author  wenzelm 
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permissions  rwrr 
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(* Title: HOL/Probability/Radon_Nikodym.thy 
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Author: Johannes Hölzl, TU München 

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*) 

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header {*RadonNikod{\'y}m derivative*} 

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38656  7 
theory Radon_Nikodym 
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imports Lebesgue_Integration 

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begin 

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47694  11 
definition "diff_measure M N = 
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measure_of (space M) (sets M) (\<lambda>A. emeasure M A  emeasure N A)" 

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lemma 

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shows space_diff_measure[simp]: "space (diff_measure M N) = space M" 

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and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M" 

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by (auto simp: diff_measure_def) 

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lemma emeasure_diff_measure: 

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assumes fin: "finite_measure M" "finite_measure N" and sets_eq: "sets M = sets N" 

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assumes pos: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure N A \<le> emeasure M A" and A: "A \<in> sets M" 

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shows "emeasure (diff_measure M N) A = emeasure M A  emeasure N A" (is "_ = ?\<mu> A") 

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unfolding diff_measure_def 

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proof (rule emeasure_measure_of_sigma) 

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show "sigma_algebra (space M) (sets M)" .. 

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show "positive (sets M) ?\<mu>" 

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using pos by (simp add: positive_def ereal_diff_positive) 

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show "countably_additive (sets M) ?\<mu>" 

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proof (rule countably_additiveI) 

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fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> sets M" and "disjoint_family A" 

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then have suminf: 

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"(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)" 

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"(\<Sum>i. emeasure N (A i)) = emeasure N (\<Union>i. A i)" 

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by (simp_all add: suminf_emeasure sets_eq) 

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with A have "(\<Sum>i. emeasure M (A i)  emeasure N (A i)) = 

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(\<Sum>i. emeasure M (A i))  (\<Sum>i. emeasure N (A i))" 

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using fin 

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by (intro suminf_ereal_minus pos emeasure_nonneg) 

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(auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure) 

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then show "(\<Sum>i. emeasure M (A i)  emeasure N (A i)) = 

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emeasure M (\<Union>i. A i)  emeasure N (\<Union>i. A i) " 

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by (simp add: suminf) 

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qed 

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qed fact 

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38656  46 
lemma (in sigma_finite_measure) Ex_finite_integrable_function: 
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shows "\<exists>h\<in>borel_measurable M. integral\<^isup>P M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>) \<and> (\<forall>x. 0 \<le> h x)" 
38656  48 
proof  
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obtain A :: "nat \<Rightarrow> 'a set" where 

50003  50 
range[measurable]: "range A \<subseteq> sets M" and 
38656  51 
space: "(\<Union>i. A i) = space M" and 
47694  52 
measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and 
38656  53 
disjoint: "disjoint_family A" 
47694  54 
using sigma_finite_disjoint by auto 
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let ?B = "\<lambda>i. 2^Suc i * emeasure M (A i)" 

38656  56 
have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)" 
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proof 

47694  58 
fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)" 
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using measure[of i] emeasure_nonneg[of M "A i"] 

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by (auto intro!: dense simp: ereal_0_gt_inverse ereal_zero_le_0_iff) 
38656  61 
qed 
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from choice[OF this] obtain n where n: "\<And>i. 0 < n i" 

47694  63 
"\<And>i. n i < inverse (2^Suc i * emeasure M (A i))" by auto 
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{ fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this 
46731  65 
let ?h = "\<lambda>x. \<Sum>i. n i * indicator (A i) x" 
38656  66 
show ?thesis 
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proof (safe intro!: bexI[of _ ?h] del: notI) 

39092  68 
have "\<And>i. A i \<in> sets M" 
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using range by fastforce+ 
47694  70 
then have "integral\<^isup>P M ?h = (\<Sum>i. n i * emeasure M (A i))" using pos 
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by (simp add: positive_integral_suminf positive_integral_cmult_indicator) 
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also have "\<dots> \<le> (\<Sum>i. (1 / 2)^Suc i)" 
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proof (rule suminf_le_pos) 
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fix N 
47694  75 
have "n N * emeasure M (A N) \<le> inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)" 
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using n[of N] 

43920  77 
by (intro ereal_mult_right_mono) auto 
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also have "\<dots> \<le> (1 / 2) ^ Suc N" 
38656  79 
using measure[of N] n[of N] 
47694  80 
by (cases rule: ereal2_cases[of "n N" "emeasure M (A N)"]) 
43920  81 
(simp_all add: inverse_eq_divide power_divide one_ereal_def ereal_power_divide) 
47694  82 
finally show "n N * emeasure M (A N) \<le> (1 / 2) ^ Suc N" . 
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show "0 \<le> n N * emeasure M (A N)" using n[of N] `A N \<in> sets M` by (simp add: emeasure_nonneg) 

38656  84 
qed 
43920  85 
finally show "integral\<^isup>P M ?h \<noteq> \<infinity>" unfolding suminf_half_series_ereal by auto 
38656  86 
next 
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{ fix x assume "x \<in> space M" 
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then obtain i where "x \<in> A i" using space[symmetric] by auto 
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with disjoint n have "?h x = n i" 
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by (auto intro!: suminf_cmult_indicator intro: less_imp_le) 
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then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by auto } 
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note pos = this 
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fix x show "0 \<le> ?h x" 
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proof cases 
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assume "x \<in> space M" then show "0 \<le> ?h x" using pos by (auto intro: less_imp_le) 
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next 
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assume "x \<notin> space M" then have "\<And>i. x \<notin> A i" using space by auto 
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then show "0 \<le> ?h x" by auto 
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qed 
50003  100 
qed measurable 
38656  101 
qed 
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40871  103 
subsection "Absolutely continuous" 
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47694  105 
definition absolutely_continuous :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where 
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"absolutely_continuous M N \<longleftrightarrow> null_sets M \<subseteq> null_sets N" 

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lemma absolutely_continuousI_count_space: "absolutely_continuous (count_space A) M" 

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unfolding absolutely_continuous_def by (auto simp: null_sets_count_space) 

38656  110 

47694  111 
lemma absolutely_continuousI_density: 
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"f \<in> borel_measurable M \<Longrightarrow> absolutely_continuous M (density M f)" 

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by (force simp add: absolutely_continuous_def null_sets_density_iff dest: AE_not_in) 

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lemma absolutely_continuousI_point_measure_finite: 

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"(\<And>x. \<lbrakk> x \<in> A ; f x \<le> 0 \<rbrakk> \<Longrightarrow> g x \<le> 0) \<Longrightarrow> absolutely_continuous (point_measure A f) (point_measure A g)" 

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unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff) 

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lemma absolutely_continuous_AE: 

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assumes sets_eq: "sets M' = sets M" 

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and "absolutely_continuous M M'" "AE x in M. P x" 

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shows "AE x in M'. P x" 
40859  123 
proof  
47694  124 
from `AE x in M. P x` obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N" 
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unfolding eventually_ae_filter by auto 

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show "AE x in M'. P x" 
47694  127 
proof (rule AE_I') 
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show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp 

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from `absolutely_continuous M M'` show "N \<in> null_sets M'" 

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using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto 

40859  131 
qed 
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qed 

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40871  134 
subsection "Existence of the RadonNikodym derivative" 
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38656  136 
lemma (in finite_measure) Radon_Nikodym_aux_epsilon: 
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fixes e :: real assumes "0 < e" 

47694  138 
assumes "finite_measure N" and sets_eq: "sets N = sets M" 
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shows "\<exists>A\<in>sets M. measure M (space M)  measure N (space M) \<le> measure M A  measure N A \<and> 

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(\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow>  e < measure M B  measure N B)" 

38656  141 
proof  
47694  142 
interpret M': finite_measure N by fact 
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let ?d = "\<lambda>A. measure M A  measure N A" 

46731  144 
let ?A = "\<lambda>A. if (\<forall>B\<in>sets M. B \<subseteq> space M  A \<longrightarrow> e < ?d B) 
38656  145 
then {} 
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else (SOME B. B \<in> sets M \<and> B \<subseteq> space M  A \<and> ?d B \<le> e)" 

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def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}" 

148 
have A_simps[simp]: 

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"A 0 = {}" 

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"\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all 

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{ fix A assume "A \<in> sets M" 

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have "?A A \<in> sets M" 

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by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) } 

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note A'_in_sets = this 

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{ fix n have "A n \<in> sets M" 

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proof (induct n) 

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case (Suc n) thus "A (Suc n) \<in> sets M" 

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using A'_in_sets[of "A n"] by (auto split: split_if_asm) 

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qed (simp add: A_def) } 

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note A_in_sets = this 

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hence "range A \<subseteq> sets M" by auto 

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{ fix n B 

163 
assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M  A n \<and> ?d B \<le> e" 

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hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M  A n \<longrightarrow> e < ?d B)" by (auto simp: not_less) 

165 
have "?d (A (Suc n)) \<le> ?d (A n)  e" unfolding A_simps if_not_P[OF False] 

166 
proof (rule someI2_ex[OF Ex]) 

167 
fix B assume "B \<in> sets M \<and> B \<subseteq> space M  A n \<and> ?d B \<le>  e" 

168 
hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> e" by auto 

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hence "?d (A n \<union> B) = ?d (A n) + ?d B" 

47694  170 
using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by (simp add: sets_eq) 
38656  171 
also have "\<dots> \<le> ?d (A n)  e" using dB by simp 
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finally show "?d (A n \<union> B) \<le> ?d (A n)  e" . 

