sets lebesgue \<Longrightarrow> A \<in> null_sets lebesgue" |
458 |
unfolding negligible_iff_lebesgue_null_sets[symmetric] by (auto simp: negligible_subset) |
|
459 |
||
40859 | 460 |
lemma |
461 |
fixes a b ::"'a::ordered_euclidean_space" |
|
47694 | 462 |
shows lmeasure_atLeastAtMost[simp]: "emeasure lebesgue {a..b} = ereal (content {a..b})" |
41654 | 463 |
proof - |
464 |
have "(indicator (UNIV \<inter> {a..b})::_\<Rightarrow>real) integrable_on UNIV" |
|
46905 | 465 |
unfolding integrable_indicator_UNIV by (simp add: integrable_const indicator_def [abs_def]) |
41654 | 466 |
from lmeasure_eq_integral[OF this] show ?thesis unfolding integral_indicator_UNIV |
46905 | 467 |
by (simp add: indicator_def [abs_def]) |
40859 | 468 |
qed |
469 |
||
470 |
lemma atLeastAtMost_singleton_euclidean[simp]: |
|
471 |
fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}" |
|
472 |
by (force simp: eucl_le[where 'a='a] euclidean_eq[where 'a='a]) |
|
473 |
||
474 |
lemma content_singleton[simp]: "content {a} = 0" |
|
475 |
proof - |
|
476 |
have "content {a .. a} = 0" |
|
477 |
by (subst content_closed_interval) auto |
|
478 |
then show ?thesis by simp |
|
479 |
qed |
|
480 |
||
481 |
lemma lmeasure_singleton[simp]: |
|
47694 | 482 |
fixes a :: "'a::ordered_euclidean_space" shows "emeasure lebesgue {a} = 0" |
41654 | 483 |
using lmeasure_atLeastAtMost[of a a] by simp |
40859 | 484 |
|
49777 | 485 |
lemma AE_lebesgue_singleton: |
486 |
fixes a :: "'a::ordered_euclidean_space" shows "AE x in lebesgue. x \<noteq> a" |
|
487 |
by (rule AE_I[where N="{a}"]) auto |
|
488 |
||
40859 | 489 |
declare content_real[simp] |
490 |
||
491 |
lemma |
|
492 |
fixes a b :: real |
|
493 |
shows lmeasure_real_greaterThanAtMost[simp]: |
|
47694 | 494 |
"emeasure lebesgue {a <.. b} = ereal (if a \<le> b then b - a else 0)" |
49777 | 495 |
proof - |
496 |
have "emeasure lebesgue {a <.. b} = emeasure lebesgue {a .. b}" |
|
497 |
using AE_lebesgue_singleton[of a] |
|
498 |
by (intro emeasure_eq_AE) auto |
|
40859 | 499 |
then show ?thesis by auto |
49777 | 500 |
qed |
40859 | 501 |
|
502 |
lemma |
|
503 |
fixes a b :: real |
|
504 |
shows lmeasure_real_atLeastLessThan[simp]: |
|
47694 | 505 |
"emeasure lebesgue {a ..< b} = ereal (if a \<le> b then b - a else 0)" |
49777 | 506 |
proof - |
507 |
have "emeasure lebesgue {a ..< b} = emeasure lebesgue {a .. b}" |
|
508 |
using AE_lebesgue_singleton[of b] |
|
509 |
by (intro emeasure_eq_AE) auto |
|
41654 | 510 |
then show ?thesis by auto |
49777 | 511 |
qed |
41654 | 512 |
|
513 |
lemma |
|
514 |
fixes a b :: real |
|
515 |
shows lmeasure_real_greaterThanLessThan[simp]: |
|
47694 | 516 |
"emeasure lebesgue {a <..< b} = ereal (if a \<le> b then b - a else 0)" |
49777 | 517 |
proof - |
518 |
have "emeasure lebesgue {a <..< b} = emeasure lebesgue {a .. b}" |
|
519 |
using AE_lebesgue_singleton[of a] AE_lebesgue_singleton[of b] |
|
520 |
by (intro emeasure_eq_AE) auto |
|
40859 | 521 |
then show ?thesis by auto |
49777 | 522 |
qed |
40859 | 523 |
|
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset
|
524 |
subsection {* Lebesgue-Borel measure *} |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset
|
525 |
|
47694 | 526 |
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
|
527 |
|
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
|
528 |
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
|
529 |
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
|
530 |
and sets_lborel[simp]: "sets lborel = sets borel" |
47694 | 531 |
and measurable_lborel1[simp]: "measurable lborel = measurable borel" |
532 |
and measurable_lborel2[simp]: "measurable A lborel = measurable A borel" |
|
533 |
using sigma_sets_eq[of borel] |
|
534 |
by (auto simp add: lborel_def measurable_def[abs_def]) |
|
40859 | 535 |
|
47694 | 536 |
lemma emeasure_lborel[simp]: "A \<in> sets borel \<Longrightarrow> emeasure lborel A = emeasure lebesgue A" |
537 |
by (rule emeasure_measure_of[OF lborel_def]) |
|
538 |
(auto simp: positive_def emeasure_nonneg countably_additive_def intro!: suminf_emeasure) |
|
40859 | 539 |
|
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
|
540 |
interpretation lborel: sigma_finite_measure lborel |
47694 | 541 |
proof (default, intro conjI exI[of _ "\<lambda>n. cube n"]) |
542 |
show "range cube \<subseteq> sets lborel" by (auto intro: borel_closed) |
|
543 |
{ fix x :: 'a have "\<exists>n. x\<in>cube n" using mem_big_cube by auto } |
|
544 |
then show "(\<Union>i. cube i) = (space lborel :: 'a set)" using mem_big_cube by auto |
|
545 |
show "\<forall>i. emeasure lborel (cube i) \<noteq> \<infinity>" by (simp add: cube_def) |
|
546 |
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
|
547 |
|
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
|
548 |
interpretation lebesgue: sigma_finite_measure lebesgue |
40859 | 549 |
proof |
47694 | 550 |
from lborel.sigma_finite guess A :: "nat \<Rightarrow> 'a set" .. |
551 |
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>)" |
|
552 |
by (intro exI[of _ A]) (auto simp: subset_eq) |
|
40859 | 553 |
qed |
554 |
||
49777 | 555 |
lemma Int_stable_atLeastAtMost: |
556 |
fixes x::"'a::ordered_euclidean_space" |
|
557 |
shows "Int_stable (range (\<lambda>(a, b::'a). {a..b}))" |
|
558 |
by (auto simp: inter_interval Int_stable_def) |
|
559 |
||
560 |
lemma lborel_eqI: |
|
561 |
fixes M :: "'a::ordered_euclidean_space measure" |
|
562 |
assumes emeasure_eq: "\<And>a b. emeasure M {a .. b} = content {a .. b}" |
|
563 |
assumes sets_eq: "sets M = sets borel" |
|
564 |
shows "lborel = M" |
|
565 |
proof (rule measure_eqI_generator_eq[OF Int_stable_atLeastAtMost]) |
|
566 |
let ?P = "\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel" |
|
567 |
let ?E = "range (\<lambda>(a, b). {a..b} :: 'a set)" |
|