173 
qed } 

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note dA_epsilon = this 

175 
{ fix n have "?d (A (Suc n)) \<le> ?d (A n)" 

176 
proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M  A n \<and> ?d B \<le>  e") 

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case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp 

178 
next 

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case False 

180 
hence "\<forall>B\<in>sets M. B \<subseteq> space M  A n \<longrightarrow> e < ?d B" by (auto simp: not_le) 

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thus ?thesis by simp 

182 
qed } 

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note dA_mono = this 

184 
show ?thesis 

185 
proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M  A n \<longrightarrow> e < ?d B") 

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case True then obtain n where B: "\<And>B. \<lbrakk> B \<in> sets M; B \<subseteq> space M  A n\<rbrakk> \<Longrightarrow> e < ?d B" by blast 

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show ?thesis 

188 
proof (safe intro!: bexI[of _ "space M  A n"]) 

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fix B assume "B \<in> sets M" "B \<subseteq> space M  A n" 

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from B[OF this] show "e < ?d B" . 

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next 

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show "space M  A n \<in> sets M" by (rule sets.compl_sets) fact 
38656  193 
next 
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show "?d (space M) \<le> ?d (space M  A n)" 

195 
proof (induct n) 

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fix n assume "?d (space M) \<le> ?d (space M  A n)" 

197 
also have "\<dots> \<le> ?d (space M  A (Suc n))" 

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using A_in_sets sets.sets_into_space dA_mono[of n] finite_measure_compl M'.finite_measure_compl 
47694  199 
by (simp del: A_simps add: sets_eq sets_eq_imp_space_eq[OF sets_eq]) 
38656  200 
finally show "?d (space M) \<le> ?d (space M  A (Suc n))" . 
201 
qed simp 

202 
qed 

203 
next 

204 
case False hence B: "\<And>n. \<exists>B. B\<in>sets M \<and> B \<subseteq> space M  A n \<and> ?d B \<le>  e" 

205 
by (auto simp add: not_less) 

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{ fix n have "?d (A n) \<le>  real n * e" 

207 
proof (induct n) 

208 
case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps) 

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next 
47694  210 
case 0 with measure_empty show ?case by (simp add: zero_ereal_def) 
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qed } note dA_less = this 
38656  212 
have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq 
213 
proof (rule incseq_SucI) 

214 
fix n show " ?d (A n) \<le>  ?d (A (Suc n))" using dA_mono[of n] by auto 

215 
qed 

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have A: "incseq A" by (auto intro!: incseq_SucI) 
47694  217 
from finite_Lim_measure_incseq[OF _ A] `range A \<subseteq> sets M` 
218 
M'.finite_Lim_measure_incseq[OF _ A] 

38656  219 
have convergent: "(\<lambda>i. ?d (A i)) > ?d (\<Union>i. A i)" 
47694  220 
by (auto intro!: tendsto_diff simp: sets_eq) 
38656  221 
obtain n :: nat where " ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto 
222 
moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less] 

223 
have "real n \<le>  ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps) 

224 
ultimately show ?thesis by auto 

225 
qed 

226 
qed 

227 

47694  228 
lemma (in finite_measure) Radon_Nikodym_aux: 
229 
assumes "finite_measure N" and sets_eq: "sets N = sets M" 

230 
shows "\<exists>A\<in>sets M. measure M (space M)  measure N (space M) \<le> 

231 
measure M A  measure N A \<and> 

232 
(\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> measure M B  measure N B)" 

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proof  
47694  234 
interpret N: finite_measure N by fact 
235 
let ?d = "\<lambda>A. measure M A  measure N A" 

46731  236 
let ?P = "\<lambda>A B n. A \<in> sets M \<and> A \<subseteq> B \<and> ?d B \<le> ?d A \<and> (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow>  1 / real (Suc n) < ?d C)" 
237 
let ?r = "\<lambda>S. restricted_space S" 

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{ fix S n assume S: "S \<in> sets M" 
47694  239 
then have "finite_measure (density M (indicator S))" "0 < 1 / real (Suc n)" 
240 
"finite_measure (density N (indicator S))" "sets (density N (indicator S)) = sets (density M (indicator S))" 

241 
by (auto simp: finite_measure_restricted N.finite_measure_restricted sets_eq) 

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from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. note X = this 
47694  243 
with S have "?P (S \<inter> X) S n" 
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by (simp add: measure_restricted sets_eq sets.Int) (metis inf_absorb2) 
47694  245 
hence "\<exists>A. ?P A S n" .. } 
38656  246 
note Ex_P = this 
247 
def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)" 

248 
have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def) 

249 
have A_0[simp]: "A 0 = space M" unfolding A_def by simp 

250 
{ fix i have "A i \<in> sets M" unfolding A_def 

251 
proof (induct i) 

252 
case (Suc i) 

253 
from Ex_P[OF this, of i] show ?case unfolding nat_rec_Suc 

254 
by (rule someI2_ex) simp 

255 
qed simp } 

256 
note A_in_sets = this 

257 
{ fix n have "?P (A (Suc n)) (A n) n" 

258 
using Ex_P[OF A_in_sets] unfolding A_Suc 

259 
by (rule someI2_ex) simp } 

260 
note P_A = this 

261 
have "range A \<subseteq> sets M" using A_in_sets by auto 

262 
have A_mono: "\<And>i. A (Suc i) \<subseteq> A i" using P_A by simp 

263 
have mono_dA: "mono (\<lambda>i. ?d (A i))" using P_A by (simp add: mono_iff_le_Suc) 

264 
have epsilon: "\<And>C i. \<lbrakk> C \<in> sets M; C \<subseteq> A (Suc i) \<rbrakk> \<Longrightarrow>  1 / real (Suc i) < ?d C" 

265 
using P_A by auto 

266 
show ?thesis 

267 
proof (safe intro!: bexI[of _ "\<Inter>i. A i"]) 

268 
show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto 

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have A: "decseq A" using A_mono by (auto intro!: decseq_SucI) 
47694  270 
from `range A \<subseteq> sets M` 
271 
finite_Lim_measure_decseq[OF _ A] N.finite_Lim_measure_decseq[OF _ A] 

272 
have "(\<lambda>i. ?d (A i)) > ?d (\<Inter>i. A i)" by (auto intro!: tendsto_diff simp: sets_eq) 

38656  273 
thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _] 
47694  274 
by (rule_tac LIMSEQ_le_const) auto 
38656  275 
next 
276 
fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)" 

277 
show "0 \<le> ?d B" 

278 
proof (rule ccontr) 

279 
assume "\<not> 0 \<le> ?d B" 

280 
hence "0 <  ?d B" by auto 

281 
from ex_inverse_of_nat_Suc_less[OF this] 

282 
obtain n where *: "?d B <  1 / real (Suc n)" 

283 
by (auto simp: real_eq_of_nat inverse_eq_divide field_simps) 

284 
have "B \<subseteq> A (Suc n)" using B by (auto simp del: nat_rec_Suc) 