568 |
show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E" |
|
569 |
by (simp_all add: borel_eq_atLeastAtMost sets_eq) |
|
570 |
||
571 |
show "range cube \<subseteq> ?E" unfolding cube_def [abs_def] by auto |
|
572 |
{ fix x :: 'a have "\<exists>n. x \<in> cube n" using mem_big_cube[of x] by fastforce } |
|
573 |
then show "(\<Union>i. cube i :: 'a set) = UNIV" by auto |
|
574 |
||
575 |
{ fix i show "emeasure lborel (cube i) \<noteq> \<infinity>" unfolding cube_def by auto } |
|
576 |
{ fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X" |
|
577 |
by (auto simp: emeasure_eq) } |
|
578 |
qed |
|
579 |
||
580 |
lemma lebesgue_real_affine: |
|
581 |
fixes c :: real assumes "c \<noteq> 0" |
|
582 |
shows "lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. \<bar>c\<bar>)" (is "_ = ?D") |
|
583 |
proof (rule lborel_eqI) |
|
584 |
fix a b show "emeasure ?D {a..b} = content {a .. b}" |
|
585 |
proof cases |
|
586 |
assume "0 < c" |
|
587 |
then have "(\<lambda>x. t + c * x) -` {a..b} = {(a - t) / c .. (b - t) / c}" |
|
588 |
by (auto simp: field_simps) |
|
589 |
with `0 < c` show ?thesis |
|
590 |
by (cases "a \<le> b") |
|
591 |
(auto simp: field_simps emeasure_density positive_integral_distr positive_integral_cmult |
|
592 |
borel_measurable_indicator' emeasure_distr) |
|
593 |
next |
|
594 |
assume "\<not> 0 < c" with `c \<noteq> 0` have "c < 0" by auto |
|
595 |
then have *: "(\<lambda>x. t + c * x) -` {a..b} = {(b - t) / c .. (a - t) / c}" |
|
596 |
by (auto simp: field_simps) |
|
597 |
with `c < 0` show ?thesis |
|
598 |
by (cases "a \<le> b") |
|
599 |
(auto simp: field_simps emeasure_density positive_integral_distr |
|
600 |
positive_integral_cmult borel_measurable_indicator' emeasure_distr) |
|
601 |
qed |
|
602 |
qed simp |
|
603 |
||
604 |
lemma lebesgue_integral_real_affine: |
|
605 |
fixes c :: real assumes c: "c \<noteq> 0" and f: "f \<in> borel_measurable borel" |
|
606 |
shows "(\<integral> x. f x \<partial> lborel) = \<bar>c\<bar> * (\<integral> x. f (t + c * x) \<partial>lborel)" |
|
607 |
by (subst lebesgue_real_affine[OF c, of t]) |
|
608 |
(simp add: f integral_density integral_distr lebesgue_integral_cmult) |
|
609 |
||
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset
|
610 |
subsection {* Lebesgue integrable implies Gauge integrable *} |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41704
diff
changeset
|
611 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
612 |
lemma has_integral_cmult_real: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
613 |
fixes c :: real |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
614 |
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
|
615 |
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
|
616 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
617 |
assume "c \<noteq> 0" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
618 |
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
|
619 |
unfolding real_scaleR_def . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
620 |
qed simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
621 |
|
40859 | 622 |
lemma simple_function_has_integral: |
43920 | 623 |
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
|
624 |
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
|
625 |
and f':"range f \<subseteq> {0..<\<infinity>}" |
47694 | 626 |
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
|
627 |
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
|
628 |
unfolding simple_integral_def space_lebesgue |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
629 |
proof (subst lebesgue_simple_function_indicator) |
47694 | 630 |
let ?M = "\<lambda>x. emeasure lebesgue (f -` {x} \<inter> UNIV)" |
46731 | 631 |
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
|
632 |
{ fix x y assume "y \<in> range f" |
43920 | 633 |
from subsetD[OF f' this] have "y * ?F y x = ereal (real y * ?F y x)" |
634 |
by (cases rule: ereal2_cases[of y "?F y x"]) |
|
635 |
(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
|
636 |
moreover |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
637 |
{ 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
|
638 |
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
|
639 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
640 |
assume "x \<noteq> 0" with om[OF x] have "?M x \<noteq> \<infinity>" by auto |
47694 | 641 |
with subsetD[OF f' x] f[THEN simple_functionD(2)] show ?thesis |
43920 | 642 |
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
|
643 |
qed simp } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
644 |
ultimately |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
645 |
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
|
646 |
((\<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
|
647 |
by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
648 |
also have \<dots> |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
649 |
proof (intro has_integral_setsum has_integral_cmult_real lmeasure_finite_has_integral |
47694 | 650 |
real_of_ereal_pos emeasure_nonneg ballI) |
651 |
show *: "finite (range f)" "\<And>y. f -` {y} \<in> sets lebesgue" |
|
652 |
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
|
653 |
fix y assume "real y \<noteq> 0" "y \<in> range f" |
47694 | 654 |
with * om[OF this(2)] show "emeasure lebesgue (f -` {y}) = ereal (real (?M y))" |
43920 | 655 |
by (auto simp: ereal_real) |
41654 | 656 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
657 |
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
|
658 |
qed fact |
40859 | 659 |
|
660 |
lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s" |
|
661 |
unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI) |
|
662 |
using assms by auto |
|
663 |
||
664 |
lemma simple_function_has_integral': |
|
43920 | 665 |
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
|
666 |
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
|
667 |
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
|
668 |
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
|
669 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
670 |
let ?f = "\<lambda>x. if x \<in> f -` {\<infinity>} then 0 else f x" |
47694 | 671 |
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
|
672 |
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
|
673 |
have f': "simple_function lebesgue ?f" |
47694 | 674 |
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
|
675 |
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
|
676 |
have "AE x in lebesgue. f x = ?f x" |
47694 | 677 |
using simple_integral_PInf[OF f i] |
678 |
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
|
679 |
from f(1) f' this have eq: "integral\<^isup>S lebesgue f = integral\<^isup>S lebesgue ?f" |
47694 | 680 |
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
|
681 |
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
|
682 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
683 |
show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
684 |
unfolding eq real_eq |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
685 |
proof (rule simple_function_has_integral[OF f' rng]) |
47694 | 686 |
fix x assume x: "x \<in> range ?f" and inf: "emeasure lebesgue (?f -` {x} \<inter> UNIV) = \<infinity>" |
687 |
have "x * emeasure lebesgue (?f -` {x} \<inter> UNIV) = (\<integral>\<^isup>S y. x * indicator (?f -` {x}) y \<partial>lebesgue)" |
|
688 |
using f'[THEN simple_functionD(2)] |
|
689 |
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
|
690 |
also have "\<dots> \<le> integral\<^isup>S lebesgue f" |
47694 | 691 |
using f'[THEN simple_functionD(2)] f |
692 |
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
|
693 |
(auto split: split_indicator) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
694 |
finally show "x = 0" unfolding inf using i subsetD[OF rng x] by (auto split: split_if_asm) |
40859 | 695 |
qed |
696 |
qed |
|
697 |
||
49801 | 698 |
lemma borel_measurable_real_induct[consumes 2, case_names cong set mult add seq, induct set: borel_measurable]: |
699 |
fixes u :: "'a \<Rightarrow> real" |
|
700 |
assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x" |
|
701 |
assumes cong: "\<And>f g. f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P g \<Longrightarrow> P f" |
|
702 |
assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)" |
|
703 |
assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)" |
|
704 |
assumes add: "\<And>u v. u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)" |
|
705 |
assumes seq: "\<And>U u. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow> (\<And>i x. 0 \<le> U i x) \<Longrightarrow> (\<And>x. (\<lambda>i. U i x) ----> u x) \<Longrightarrow> (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> P u" |
|
706 |
shows "P u" |
|
707 |
proof - |
|
708 |
have "(\<lambda>x. ereal (u x)) \<in> borel_measurable M" |
|
709 |
using u by auto |
|
710 |
then obtain U where U: "\<And>i. simple_function M (U i)" "incseq U" "\<And>i. \<infinity> \<notin> range (U i)" and |
|
711 |
"\<And>x. (SUP i. U i x) = max 0 (ereal (u x))" and nn: "\<And>i x. 0 \<le> U i x" |
|
712 |
using borel_measurable_implies_simple_function_sequence'[of u M] by auto |
|
713 |
then have u_eq: "\<And>x. ereal (u x) = (SUP i. U i x)" |
|
714 |
using u by (auto simp: max_def) |
|
715 |
||
716 |
have [simp]: "\<And>i x. U i x \<noteq> \<infinity>" using U by (auto simp: image_def eq_commute) |
|
717 |
||
718 |
{ fix i x have [simp]: "U i x \<noteq> -\<infinity>" using nn[of i x] by auto } |
|
719 |
note this[simp] |
|
720 |
||
721 |
show "P u" |
|
722 |
proof (rule seq) |
|
723 |
show "\<And>i. (\<lambda>x. real (U i x)) \<in> borel_measurable M" |
|
724 |
using U by (auto intro: borel_measurable_simple_function) |
|
725 |
show "\<And>i x. 0 \<le> real (U i x)" |
|
726 |
using nn by (auto simp: real_of_ereal_pos) |
|
727 |
show "incseq (\<lambda>i x. real (U i x))" |
|
728 |
using U(2) by (auto simp: incseq_def image_iff le_fun_def intro!: real_of_ereal_positive_mono nn) |
|
729 |
then show "\<And>x. (\<lambda>i. real (U i x)) ----> u x" |
|
730 |
by (intro SUP_eq_LIMSEQ[THEN iffD1]) |
|
731 |
(auto simp: incseq_mono incseq_def le_fun_def u_eq ereal_real |
|
732 |
intro!: arg_cong2[where f=SUPR] ext) |
|
733 |
show "\<And>i. P (\<lambda>x. real (U i x))" |
|
734 |
proof (rule cong) |
|
735 |
fix x i assume x: "x \<in> space M" |
|
736 |
have [simp]: "\<And>A x. real (indicator A x :: ereal) = indicator A x" |
|
737 |
by (auto simp: indicator_def one_ereal_def) |
|
738 |
{ fix y assume "y \<in> U i ` space M" |
|
739 |
then have "0 \<le> y" "y \<noteq> \<infinity>" using nn by auto |
|
740 |
then have "\<bar>y * indicator (U i -` {y} \<inter> space M) x\<bar> \<noteq> \<infinity>" |
|
741 |
by (auto simp: indicator_def) } |
|
742 |
then show "real (U i x) = (\<Sum>y \<in> U i ` space M. real y * indicator (U i -` {y} \<inter> space M) x)" |
|
743 |
unfolding simple_function_indicator_representation[OF U(1) x] |
|
744 |
by (subst setsum_real_of_ereal[symmetric]) auto |
|
745 |
next |
|
746 |
fix i |
|
747 |
have "finite (U i ` space M)" "\<And>x. x \<in> U i ` space M \<Longrightarrow> 0 \<le> x""\<And>x. x \<in> U i ` space M \<Longrightarrow> x \<noteq> \<infinity>" |
|
748 |
using U(1) nn by (auto simp: simple_function_def) |
|
749 |
then show "P (\<lambda>x. \<Sum>y \<in> U i ` space M. real y * indicator (U i -` {y} \<inter> space M) x)" |
|
750 |
proof induct |
|
751 |
case empty then show ?case |
|
752 |
using set[of "{}"] by (simp add: indicator_def[abs_def]) |