285 
from epsilon[OF B(1) this] * 

286 
show False by auto 

287 
qed 

288 
qed 

289 
qed 

290 

291 
lemma (in finite_measure) Radon_Nikodym_finite_measure: 

47694  292 
assumes "finite_measure N" and sets_eq: "sets N = sets M" 
293 
assumes "absolutely_continuous M N" 

294 
shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N" 

38656  295 
proof  
47694  296 
interpret N: finite_measure N by fact 
297 
def G \<equiv> "{g \<in> borel_measurable M. (\<forall>x. 0 \<le> g x) \<and> (\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> N A)}" 

50003  298 
{ fix f have "f \<in> G \<Longrightarrow> f \<in> borel_measurable M" by (auto simp: G_def) } 
299 
note this[measurable_dest] 

38656  300 
have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto 
301 
hence "G \<noteq> {}" by auto 

302 
{ fix f g assume f: "f \<in> G" and g: "g \<in> G" 

303 
have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def 

304 
proof safe 

305 
show "?max \<in> borel_measurable M" using f g unfolding G_def by auto 

306 
let ?A = "{x \<in> space M. f x \<le> g x}" 

307 
have "?A \<in> sets M" using f g unfolding G_def by auto 

308 
fix A assume "A \<in> sets M" 

309 
hence sets: "?A \<inter> A \<in> sets M" "(space M  ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto 

47694  310 
hence sets': "?A \<inter> A \<in> sets N" "(space M  ?A) \<inter> A \<in> sets N" by (auto simp: sets_eq) 
38656  311 
have union: "((?A \<inter> A) \<union> ((space M  ?A) \<inter> A)) = A" 
50244
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312 
using sets.sets_into_space[OF `A \<in> sets M`] by auto 
38656  313 
have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x = 
314 
g x * indicator (?A \<inter> A) x + f x * indicator ((space M  ?A) \<inter> A) x" 

315 
by (auto simp: indicator_def max_def) 

41689
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changeset

316 
hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) = 
3e39b0e730d6
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hoelzl
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changeset

317 
(\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x \<partial>M) + 
3e39b0e730d6
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hoelzl
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318 
(\<integral>\<^isup>+x. f x * indicator ((space M  ?A) \<inter> A) x \<partial>M)" 
38656  319 
using f g sets unfolding G_def 
46731  320 
by (auto cong: positive_integral_cong intro!: positive_integral_add) 
47694  321 
also have "\<dots> \<le> N (?A \<inter> A) + N ((space M  ?A) \<inter> A)" 
38656  322 
using f g sets unfolding G_def by (auto intro!: add_mono) 
47694  323 
also have "\<dots> = N A" 
324 
using plus_emeasure[OF sets'] union by auto 

325 
finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> N A" . 

41981
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changeset

326 
next 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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changeset

327 
fix x show "0 \<le> max (g x) (f x)" using f g by (auto simp: G_def split: split_max) 
38656  328 
qed } 
329 
note max_in_G = this 

41981
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changeset

330 
{ fix f assume "incseq f" and f: "\<And>i. f i \<in> G" 
50003  331 
then have [measurable]: "\<And>i. f i \<in> borel_measurable M" by (auto simp: G_def) 
41981
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hoelzl
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diff
changeset

332 
have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def 
38656  333 
proof safe 
50003  334 
show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M" by measurable 
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

335 
{ fix x show "0 \<le> (SUP i. f i x)" 
44928
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hoelzl
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44918
diff
changeset

336 
using f by (auto simp: G_def intro: SUP_upper2) } 
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

337 
next 
38656  338 
fix A assume "A \<in> sets M" 
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

339 
have "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) = 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

340 
(\<integral>\<^isup>+x. (SUP i. f i x * indicator A x) \<partial>M)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff
changeset

341 
by (intro positive_integral_cong) (simp split: split_indicator) 
cdf7693bbe08
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diff
changeset

342 
also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x \<partial>M))" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff
changeset

343 
using `incseq f` f `A \<in> sets M` 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff
changeset

344 
by (intro positive_integral_monotone_convergence_SUP) 
cdf7693bbe08
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hoelzl
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diff
changeset

345 
(auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator) 
47694  346 
finally show "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A" 
44928
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hoelzl
parents:
44918
diff
changeset

347 
using f `A \<in> sets M` by (auto intro!: SUP_least simp: G_def) 
38656  348 
qed } 
349 
note SUP_in_G = this 

41689
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hoelzl
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41661
diff
changeset

350 
let ?y = "SUP g : G. integral\<^isup>P M g" 
47694  351 
have y_le: "?y \<le> N (space M)" unfolding G_def 
44928
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hoelzl
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44918
diff
changeset

352 
proof (safe intro!: SUP_least) 
47694  353 
fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> N A" 
50244
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immler
parents:
50021
diff
changeset

354 
from this[THEN bspec, OF sets.top] show "integral\<^isup>P M g \<le> N (space M)" 
38656  355 
by (simp cong: positive_integral_cong) 
356 
qed 

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

357 
from SUPR_countable_SUPR[OF `G \<noteq> {}`, of "integral\<^isup>P M"] guess ys .. note ys = this 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset

358 
then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n" 
38656  359 
proof safe 
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

360 
fix n assume "range ys \<subseteq> integral\<^isup>P M ` G" 
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

361 
hence "ys n \<in> integral\<^isup>P M ` G" by auto 
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

362 
thus "\<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n" by auto 
38656  363 
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

364 
from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^isup>P M (gs n) = ys n" by auto 
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

365 
hence y_eq: "?y = (SUP i. integral\<^isup>P M (gs i))" using ys by auto 
46731  366 
let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})" 
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

367 
def f \<equiv> "\<lambda>x. SUP i. ?g i x" 
46731  368 
let ?F = "\<lambda>A x. f x * indicator A x" 
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

369 
have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto 
38656  370 
{ fix i have "?g i \<in> G" 
371 
proof (induct i) 

372 
case 0 thus ?case by simp fact 

373 
next 

374 
case (Suc i) 

375 
with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case 

376 
by (auto simp add: atMost_Suc intro!: max_in_G) 

377 
qed } 

378 
note g_in_G = this 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

379 
have "incseq ?g" using gs_not_empty 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

380 
by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc) 
50003  381 
from SUP_in_G[OF this g_in_G] have [measurable]: "f \<in> G" unfolding f_def . 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

382 
then have [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

383 
have "integral\<^isup>P M f = (SUP i. integral\<^isup>P M (?g i))" unfolding f_def 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

384 
using g_in_G `incseq ?g` 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

385 
by (auto intro!: positive_integral_monotone_convergence_SUP simp: G_def) 
38656  386 
also have "\<dots> = ?y" 
387 
proof (rule antisym) 

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

388 
show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y" 
47694  389 
using g_in_G by (auto intro: Sup_mono simp: SUP_def) 
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

390 
show "?y \<le> (SUP i. integral\<^isup>P M (?g i))" unfolding y_eq 
38656  391 
by (auto intro!: SUP_mono positive_integral_mono Max_ge) 
392 
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

393 
finally have int_f_eq_y: "integral\<^isup>P M f = ?y" . 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

394 
have "\<And>x. 0 \<le> f x" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

395 
unfolding f_def using `\<And>i. gs i \<in> G` 
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44918
diff
changeset

396 
by (auto intro!: SUP_upper2 Max_ge_iff[THEN iffD2] simp: G_def) 
47694  397 
let ?t = "\<lambda>A. N A  (\<integral>\<^isup>+x. ?F A x \<partial>M)" 
398 
let ?M = "diff_measure N (density M f)" 

399 
have f_le_N: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> N A" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

400 
using `f \<in> G` unfolding G_def by auto 
47694  401 
have emeasure_M: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure ?M A = ?t A" 
402 
proof (subst emeasure_diff_measure) 

403 
from f_le_N[of "space M"] show "finite_measure N" "finite_measure (density M f)" 

404 
by (auto intro!: finite_measureI simp: emeasure_density cong: positive_integral_cong) 

405 
next 

406 
fix B assume "B \<in> sets N" with f_le_N[of B] show "emeasure (density M f) B \<le> emeasure N B" 

407 
by (auto simp: sets_eq emeasure_density cong: positive_integral_cong) 

408 
qed (auto simp: sets_eq emeasure_density) 

409 
from emeasure_M[of "space M"] N.finite_emeasure_space positive_integral_positive[of M "?F (space M)"] 