|
753 |
qed (auto intro!: add mult set simple_functionD U setsum_nonneg borel_measurable_setsum mult_nonneg_nonneg real_of_ereal_pos) |
|
754 |
qed (auto intro: borel_measurable_simple_function U simple_functionD intro!: borel_measurable_setsum borel_measurable_times) |
|
755 |
qed |
|
756 |
qed |
|
757 |
||
758 |
lemma ereal_indicator: "ereal (indicator A x) = indicator A x" |
|
759 |
by (auto simp: indicator_def one_ereal_def) |
|
760 |
||
40859 | 761 |
lemma positive_integral_has_integral: |
43920 | 762 |
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
|
763 |
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
|
764 |
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
|
765 |
proof - |
47694 | 766 |
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
|
767 |
guess u . note u = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
768 |
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
|
769 |
using u(4) f(2)[THEN subsetD] by (auto split: split_max) |
46731 | 770 |
let ?u = "\<lambda>i x. real (u i x)" |
47694 | 771 |
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
|
772 |
{ fix i |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
773 |
note u_eq |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
774 |
also have "integral\<^isup>P lebesgue (u i) \<le> (\<integral>\<^isup>+x. max 0 (f x) \<partial>lebesgue)" |
47694 | 775 |
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
|
776 |
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
|
777 |
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
|
778 |
note u_fin = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
779 |
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
|
780 |
by (rule simple_function_has_integral'[OF u(1,5)]) |
43920 | 781 |
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
|
782 |
proof |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
783 |
fix x from f(2) have "0 \<le> f x" "f x \<noteq> \<infinity>" by (auto simp: subset_eq) |
43920 | 784 |
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
|
785 |
qed |
43920 | 786 |
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
|
787 |
|
43920 | 788 |
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
|
789 |
proof |
43920 | 790 |
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
|
791 |
proof (intro choice allI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
792 |
fix i x have "u i x \<noteq> \<infinity>" using u(3)[of i] by (auto simp: image_iff) metis |
43920 | 793 |
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
|
794 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
795 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
796 |
from choice[OF this] obtain u' where |
43920 | 797 |
u': "u = (\<lambda>i x. ereal (u' i x))" "\<And>i x. 0 \<le> u' i x" by (auto simp: fun_eq_iff) |
40859 | 798 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
799 |
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
|
800 |
(\<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
|
801 |
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
|
802 |
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
|
803 |
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
|
804 |
show "\<And>k x. u' k x \<le> u' (Suc k) x" using u(2,3,5) |
43920 | 805 |
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
|
806 |
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
|
807 |
using SUP_eq u(2) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
808 |
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
|
809 |
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
|
810 |
proof (safe intro!: bounded_realI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
811 |
fix k |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
812 |
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
|
813 |
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
|
814 |
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
|
815 |
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
|
816 |
also have "\<dots> = real (integral\<^isup>P lebesgue (u k))" |
47694 | 817 |
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
|
818 |
also have "\<dots> \<le> real (integral\<^isup>P lebesgue f)" using f |
47694 | 819 |
by (auto intro!: real_of_ereal_positive_mono positive_integral_positive |
820 |
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
|
821 |
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
|
822 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
823 |
qed |
40859 | 824 |
|
43920 | 825 |
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
|
826 |
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
|
827 |
have "(\<lambda>i. integral\<^isup>S lebesgue (u i)) ----> (SUP i. integral\<^isup>P lebesgue (u i))" |
47694 | 828 |
unfolding u_eq by (intro LIMSEQ_ereal_SUPR incseq_positive_integral u) |
829 |
also note positive_integral_monotone_convergence_SUP |
|
830 |
[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
|
831 |
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
|
832 |
unfolding SUP_eq . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
833 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
834 |
{ fix k |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
835 |
have "0 \<le> integral\<^isup>S lebesgue (u k)" |
47694 | 836 |
using u by (auto intro!: simple_integral_positive) |
43920 | 837 |
then have "integral\<^isup>S lebesgue (u k) = ereal (real (integral\<^isup>S lebesgue (u k)))" |
838 |
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
|
839 |
note * = this |
43920 | 840 |
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
|
841 |
using convergent using u_int[THEN integral_unique, symmetric] |
47694 | 842 |
by (subst *) (simp add: u') |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
843 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
844 |
then show ?thesis using convergent by (simp add: f' integrable_integral) |
40859 | 845 |
qed |
846 |
||
847 |
lemma lebesgue_integral_has_integral: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
848 |
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
|
849 |
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
|
850 |
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
|
851 |
proof - |
43920 | 852 |
let ?n = "\<lambda>x. real (ereal (max 0 (- f x)))" and ?p = "\<lambda>x. real (ereal (max 0 (f x)))" |
853 |
have *: "f = (\<lambda>x. ?p x - ?n x)" by (auto simp del: ereal_max) |
|
47694 | 854 |
{ fix f :: "'a \<Rightarrow> real" have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (\<integral>\<^isup>+ x. ereal (max 0 (f x)) \<partial>lebesgue)" |
855 |
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
|
856 |
note eq = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
857 |
show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
858 |
unfolding lebesgue_integral_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
859 |
apply (subst *) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
860 |
apply (rule has_integral_sub) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
861 |
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
|
862 |
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
|
863 |
using integrableD[OF f] |
43920 | 864 |
by (auto simp: zero_ereal_def[symmetric] positive_integral_max_0 split: split_max |
47694 | 865 |
intro!: measurable_If) |
40859 | 866 |
qed |
867 |
||
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
868 |
lemma lebesgue_simple_integral_eq_borel: |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
869 |
assumes f: "f \<in> borel_measurable borel" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
870 |
shows "integral\<^isup>S lebesgue f = integral\<^isup>S lborel f" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
871 |
using f[THEN measurable_sets] |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
872 |
by (auto intro!: setsum_cong arg_cong2[where f="op *"] emeasure_lborel[symmetric] |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
873 |
simp: simple_integral_def) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
874 |
|
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
875 |
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
|
876 |
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
|
877 |
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
|
878 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
879 |
from f have "integral\<^isup>P lebesgue (\<lambda>x. max 0 (f x)) = integral\<^isup>P lborel (\<lambda>x. max 0 (f x))" |
47694 | 880 |
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
|
881 |
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
|
882 |
qed |
41546
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
883 |
|
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
884 |
lemma lebesgue_integral_eq_borel: |
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
885 |
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
|
886 |
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
|
887 |
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
|
888 |
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
|
889 |
have "sets lborel \<subseteq> sets lebesgue" by auto |
47694 | 890 |
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
|
891 |
show ?P ?I by auto |
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
892 |
qed |
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
893 |
|
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
894 |
lemma borel_integral_has_integral: |
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
895 |
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
|
896 |
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
|
897 |
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
|
898 |
proof - |
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
899 |
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
|
900 |
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
|
901 |
from f show ?thesis |
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
902 |
using lebesgue_integral_has_integral[of f] |
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
903 |
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
|
904 |
qed |
2a12c23b7a34
integral on lebesgue measure is extension of integral on borel measure
hoelzl
parents:
41097
diff
changeset
|
905 |
|
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
906 |
lemma integrable_on_cmult_iff: |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
907 |
fixes c :: real assumes "c \<noteq> 0" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
908 |
shows "(\<lambda>x. c * f x) integrable_on s \<longleftrightarrow> f integrable_on s" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
909 |
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 MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
910 |
by auto |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
911 |
|
49777 | 912 |
lemma positive_integral_lebesgue_has_integral: |
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
913 |
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" |
49777 | 914 |
assumes f_borel: "f \<in> borel_measurable lebesgue" and nonneg: "\<And>x. 0 \<le> f x" |
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
915 |
assumes I: "(f has_integral I) UNIV" |
49777 | 916 |
shows "(\<integral>\<^isup>+x. f x \<partial>lebesgue) = I" |
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
917 |
proof - |
49777 | 918 |