410 
interpret M': finite_measure ?M 

411 
by (auto intro!: finite_measureI simp: sets_eq_imp_space_eq[OF sets_eq] N.emeasure_eq_measure ereal_minus_eq_PInfty_iff) 

412 

413 
have ac: "absolutely_continuous M ?M" unfolding absolutely_continuous_def 

45777
c36637603821
remove unnecessary sublocale instantiations in HOLProbability (for clarity and speedup); remove Infinite_Product_Measure.product_prob_space which was a duplicate of Probability_Measure.product_prob_space
hoelzl
parents:
45769
diff
changeset

414 
proof 
47694  415 
fix A assume A: "A \<in> null_sets M" 
416 
with `absolutely_continuous M N` have "A \<in> null_sets N" 

417 
unfolding absolutely_continuous_def by auto 

418 
moreover with A have "(\<integral>\<^isup>+ x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` by (auto simp: G_def) 

419 
ultimately have "N A  (\<integral>\<^isup>+ x. ?F A x \<partial>M) = 0" 

420 
using positive_integral_positive[of M] by (auto intro!: antisym) 

421 
then show "A \<in> null_sets ?M" 

422 
using A by (simp add: emeasure_M null_sets_def sets_eq) 

38656  423 
qed 
47694  424 
have upper_bound: "\<forall>A\<in>sets M. ?M A \<le> 0" 
38656  425 
proof (rule ccontr) 
426 
assume "\<not> ?thesis" 

47694  427 
then obtain A where A: "A \<in> sets M" and pos: "0 < ?M A" 
38656  428 
by (auto simp: not_le) 
429 
note pos 

47694  430 
also have "?M A \<le> ?M (space M)" 
431 
using emeasure_space[of ?M A] by (simp add: sets_eq[THEN sets_eq_imp_space_eq]) 

432 
finally have pos_t: "0 < ?M (space M)" by simp 

38656  433 
moreover 
47694  434 
then have "emeasure M (space M) \<noteq> 0" 
435 
using ac unfolding absolutely_continuous_def by (auto simp: null_sets_def) 

436 
then have pos_M: "0 < emeasure M (space M)" 

437 
using emeasure_nonneg[of M "space M"] by (simp add: le_less) 

38656  438 
moreover 
47694  439 
have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> N (space M)" 
38656  440 
using `f \<in> G` unfolding G_def by auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

441 
hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>" 
47694  442 
using M'.finite_emeasure_space by auto 
38656  443 
moreover 
47694  444 
def b \<equiv> "?M (space M) / emeasure M (space M) / 2" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

445 
ultimately have b: "b \<noteq> 0 \<and> 0 \<le> b \<and> b \<noteq> \<infinity>" 
47694  446 
by (auto simp: ereal_divide_eq) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

447 
then have b: "b \<noteq> 0" "0 \<le> b" "0 < b" "b \<noteq> \<infinity>" by auto 
47694  448 
let ?Mb = "density M (\<lambda>_. b)" 
449 
have Mb: "finite_measure ?Mb" "sets ?Mb = sets ?M" 

450 
using b by (auto simp: emeasure_density_const sets_eq intro!: finite_measureI) 

451 
from M'.Radon_Nikodym_aux[OF this] guess A0 .. 

452 
then have "A0 \<in> sets M" 

453 
and space_less_A0: "measure ?M (space M)  real b * measure M (space M) \<le> measure ?M A0  real b * measure M A0" 

454 
and *: "\<And>B. B \<in> sets M \<Longrightarrow> B \<subseteq> A0 \<Longrightarrow> 0 \<le> measure ?M B  real b * measure M B" 

455 
using b by (simp_all add: measure_density_const sets_eq_imp_space_eq[OF sets_eq] sets_eq) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

456 
{ fix B assume B: "B \<in> sets M" "B \<subseteq> A0" 
47694  457 
with *[OF this] have "b * emeasure M B \<le> ?M B" 
458 
using b unfolding M'.emeasure_eq_measure emeasure_eq_measure by (cases b) auto } 

38656  459 
note bM_le_t = this 
46731  460 
let ?f0 = "\<lambda>x. f x + b * indicator A0 x" 
38656  461 
{ fix A assume A: "A \<in> sets M" 
462 
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto 

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

463 
have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) = 
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

464 
(\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

465 
by (auto intro!: positive_integral_cong split: split_indicator) 
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

466 
hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) = 
47694  467 
(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * emeasure M (A \<inter> A0)" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

468 
using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A b `f \<in> G` 
50003  469 
by (simp add: positive_integral_add positive_integral_cmult_indicator G_def) } 
38656  470 
note f0_eq = this 
471 
{ fix A assume A: "A \<in> sets M" 

472 
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto 

47694  473 
have f_le_v: "(\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> N A" using `f \<in> G` A unfolding G_def by auto 
38656  474 
note f0_eq[OF A] 
47694  475 
also have "(\<integral>\<^isup>+x. ?F A x \<partial>M) + b * emeasure M (A \<inter> A0) \<le> (\<integral>\<^isup>+x. ?F A x \<partial>M) + ?M (A \<inter> A0)" 
38656  476 
using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M` 
477 
by (auto intro!: add_left_mono) 

47694  478 
also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?M A" 
479 
using emeasure_mono[of "A \<inter> A0" A ?M] `A \<in> sets M` `A0 \<in> sets M` 

480 
by (auto intro!: add_left_mono simp: sets_eq) 

481 
also have "\<dots> \<le> N A" 

482 
unfolding emeasure_M[OF `A \<in> sets M`] 

483 
using f_le_v N.emeasure_eq_measure[of A] positive_integral_positive[of M "?F A"] 

484 
by (cases "\<integral>\<^isup>+x. ?F A x \<partial>M", cases "N A") auto 

485 
finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> N A" . } 

50003  486 
hence "?f0 \<in> G" using `A0 \<in> sets M` b `f \<in> G` 
487 
by (auto intro!: ereal_add_nonneg_nonneg simp: G_def) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

488 
have int_f_finite: "integral\<^isup>P M f \<noteq> \<infinity>" 
47694  489 
by (metis N.emeasure_finite ereal_infty_less_eq2(1) int_f_eq_y y_le) 
490 
have "0 < ?M (space M)  emeasure ?Mb (space M)" 

491 
using pos_t 

492 
by (simp add: b emeasure_density_const) 

493 
(simp add: M'.emeasure_eq_measure emeasure_eq_measure pos_M b_def) 

494 
also have "\<dots> \<le> ?M A0  b * emeasure M A0" 

495 
using space_less_A0 `A0 \<in> sets M` b 

496 
by (cases b) (auto simp add: b emeasure_density_const sets_eq M'.emeasure_eq_measure emeasure_eq_measure) 

497 
finally have 1: "b * emeasure M A0 < ?M A0" 

498 
by (metis M'.emeasure_real `A0 \<in> sets M` bM_le_t diff_self ereal_less(1) ereal_minus(1) 

499 
less_eq_ereal_def mult_zero_left not_square_less_zero subset_refl zero_ereal_def) 

500 
with b have "0 < ?M A0" 

501 
by (metis M'.emeasure_real MInfty_neq_PInfty(1) emeasure_real ereal_less_eq(5) ereal_zero_times 

502 
ereal_mult_eq_MInfty ereal_mult_eq_PInfty ereal_zero_less_0_iff less_eq_ereal_def) 

503 
then have "emeasure M A0 \<noteq> 0" using ac `A0 \<in> sets M` 

504 
by (auto simp: absolutely_continuous_def null_sets_def) 

505 
then have "0 < emeasure M A0" using emeasure_nonneg[of M A0] by auto 

506 
hence "0 < b * emeasure M A0" using b by (auto simp: ereal_zero_less_0_iff) 

507 
with int_f_finite have "?y + 0 < integral\<^isup>P M f + b * emeasure M A0" unfolding int_f_eq_y 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

508 
using `f \<in> G` 
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44918
diff
changeset

509 
by (intro ereal_add_strict_mono) (auto intro!: SUP_upper2 positive_integral_positive) 
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

510 
also have "\<dots> = integral\<^isup>P M ?f0" using f0_eq[OF sets.top] `A0 \<in> sets M` sets.sets_into_space 
38656  511 
by (simp cong: positive_integral_cong) 
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