from f_borel have "(\<lambda>x. ereal (f x)) \<in> borel_measurable lebesgue" by auto |
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
919 |
from borel_measurable_implies_simple_function_sequence'[OF this] guess F . note F = this |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
920 |
|
49777 | 921 |
have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) = (SUP i. integral\<^isup>S lebesgue (F i))" |
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
922 |
using F |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
923 |
by (subst positive_integral_monotone_convergence_simple) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
924 |
(simp_all add: positive_integral_max_0 simple_function_def) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
925 |
also have "\<dots> \<le> ereal I" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
926 |
proof (rule SUP_least) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
927 |
fix i :: nat |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
928 |
|
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
929 |
{ fix z |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
930 |
from F(4)[of z] have "F i z \<le> ereal (f z)" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
931 |
by (metis SUP_upper UNIV_I ereal_max_0 min_max.sup_absorb2 nonneg) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
932 |
with F(5)[of i z] have "real (F i z) \<le> f z" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
933 |
by (cases "F i z") simp_all } |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
934 |
note F_bound = this |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
935 |
|
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
936 |
{ fix x :: ereal assume x: "x \<noteq> 0" "x \<in> range (F i)" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
937 |
with F(3,5)[of i] have [simp]: "real x \<noteq> 0" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
938 |
by (metis image_iff order_eq_iff real_of_ereal_le_0) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
939 |
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 MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
940 |
have "(\<lambda>z::'a. real x * indicator (F i -` {x}) z) integrable_on UNIV" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
941 |
proof (rule dominated_convergence(1)) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
942 |
fix n :: nat |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
943 |
have "(\<lambda>z. indicator (F i -` {x} \<inter> cube n) z :: real) integrable_on cube n" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
944 |
using x F(1)[of i] |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
945 |
by (intro lebesgueD) (auto simp: simple_function_def) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
946 |
then have cube: "?s n integrable_on cube n" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
947 |
by (simp add: integrable_on_cmult_iff) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
948 |
show "?s n integrable_on UNIV" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
949 |
by (rule integrable_on_superset[OF _ _ cube]) auto |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
950 |
next |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
951 |
show "f integrable_on UNIV" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
952 |
unfolding integrable_on_def using I by auto |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
953 |
next |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
954 |
fix n from F_bound show "\<forall>x\<in>UNIV. norm (?s n x) \<le> f x" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
955 |
using nonneg F(5) by (auto split: split_indicator) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
956 |
next |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
957 |
show "\<forall>z\<in>UNIV. (\<lambda>n. ?s n z) ----> real x * indicator (F i -` {x}) z" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
958 |
proof |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
959 |
fix z :: 'a |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
960 |
from mem_big_cube[of z] guess j . |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
961 |
then have x: "eventually (\<lambda>n. ?s n z = real x * indicator (F i -` {x}) z) sequentially" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
962 |
by (auto intro!: eventually_sequentiallyI[where c=j] dest!: cube_subset split: split_indicator) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
963 |
then show "(\<lambda>n. ?s n z) ----> real x * indicator (F i -` {x}) z" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
964 |
by (rule Lim_eventually) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
965 |
qed |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
966 |
qed |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
967 |
then have "(indicator (F i -` {x}) :: 'a \<Rightarrow> real) integrable_on UNIV" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
968 |
by (simp add: integrable_on_cmult_iff) } |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
969 |
note F_finite = lmeasure_finite[OF this] |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
970 |
|
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
971 |
have "((\<lambda>x. real (F i x)) has_integral real (integral\<^isup>S lebesgue (F i))) UNIV" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
972 |
proof (rule simple_function_has_integral[of "F i"]) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
973 |
show "simple_function lebesgue (F i)" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
974 |
using F(1) by (simp add: simple_function_def) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
975 |
show "range (F i) \<subseteq> {0..<\<infinity>}" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
976 |
using F(3,5)[of i] by (auto simp: image_iff) metis |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
977 |
next |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
978 |
fix x assume "x \<in> range (F i)" "emeasure lebesgue (F i -` {x} \<inter> UNIV) = \<infinity>" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
979 |
with F_finite[of x] show "x = 0" by auto |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
980 |
qed |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
981 |
from this I have "real (integral\<^isup>S lebesgue (F i)) \<le> I" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
982 |
by (rule has_integral_le) (intro ballI F_bound) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
983 |
moreover |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
984 |
{ fix x assume x: "x \<in> range (F i)" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
985 |
with F(3,5)[of i] have "x = 0 \<or> (0 < x \<and> x \<noteq> \<infinity>)" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
986 |
by (auto simp: image_iff le_less) metis |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
987 |
with F_finite[OF _ x] x have "x * emeasure lebesgue (F i -` {x} \<inter> UNIV) \<noteq> \<infinity>" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
988 |
by auto } |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
989 |
then have "integral\<^isup>S lebesgue (F i) \<noteq> \<infinity>" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
990 |
unfolding simple_integral_def setsum_Pinfty space_lebesgue by blast |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
991 |
moreover have "0 \<le> integral\<^isup>S lebesgue (F i)" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
992 |
using F(1,5) by (intro simple_integral_positive) (auto simp: simple_function_def) |
49777 | 993 |
ultimately show "integral\<^isup>S lebesgue (F i) \<le> ereal I" |
994 |
by (cases "integral\<^isup>S lebesgue (F i)") auto |
|
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
995 |
qed |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
996 |
also have "\<dots> < \<infinity>" by simp |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
997 |
finally have finite: "(\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue) \<noteq> \<infinity>" by simp |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
998 |
have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable lebesgue" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
999 |
using f_borel by (auto intro: borel_measurable_lebesgueI) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1000 |
from positive_integral_has_integral[OF borel _ finite] |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1001 |
have "(f has_integral real (\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue)) UNIV" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1002 |
using nonneg by (simp add: subset_eq) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1003 |
with I have "I = real (\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue)" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1004 |
by (rule has_integral_unique) |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1005 |
with finite positive_integral_positive[of _ "\<lambda>x. ereal (f x)"] show ?thesis |
49777 | 1006 |
by (cases "\<integral>\<^isup>+ x. ereal (f x) \<partial>lebesgue") auto |
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1007 |
qed |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1008 |
|
49777 | 1009 |
lemma has_integral_iff_positive_integral_lebesgue: |
1010 |
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" |
|
1011 |
assumes f: "f \<in> borel_measurable lebesgue" "\<And>x. 0 \<le> f x" |
|
1012 |
shows "(f has_integral I) UNIV \<longleftrightarrow> integral\<^isup>P lebesgue f = I" |
|
1013 |
using f positive_integral_lebesgue_has_integral[of f I] positive_integral_has_integral[of f] |
|
1014 |
by (auto simp: subset_eq) |
|
1015 |
||
1016 |
lemma has_integral_iff_positive_integral_lborel: |
|
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1017 |
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1018 |
assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" |
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1019 |
shows "(f has_integral I) UNIV \<longleftrightarrow> integral\<^isup>P lborel f = I" |
49777 | 1020 |
using assms |
1021 |
by (subst has_integral_iff_positive_integral_lebesgue) |
|
1022 |
(auto simp: borel_measurable_lebesgueI lebesgue_positive_integral_eq_borel) |
|
1023 |
||
1024 |
subsection {* Equivalence between product spaces and euclidean spaces *} |
|
1025 |
||
1026 |
definition e2p :: "'a::ordered_euclidean_space \<Rightarrow> (nat \<Rightarrow> real)" where |
|
1027 |
"e2p x = (\<lambda>i\<in>{..<DIM('a)}. x$$i)" |
|
1028 |
||
1029 |
definition p2e :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::ordered_euclidean_space" where |
|
1030 |
"p2e x = (\<chi>\<chi> i. x i)" |
|
1031 |
||
1032 |
lemma e2p_p2e[simp]: |
|
1033 |
"x \<in> extensional {..<DIM('a)} \<Longrightarrow> e2p (p2e x::'a::ordered_euclidean_space) = x" |
|
1034 |
by (auto simp: fun_eq_iff extensional_def p2e_def e2p_def) |
|
1035 |
||
1036 |
lemma p2e_e2p[simp]: |
|
1037 |
"p2e (e2p x) = (x::'a::ordered_euclidean_space)" |
|
1038 |
by (auto simp: euclidean_eq[where 'a='a] p2e_def e2p_def) |
|
1039 |
||
1040 |
interpretation lborel_product: product_sigma_finite "\<lambda>x. lborel::real measure" |
|
1041 |
by default |
|
1042 |
||
1043 |
interpretation lborel_space: finite_product_sigma_finite "\<lambda>x. lborel::real measure" "{..<n}" for n :: nat |
|
1044 |
by default auto |
|
1045 |
||
1046 |