512 
finally have "?y < integral\<^isup>P M ?f0" by simp 
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44918
diff
changeset

513 
moreover from `?f0 \<in> G` have "integral\<^isup>P M ?f0 \<le> ?y" by (auto intro!: SUP_upper) 
38656  514 
ultimately show False by auto 
515 
qed 

47694  516 
let ?f = "\<lambda>x. max 0 (f x)" 
38656  517 
show ?thesis 
47694  518 
proof (intro bexI[of _ ?f] measure_eqI conjI) 
519 
show "sets (density M ?f) = sets N" 

520 
by (simp add: sets_eq) 

521 
fix A assume A: "A\<in>sets (density M ?f)" 

522 
then show "emeasure (density M ?f) A = emeasure N A" 

523 
using `f \<in> G` A upper_bound[THEN bspec, of A] N.emeasure_eq_measure[of A] 

524 
by (cases "integral\<^isup>P M (?F A)") 

525 
(auto intro!: antisym simp add: emeasure_density G_def emeasure_M density_max_0[symmetric]) 

526 
qed auto 

38656  527 
qed 
528 

40859  529 
lemma (in finite_measure) split_space_into_finite_sets_and_rest: 
47694  530 
assumes ac: "absolutely_continuous M N" and sets_eq: "sets N = sets M" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

531 
shows "\<exists>A0\<in>sets M. \<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> A0 = space M  (\<Union>i. B i) \<and> 
47694  532 
(\<forall>A\<in>sets M. A \<subseteq> A0 \<longrightarrow> (emeasure M A = 0 \<and> N A = 0) \<or> (emeasure M A > 0 \<and> N A = \<infinity>)) \<and> 
533 
(\<forall>i. N (B i) \<noteq> \<infinity>)" 

38656  534 
proof  
47694  535 
let ?Q = "{Q\<in>sets M. N Q \<noteq> \<infinity>}" 
536 
let ?a = "SUP Q:?Q. emeasure M Q" 

537 
have "{} \<in> ?Q" by auto 

38656  538 
then have Q_not_empty: "?Q \<noteq> {}" by blast 
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

539 
have "?a \<le> emeasure M (space M)" using sets.sets_into_space 
47694  540 
by (auto intro!: SUP_least emeasure_mono) 
541 
then have "?a \<noteq> \<infinity>" using finite_emeasure_space 

38656  542 
by auto 
47694  543 
from SUPR_countable_SUPR[OF Q_not_empty, of "emeasure M"] 
544 
obtain Q'' where "range Q'' \<subseteq> emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)" 

38656  545 
by auto 
47694  546 
then have "\<forall>i. \<exists>Q'. Q'' i = emeasure M Q' \<and> Q' \<in> ?Q" by auto 
547 
from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = emeasure M (Q' i)" "\<And>i. Q' i \<in> ?Q" 

38656  548 
by auto 
47694  549 
then have a_Lim: "?a = (SUP i::nat. emeasure M (Q' i))" using a by simp 
46731  550 
let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i" 
47694  551 
have Union: "(SUP i. emeasure M (?O i)) = emeasure M (\<Union>i. ?O i)" 
552 
proof (rule SUP_emeasure_incseq[of ?O]) 

553 
show "range ?O \<subseteq> sets M" using Q' by auto 

44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
44568
diff
changeset

554 
show "incseq ?O" by (fastforce intro!: incseq_SucI) 
38656  555 
qed 
556 
have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto 

47694  557 
have O_sets: "\<And>i. ?O i \<in> sets M" using Q' by auto 
38656  558 
then have O_in_G: "\<And>i. ?O i \<in> ?Q" 
559 
proof (safe del: notI) 

47694  560 
fix i have "Q' ` {..i} \<subseteq> sets M" using Q' by auto 
561 
then have "N (?O i) \<le> (\<Sum>i\<le>i. N (Q' i))" 

562 
by (simp add: sets_eq emeasure_subadditive_finite) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

563 
also have "\<dots> < \<infinity>" using Q' by (simp add: setsum_Pinfty) 
47694  564 
finally show "N (?O i) \<noteq> \<infinity>" by simp 
38656  565 
qed auto 
44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
44568
diff
changeset

566 
have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce 
47694  567 
have a_eq: "?a = emeasure M (\<Union>i. ?O i)" unfolding Union[symmetric] 
38656  568 
proof (rule antisym) 
47694  569 
show "?a \<le> (SUP i. emeasure M (?O i))" unfolding a_Lim 
570 
using Q' by (auto intro!: SUP_mono emeasure_mono) 

571 
show "(SUP i. emeasure M (?O i)) \<le> ?a" unfolding SUP_def 

38656  572 
proof (safe intro!: Sup_mono, unfold bex_simps) 
573 
fix i 

574 
have *: "(\<Union>Q' ` {..i}) = ?O i" by auto 

47694  575 
then show "\<exists>x. (x \<in> sets M \<and> N x \<noteq> \<infinity>) \<and> 
576 
emeasure M (\<Union>Q' ` {..i}) \<le> emeasure M x" 

38656  577 
using O_in_G[of i] by (auto intro!: exI[of _ "?O i"]) 
578 
qed 

579 
qed 

46731  580 
let ?O_0 = "(\<Union>i. ?O i)" 
38656  581 
have "?O_0 \<in> sets M" using Q' by auto 
40859  582 
def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0  Suc n \<Rightarrow> ?O (Suc n)  ?O n" 
38656  583 
{ fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) } 
584 
note Q_sets = this 

40859  585 
show ?thesis 
586 
proof (intro bexI exI conjI ballI impI allI) 

587 
show "disjoint_family Q" 

44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
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parents:
44568
diff
changeset

588 
by (fastforce simp: disjoint_family_on_def Q_def 
40859  589 
split: nat.split_asm) 
590 
show "range Q \<subseteq> sets M" 

591 
using Q_sets by auto 

592 
{ fix A assume A: "A \<in> sets M" "A \<subseteq> space M  ?O_0" 

47694  593 
show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>" 
40859  594 
proof (rule disjCI, simp) 
47694  595 
assume *: "0 < emeasure M A \<longrightarrow> N A \<noteq> \<infinity>" 
596 
show "emeasure M A = 0 \<and> N A = 0" 

40859  597 
proof cases 
47694  598 
assume "emeasure M A = 0" moreover with ac A have "N A = 0" 
40859  599 
unfolding absolutely_continuous_def by auto 
600 
ultimately show ?thesis by simp 

601 
next 

47694  602 
assume "emeasure M A \<noteq> 0" with * have "N A \<noteq> \<infinity>" using emeasure_nonneg[of M A] by auto 
603 
with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

604 
using Q' by (auto intro!: plus_emeasure sets.countable_UN) 
47694  605 
also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))" 
606 
proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified]) 

40859  607 
show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M" 
47694  608 
using `N A \<noteq> \<infinity>` O_sets A by auto 
44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
44568
diff
changeset

609 
qed (fastforce intro!: incseq_SucI) 
40859  610 
also have "\<dots> \<le> ?a" 
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44918
diff
changeset

611 
proof (safe intro!: SUP_least) 
40859  612 
fix i have "?O i \<union> A \<in> ?Q" 
613 
proof (safe del: notI) 

614 
show "?O i \<union> A \<in> sets M" using O_sets A by auto 

47694  615 
from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A" 
616 
using emeasure_subadditive[of "?O i" N A] A O_sets by (auto simp: sets_eq) 

617 
with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>" 

618 
using `N A \<noteq> \<infinity>` by auto 

40859  619 
qed 
47694  620 
then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper) 
40859  621 
qed 
47694  622 
finally have "emeasure M A = 0" 
623 
unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure) 

624 
with `emeasure M A \<noteq> 0` show ?thesis by auto 

40859  625 
qed 
626 
qed } 

47694  627 
{ fix i show "N (Q i) \<noteq> \<infinity>" 
40859  628 
proof (cases i) 
629 
case 0 then show ?thesis 

630 
unfolding Q_def using Q'[of 0] by simp 

631 
next 

632 
case (Suc n) 

47694  633 
with `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q` 
634 
emeasure_Diff[OF _ _ _ O_mono, of N n] emeasure_nonneg[of N "(\<Union> x\<le>n. Q' x)"] 