lemma bchoice_iff: "(\<forall>x\<in>A. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>A. P x (f x))" |
|
1047 |
by metis |
|
1048 |
||
1049 |
lemma sets_product_borel: |
|
1050 |
assumes I: "finite I" |
|
1051 |
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") |
|
1052 |
proof (subst sigma_prod_algebra_sigma_eq[where S="\<lambda>_ i::nat. {..<real i}" and E="\<lambda>_. range lessThan", OF I]) |
|
1053 |
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" |
|
1054 |
by (intro arg_cong2[where f=sigma_sets]) (auto simp: space_PiM image_iff bchoice_iff) |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49777
diff
changeset
|
1055 |
qed (auto simp: borel_eq_lessThan reals_Archimedean2) |
49777 | 1056 |
|
1057 |
lemma measurable_e2p: |
|
1058 |
"e2p \<in> measurable (borel::'a::ordered_euclidean_space measure) (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure))" |
|
1059 |
proof (rule measurable_sigma_sets[OF sets_product_borel]) |
|
1060 |
fix A :: "(nat \<Rightarrow> real) set" assume "A \<in> {\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..<x i} |x. True} " |
|
1061 |
then obtain x where "A = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {..<x i})" by auto |
|
1062 |
then have "e2p -` A = {..< (\<chi>\<chi> i. x i) :: 'a}" |
|
1063 |
using DIM_positive by (auto simp add: Pi_iff set_eq_iff e2p_def |
|
1064 |
euclidean_eq[where 'a='a] eucl_less[where 'a='a]) |
|
1065 |
then show "e2p -` A \<inter> space (borel::'a measure) \<in> sets borel" by simp |
|
1066 |
qed (auto simp: e2p_def) |
|
1067 |
||
1068 |
lemma measurable_p2e: |
|
1069 |
"p2e \<in> measurable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. (lborel :: real measure)) |
|
1070 |
(borel :: 'a::ordered_euclidean_space measure)" |
|
1071 |
(is "p2e \<in> measurable ?P _") |
|
1072 |
proof (safe intro!: borel_measurable_iff_halfspace_le[THEN iffD2]) |
|
1073 |
fix x i |
|
1074 |
let ?A = "{w \<in> space ?P. (p2e w :: 'a) $$ i \<le> x}" |
|
1075 |
assume "i < DIM('a)" |
|
1076 |
then have "?A = (\<Pi>\<^isub>E j\<in>{..<DIM('a)}. if i = j then {.. x} else UNIV)" |
|
1077 |
using DIM_positive by (auto simp: space_PiM p2e_def split: split_if_asm) |
|
1078 |
then show "?A \<in> sets ?P" |
|
1079 |
by auto |
|
1080 |
qed |
|
1081 |
||
1082 |
lemma lborel_eq_lborel_space: |
|
1083 |
"(lborel :: 'a measure) = distr (\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel) borel p2e" |
|
1084 |
(is "?B = ?D") |
|
1085 |
proof (rule lborel_eqI) |
|
1086 |
show "sets ?D = sets borel" by simp |
|
1087 |
let ?P = "(\<Pi>\<^isub>M i\<in>{..<DIM('a::ordered_euclidean_space)}. lborel)" |
|
1088 |
fix a b :: 'a |
|
1089 |
have *: "p2e -` {a .. b} \<inter> space ?P = (\<Pi>\<^isub>E i\<in>{..<DIM('a)}. {a $$ i .. b $$ i})" |
|
1090 |
by (auto simp: Pi_iff eucl_le[where 'a='a] p2e_def space_PiM) |
|
1091 |
have "emeasure ?P (p2e -` {a..b} \<inter> space ?P) = content {a..b}" |
|
1092 |
proof cases |
|
1093 |
assume "{a..b} \<noteq> {}" |
|
1094 |
then have "a \<le> b" |
|
1095 |
by (simp add: interval_ne_empty eucl_le[where 'a='a]) |
|
1096 |
then have "emeasure lborel {a..b} = (\<Prod>x<DIM('a). emeasure lborel {a $$ x .. b $$ x})" |
|
1097 |
by (auto simp: content_closed_interval eucl_le[where 'a='a] |
|
1098 |
intro!: setprod_ereal[symmetric]) |
|
1099 |
also have "\<dots> = emeasure ?P (p2e -` {a..b} \<inter> space ?P)" |
|
1100 |
unfolding * by (subst lborel_space.measure_times) auto |
|
1101 |
finally show ?thesis by simp |
|
1102 |
qed simp |
|
1103 |
then show "emeasure ?D {a .. b} = content {a .. b}" |
|
1104 |
by (simp add: emeasure_distr measurable_p2e) |
|
1105 |
qed |
|
1106 |
||
1107 |
lemma borel_fubini_positiv_integral: |
|
1108 |
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> ereal" |
|
1109 |
assumes f: "f \<in> borel_measurable borel" |
|
1110 |
shows "integral\<^isup>P lborel f = \<integral>\<^isup>+x. f (p2e x) \<partial>(\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel)" |
|
1111 |
by (subst lborel_eq_lborel_space) (simp add: positive_integral_distr measurable_p2e f) |
|
1112 |
||
1113 |
lemma borel_fubini_integrable: |
|
1114 |
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" |
|
1115 |
shows "integrable lborel f \<longleftrightarrow> |
|
1116 |
integrable (\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel) (\<lambda>x. f (p2e x))" |
|
1117 |
(is "_ \<longleftrightarrow> integrable ?B ?f") |
|
1118 |
proof |
|
1119 |
assume "integrable lborel f" |
|
1120 |
moreover then have f: "f \<in> borel_measurable borel" |
|
1121 |
by auto |
|
1122 |
moreover with measurable_p2e |
|
1123 |
have "f \<circ> p2e \<in> borel_measurable ?B" |
|
1124 |
by (rule measurable_comp) |
|
1125 |
ultimately show "integrable ?B ?f" |
|
1126 |
by (simp add: comp_def borel_fubini_positiv_integral integrable_def) |
|
1127 |
next |
|
1128 |
assume "integrable ?B ?f" |
|
1129 |
moreover |
|
1130 |
then have "?f \<circ> e2p \<in> borel_measurable (borel::'a measure)" |
|
1131 |
by (auto intro!: measurable_e2p) |
|
1132 |
then have "f \<in> borel_measurable borel" |
|
1133 |
by (simp cong: measurable_cong) |
|
1134 |
ultimately show "integrable lborel f" |
|
1135 |
by (simp add: borel_fubini_positiv_integral integrable_def) |
|
1136 |
qed |
|
1137 |
||
1138 |
lemma borel_fubini: |
|
1139 |
fixes f :: "'a::ordered_euclidean_space \<Rightarrow> real" |
|
1140 |
assumes f: "f \<in> borel_measurable borel" |
|
1141 |
shows "integral\<^isup>L lborel f = \<integral>x. f (p2e x) \<partial>((\<Pi>\<^isub>M i\<in>{..<DIM('a)}. lborel))" |
|
1142 |
using f by (simp add: borel_fubini_positiv_integral lebesgue_integral_def) |
|
47757
5e6fe71e2390
equate positive Lebesgue integral and MV-Analysis' Gauge integral
hoelzl
parents:
47694
diff
changeset
|
1143 |
|
38656 | 1144 |
end |