635 
show ?thesis 

636 
by (auto simp: sets_eq ereal_minus_eq_PInfty_iff Q_def) 

40859  637 
qed } 
638 
show "space M  ?O_0 \<in> sets M" using Q'_sets by auto 

639 
{ fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)" 

640 
proof (induct j) 

641 
case 0 then show ?case by (simp add: Q_def) 

642 
next 

643 
case (Suc j) 

44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
44568
diff
changeset

644 
have eq: "\<And>j. (\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q' i)" by fastforce 
40859  645 
have "{..j} \<union> {..Suc j} = {..Suc j}" by auto 
646 
then have "(\<Union>i\<le>Suc j. Q' i) = (\<Union>i\<le>j. Q' i) \<union> Q (Suc j)" 

647 
by (simp add: UN_Un[symmetric] Q_def del: UN_Un) 

648 
then show ?case using Suc by (auto simp add: eq atMost_Suc) 

649 
qed } 

650 
then have "(\<Union>j. (\<Union>i\<le>j. ?O i)) = (\<Union>j. (\<Union>i\<le>j. Q i))" by simp 

44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
44568
diff
changeset

651 
then show "space M  ?O_0 = space M  (\<Union>i. Q i)" by fastforce 
40859  652 
qed 
653 
qed 

654 

655 
lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite: 

47694  656 
assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M" 
657 
shows "\<exists>f\<in>borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N" 

40859  658 
proof  
659 
from split_space_into_finite_sets_and_rest[OF assms] 

660 
obtain Q0 and Q :: "nat \<Rightarrow> 'a set" 

661 
where Q: "disjoint_family Q" "range Q \<subseteq> sets M" 

662 
and Q0: "Q0 \<in> sets M" "Q0 = space M  (\<Union>i. Q i)" 

47694  663 
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>" 
664 
and Q_fin: "\<And>i. N (Q i) \<noteq> \<infinity>" by force 

40859  665 
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto 
47694  666 
let ?N = "\<lambda>i. density N (indicator (Q i))" and ?M = "\<lambda>i. density M (indicator (Q i))" 
667 
have "\<forall>i. \<exists>f\<in>borel_measurable (?M i). (\<forall>x. 0 \<le> f x) \<and> density (?M i) f = ?N i" 

668 
proof (intro allI finite_measure.Radon_Nikodym_finite_measure) 

38656  669 
fix i 
47694  670 
from Q show "finite_measure (?M i)" 
671 
by (auto intro!: finite_measureI cong: positive_integral_cong 

672 
simp add: emeasure_density subset_eq sets_eq) 

673 
from Q have "emeasure (?N i) (space N) = emeasure N (Q i)" 

674 
by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: positive_integral_cong) 

675 
with Q_fin show "finite_measure (?N i)" 

676 
by (auto intro!: finite_measureI) 

677 
show "sets (?N i) = sets (?M i)" by (simp add: sets_eq) 

50003  678 
have [measurable]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq) 
47694  679 
show "absolutely_continuous (?M i) (?N i)" 
680 
using `absolutely_continuous M N` `Q i \<in> sets M` 

681 
by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq 

682 
intro!: absolutely_continuous_AE[OF sets_eq]) 

38656  683 
qed 
47694  684 
from choice[OF this[unfolded Bex_def]] 
685 
obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x" 

686 
and f_density: "\<And>i. density (?M i) (f i) = ?N i" 

38656  687 
by auto 
47694  688 
{ fix A i assume A: "A \<in> sets M" 
689 
with Q borel have "(\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M) = emeasure (density (?M i) (f i)) A" 

690 
by (auto simp add: emeasure_density positive_integral_density subset_eq 

691 
intro!: positive_integral_cong split: split_indicator) 

692 
also have "\<dots> = emeasure N (Q i \<inter> A)" 

693 
using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq) 

694 
finally have "emeasure N (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)" .. } 

695 
note integral_eq = this 

46731  696 
let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator Q0 x" 
38656  697 
show ?thesis 
698 
proof (safe intro!: bexI[of _ ?f]) 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

699 
show "?f \<in> borel_measurable M" using Q0 borel Q_sets 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

700 
by (auto intro!: measurable_If) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

701 
show "\<And>x. 0 \<le> ?f x" using borel by (auto intro!: suminf_0_le simp: indicator_def) 
47694  702 
show "density M ?f = N" 
703 
proof (rule measure_eqI) 

704 
fix A assume "A \<in> sets (density M ?f)" 

705 
then have "A \<in> sets M" by simp 

706 
have Qi: "\<And>i. Q i \<in> sets M" using Q by auto 

707 
have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M" 

708 
"\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x" 

709 
using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_ereal_times) 

710 
have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator (Q0 \<inter> A) x \<partial>M)" 

711 
using borel by (intro positive_integral_cong) (auto simp: indicator_def) 

712 
also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M (Q0 \<inter> A)" 

713 
using borel Qi Q0(1) `A \<in> sets M` 

714 
by (subst positive_integral_add) (auto simp del: ereal_infty_mult 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

715 
simp add: positive_integral_cmult_indicator sets.Int intro!: suminf_0_le) 
47694  716 
also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)" 
717 
by (subst integral_eq[OF `A \<in> sets M`], subst positive_integral_suminf) auto 

718 
finally have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)" . 

719 
moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)" 

720 
using Q Q_sets `A \<in> sets M` 

721 
by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq) 

722 
moreover have "\<infinity> * emeasure M (Q0 \<inter> A) = N (Q0 \<inter> A)" 

723 
proof  

724 
have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast 

725 
from in_Q0[OF this] show ?thesis by auto 

726 
qed 

727 
moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M" 

728 
using Q_sets `A \<in> sets M` Q0(1) by auto 

729 
moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

730 
using `A \<in> sets M` sets.sets_into_space Q0 by auto 
47694  731 
ultimately have "N A = (\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M)" 
732 
using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "Q0 \<inter> A"] by (simp add: sets_eq) 

733 
with `A \<in> sets M` borel Q Q0(1) show "emeasure (density M ?f) A = N A" 

50003  734 
by (auto simp: subset_eq emeasure_density) 
47694  735 
qed (simp add: sets_eq) 
38656  736 
qed 
737 
qed 

738 

739 
lemma (in sigma_finite_measure) Radon_Nikodym: 

47694  740 
assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M" 
741 
shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N" 

38656  742 
proof  
743 
from Ex_finite_integrable_function 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

744 
obtain h where finite: "integral\<^isup>P M h \<noteq> \<infinity>" and 
38656  745 
borel: "h \<in> borel_measurable M" and 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

746 
nn: "\<And>x. 0 \<le> h x" and 
38656  747 
pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

748 
"\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto 
46731  749 
let ?T = "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M)" 
47694  750 
let ?MT = "density M h" 
751 
from borel finite nn interpret T: finite_measure ?MT 

752 
by (auto intro!: finite_measureI cong: positive_integral_cong simp: emeasure_density) 

753 
have "absolutely_continuous ?MT N" "sets N = sets ?MT" 

754 
proof (unfold absolutely_continuous_def, safe) 

755 
fix A assume "A \<in> null_sets ?MT" 

756 
with borel have "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> h x \<le> 0" 

757 
by (auto simp add: null_sets_density_iff) 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

758 
with pos sets.sets_into_space have "AE x in M. x \<notin> A" 
47694  759 
by (elim eventually_elim1) (auto simp: not_le[symmetric]) 
760 
then have "A \<in> null_sets M" 

761 
using `A \<in> sets M` by (simp add: AE_iff_null_sets) 

762 
with ac show "A \<in> null_sets N" 

763 
by (auto simp: absolutely_continuous_def) 

764 
qed (auto simp add: sets_eq) 

765 
from T.Radon_Nikodym_finite_measure_infinite[OF this] 

766 
obtain f where f_borel: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "density ?MT f = N" by auto 

767 
with nn borel show ?thesis 

768 
by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq) 

38656  769 
qed 
770 

40859  771 
section "Uniqueness of densities" 
772 

47694  773 
lemma finite_density_unique: 
40859  774 
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" 
47694  775 
assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

776 
and fin: "integral\<^isup>P M f \<noteq> \<infinity>" 
49785  777 
shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)" 
40859  778 
proof (intro iffI ballI) 
47694  779 
fix A assume eq: "AE x in M. f x = g x" 
49785  780 
with borel show "density M f = density M g" 
781 
by (auto intro: density_cong) 

40859  782 
next 
49785  783 
let ?P = "\<lambda>f A. \<integral>\<^isup>+ x. f x * indicator A x \<partial>M" 
784 
assume "density M f = density M g" 

785 
with borel have eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" 

786 
by (simp add: emeasure_density[symmetric]) 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

787 
from this[THEN bspec, OF sets.top] fin 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

788 
have g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" by (simp cong: positive_integral_cong) 
40859  789 
{ fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" 
47694  790 
and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

791 
and g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" 
40859  792 
let ?N = "{x\<in>space M. g x < f x}" 
793 
have N: "?N \<in> sets M" using borel by simp 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

794 
have "?P g ?N \<le> integral\<^isup>P M g" using pos 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

795 
by (intro positive_integral_mono_AE) (auto split: split_indicator) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

796 
then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by auto 
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

797 
have "?P (\<lambda>x. (f x  g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x  g x * indicator ?N x \<partial>M)" 
40859  798 
by (auto intro!: positive_integral_cong simp: indicator_def) 
799 
also have "\<dots> = ?P f ?N  ?P g ?N" 

800 
proof (rule positive_integral_diff) 

801 
show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M" 

802 
using borel N by auto 

47694  803 
show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x" 
804 
"AE x in M. 0 \<le> g x * indicator ?N x" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

805 
using pos by (auto split: split_indicator) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

806 
qed fact 
40859  807 
also have "\<dots> = 0" 
47694  808 
unfolding eq[THEN bspec, OF N] using positive_integral_positive[of M] Pg_fin by auto 
809 
finally have "AE x in M. f x \<le> g x" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

810 
using pos borel positive_integral_PInf_AE[OF borel(2) g_fin] 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

811 
by (subst (asm) positive_integral_0_iff_AE) 
43920  812 
(auto split: split_indicator simp: not_less ereal_minus_le_iff) } 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

813 
from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq 
47694  814 
show "AE x in M. f x = g x" by auto 
40859  815 
qed 
816 

817 
lemma (in finite_measure) density_unique_finite_measure: 

818 
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M" 

47694  819 
assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> f' x" 
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

820 
assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)" 
40859  821 
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A") 
47694  822 
shows "AE x in M. f x = f' x" 
40859  823 
proof  
47694  824 
let ?D = "\<lambda>f. density M f" 
825 
let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A" 

46731  826 
let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x" 
47694  827 

828 
have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M" 

829 
using borel by (auto intro!: absolutely_continuousI_density) 

830 
from split_space_into_finite_sets_and_rest[OF this] 

40859  831 
obtain Q0 and Q :: "nat \<Rightarrow> 'a set" 
832 
where Q: "disjoint_family Q" "range Q \<subseteq> sets M" 

833 
and Q0: "Q0 \<in> sets M" "Q0 = space M  (\<Union>i. Q i)" 

47694  834 
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> ?D f A = 0 \<or> 0 < emeasure M A \<and> ?D f A = \<infinity>" 
835 
and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force 

836 
with borel pos have in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> ?N A = 0 \<or> 0 < emeasure M A \<and> ?N A = \<infinity>" 

837 
and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq) 

838 

40859  839 
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto 
47694  840 
let ?D = "{x\<in>space M. f x \<noteq> f' x}" 
841 
have "?D \<in> sets M" using borel by auto 

43920  842 
have *: "\<And>i x A. \<And>y::ereal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x" 
40859  843 
unfolding indicator_def by auto 
47694  844 
have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos 
40859  845 
by (intro finite_density_unique[THEN iffD1] allI) 
50003  846 
(auto intro!: f measure_eqI simp: emeasure_density * subset_eq) 
47694  847 
moreover have "AE x in M. ?f Q0 x = ?f' Q0 x" 
40859  848 
proof (rule AE_I') 
43920  849 
{ fix f :: "'a \<Rightarrow> ereal" assume borel: "f \<in> borel_measurable M" 
47694  850 
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)" 
46731  851 
let ?A = "\<lambda>i. Q0 \<inter> {x \<in> space M. f x < (i::nat)}" 
47694  852 
have "(\<Union>i. ?A i) \<in> null_sets M" 
40859  853 
proof (rule null_sets_UN) 
43923  854 
fix i ::nat have "?A i \<in> sets M" 
40859  855 
using borel Q0(1) by auto 
47694  856 
have "?N (?A i) \<le> (\<integral>\<^isup>+x. (i::ereal) * indicator (?A i) x \<partial>M)" 
40859  857 
unfolding eq[OF `?A i \<in> sets M`] 
858 
by (auto intro!: positive_integral_mono simp: indicator_def) 

47694  859 
also have "\<dots> = i * emeasure M (?A i)" 
40859  860 
using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator) 
47694  861 
also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by simp 
862 
finally have "?N (?A i) \<noteq> \<infinity>" by simp 

863 
then show "?A i \<in> null_sets M" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto 

40859  864 
qed 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

865 
also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

866 
by (auto simp: less_PInf_Ex_of_nat real_eq_of_nat) 
47694  867 
finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp } 
40859  868 
from this[OF borel(1) refl] this[OF borel(2) f] 
47694  869 
have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all 
870 
then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets M" by (rule null_sets.Un) 

40859  871 
show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq> 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

872 
(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def) 
40859  873 
qed 
47694  874 
moreover have "AE x in M. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow> 
40859  875 
?f (space M) x = ?f' (space M) x" 
876 
by (auto simp: indicator_def Q0) 

47694  877 
ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x" 
878 
unfolding AE_all_countable[symmetric] 

879 
by eventually_elim (auto intro!: AE_I2 split: split_if_asm simp: indicator_def) 

880 
then show "AE x in M. f x = f' x" by auto 

40859  881 
qed 
882 

883 
lemma (in sigma_finite_measure) density_unique: 

47694  884 
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" 
885 
assumes f': "f' \<in> borel_measurable M" "AE x in M. 0 \<le> f' x" 

886 
assumes density_eq: "density M f = density M f'" 

887 
shows "AE x in M. f x = f' x" 

40859  888 
proof  
889 
obtain h where h_borel: "h \<in> borel_measurable M" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

890 
and fin: "integral\<^isup>P M h \<noteq> \<infinity>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<infinity>" "\<And>x. 0 \<le> h x" 
40859  891 
using Ex_finite_integrable_function by auto 
47694  892 
then have h_nn: "AE x in M. 0 \<le> h x" by auto 
893 
let ?H = "density M h" 

894 
interpret h: finite_measure ?H 

895 
using fin h_borel pos 

896 
by (intro finite_measureI) (simp cong: positive_integral_cong emeasure_density add: fin) 

897 
let ?fM = "density M f" 

898 
let ?f'M = "density M f'" 

40859  899 
{ fix A assume "A \<in> sets M" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

900 
then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A" 
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

901 
using pos(1) sets.sets_into_space by (force simp: indicator_def) 
47694  902 
then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

903 
using h_borel `A \<in> sets M` h_nn by (subst positive_integral_0_iff) auto } 
40859  904 
note h_null_sets = this 
905 
{ fix A assume "A \<in> sets M" 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

906 
have "(\<integral>\<^isup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

907 
using `A \<in> sets M` h_borel h_nn f f' 
47694  908 
by (intro positive_integral_density[symmetric]) auto 
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

909 
also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)" 
47694  910 
by (simp_all add: density_eq) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

911 
also have "\<dots> = (\<integral>\<^isup>+x. f' x * (h x * indicator A x) \<partial>M)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

912 
using `A \<in> sets M` h_borel h_nn f f' 
47694  913 
by (intro positive_integral_density) auto 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

914 
finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

915 
by (simp add: ac_simps) 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

916 
then have "(\<integral>\<^isup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^isup>+x. (f' x * indicator A x) \<partial>?H)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

917 
using `A \<in> sets M` h_borel h_nn f f' 
47694  918 
by (subst (asm) (1 2) positive_integral_density[symmetric]) auto } 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

919 
then have "AE x in ?H. f x = f' x" using h_borel h_nn f f' 
47694  920 
by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M]) 
921 
(auto simp add: AE_density) 

922 
then show "AE x in M. f x = f' x" 

923 
unfolding eventually_ae_filter using h_borel pos 

924 
by (auto simp add: h_null_sets null_sets_density_iff not_less[symmetric] 

50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset

925 
AE_iff_null_sets[symmetric]) blast 
40859  926 
qed 
927 

47694  928 
lemma (in sigma_finite_measure) density_unique_iff: 
929 
assumes f: "f \<in> borel_measurable M" and "AE x in M. 0 \<le> f x" 

930 
assumes f': "f' \<in> borel_measurable M" and "AE x in M. 0 \<le> f' x" 

931 
shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)" 

932 
using density_unique[OF assms] density_cong[OF f f'] by auto 

933 

49785  934 
lemma sigma_finite_density_unique: 
935 
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" 

936 
assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" 

937 
and fin: "sigma_finite_measure (density M f)" 

938 
shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)" 

939 
proof 

940 
assume "AE x in M. f x = g x" with borel show "density M f = density M g" 

941 
by (auto intro: density_cong) 

942 
next 

943 
assume eq: "density M f = density M g" 

944 
interpret f!: sigma_finite_measure "density M f" by fact 

945 
from f.sigma_finite_incseq guess A . note cover = this 

946 

947 
have "AE x in M. \<forall>i. x \<in> A i \<longrightarrow> f x = g x" 

948 
unfolding AE_all_countable 

949 
proof 

950 
fix i 

951 
have "density (density M f) (indicator (A i)) = density (density M g) (indicator (A i))" 

952 
unfolding eq .. 

953 
moreover have "(\<integral>\<^isup>+x. f x * indicator (A i) x \<partial>M) \<noteq> \<infinity>" 

954 
using cover(1) cover(3)[of i] borel by (auto simp: emeasure_density subset_eq) 

955 
ultimately have "AE x in M. f x * indicator (A i) x = g x * indicator (A i) x" 

956 
using borel pos cover(1) pos 

957 
by (intro finite_density_unique[THEN iffD1]) 

958 
(auto simp: density_density_eq subset_eq) 

959 
then show "AE x in M. x \<in> A i \<longrightarrow> f x = g x" 

960 
by auto 

961 
qed 

962 
with AE_space show "AE x in M. f x = g x" 

963 
apply eventually_elim 

964 
using cover(2)[symmetric] 

965 
apply auto 

966 
done 

967 
qed 

968 

49778
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset

969 
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite': 
47694  970 
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" 
971 
shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)" 

972 
(is "sigma_finite_measure ?N \<longleftrightarrow> _") 

40859  973 
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

974 
assume "sigma_finite_measure ?N" 
47694  975 
then interpret N: sigma_finite_measure ?N . 
976 
from N.Ex_finite_integrable_function obtain h where 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

977 
h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<infinity>" and 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

978 
h_nn: "\<And>x. 0 \<le> h x" and 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

979 
fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" by auto 
47694  980 
have "AE x in M. f x * h x \<noteq> \<infinity>" 
40859  981 
proof (rule AE_I') 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

982 
have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h h_nn 
47694  983 
by (auto intro!: positive_integral_density) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

984 
then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<infinity>" 
40859  985 
using h(2) by simp 
47694  986 
then show "(\<lambda>x. f x * h x) ` {\<infinity>} \<inter> space M \<in> null_sets M" 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

987 
using f h(1) by (auto intro!: positive_integral_PInf borel_measurable_vimage) 
40859  988 
qed auto 
47694  989 
then show "AE x in M. f x \<noteq> \<infinity>" 
41705  990 
using fin by (auto elim!: AE_Ball_mp) 
40859  991 
next 
47694  992 
assume AE: "AE x in M. f x \<noteq> \<infinity>" 
40859  993 
from sigma_finite guess Q .. note Q = this 
43923  994 
def A \<equiv> "\<lambda>i. f ` (case i of 0 \<Rightarrow> {\<infinity>}  Suc n \<Rightarrow> {.. ereal(of_nat (Suc n))}) \<inter> space M" 
40859  995 
{ fix i j have "A i \<inter> Q j \<in> sets M" 
996 
unfolding A_def using f Q 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

997 
apply (rule_tac sets.Int) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

998 
by (cases i) (auto intro: measurable_sets[OF f(1)]) } 
40859  999 
note A_in_sets = this 
46731  1000 
let ?A = "\<lambda>n. case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j" 
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

1001 
show "sigma_finite_measure ?N" 
40859  1002 
proof (default, intro exI conjI subsetI allI) 
1003 
fix x assume "x \<in> range ?A" 

1004 
then obtain n where n: "x = ?A n" by auto 

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

1005 
then show "x \<in> sets ?N" using A_in_sets by (cases "prod_decode n") auto 
40859  1006 
next 
1007 
have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)" 

1008 
proof safe 

1009 
fix x i j assume "x \<in> A i" "x \<in> Q j" 

1010 
then show "x \<in> (\<Union>i. case prod_decode i of (i, j) \<Rightarrow> A i \<inter> Q j)" 

1011 
by (intro UN_I[of "prod_encode (i,j)"]) auto 

1012 
qed auto 

1013 
also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto 

1014 
also have "(\<Union>i. A i) = space M" 

1015 
proof safe 

1016 
fix x assume x: "x \<in> space M" 

1017 
show "x \<in> (\<Union>i. A i)" 

1018 
proof (cases "f x") 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

1019 
case PInf with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0]) 
40859  1020 
next 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

1021 
case (real r) 
43923  1022 
with less_PInf_Ex_of_nat[of "f x"] obtain n :: nat where "f x < n" by (auto simp: real_eq_of_nat) 
45769
2d5b1af2426a
real is better supported than real_of_nat, use it in the nat => ereal coercion
hoelzl
parents:
44941
diff
changeset

1023 
then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"] simp: real_eq_of_nat) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

1024 
next 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

1025 
case MInf with x show ?thesis 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

1026 
unfolding A_def by (auto intro!: exI[of _ "Suc 0"]) 
40859  1027 
qed 
1028 
qed (auto simp: A_def) 

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

1029 
finally show "(\<Union>i. ?A i) = space ?N" by simp 
40859  1030 
next 
1031 
fix n obtain i j where 

1032 
[simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto 

41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

1033 
have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>" 
40859  1034 
proof (cases i) 
1035 
case 0 

47694  1036 
have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0" 
41705  1037 
using AE by (auto simp: A_def `i = 0`) 
1038 
from positive_integral_cong_AE[OF this] show ?thesis by simp 

40859  1039 
next 
1040 
case (Suc n) 

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

1041 
then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le> 
43923  1042 
(\<integral>\<^isup>+x. (Suc n :: ereal) * indicator (Q j) x \<partial>M)" 
45769
2d5b1af2426a
real is better supported than real_of_nat, use it in the nat => ereal coercion
hoelzl
parents:
44941
diff
changeset

1043 
by (auto intro!: positive_integral_mono simp: indicator_def A_def real_eq_of_nat) 
47694  1044 
also have "\<dots> = Suc n * emeasure M (Q j)" 
40859  1045 
using Q by (auto intro!: positive_integral_cmult_indicator) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

1046 
also have "\<dots> < \<infinity>" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset

1047 
using Q by (auto simp: real_eq_of_nat[symmetric]) 
40859  1048 
finally show ?thesis by simp 
1049 
qed 

47694  1050 
then show "emeasure ?N (?A n) \<noteq> \<infinity>" 
1051 
using A_in_sets Q f by (auto simp: emeasure_density) 

40859  1052 
qed 
1053 
qed 

1054 

49778
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset

1055 
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite: 
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset

1056 
"f \<in> borel_measurable M \<Longrightarrow> sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)" 
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset

1057 
apply (subst density_max_0) 
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset

1058 
apply (subst sigma_finite_iff_density_finite') 
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset

1059 
apply (auto simp: max_def intro!: measurable_If) 
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset

1060 
done 
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset

1061 

40871  1062 
section "RadonNikodym derivative" 
38656  1063 

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

1064 
definition 
47694  1065 
"RN_deriv M N \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N" 
38656  1066 

47694  1067 
lemma 
1068 
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" 

1069 
shows borel_measurable_RN_deriv_density: "RN_deriv M (density M f) \<in> borel_measurable M" (is ?borel) 

1070 
and density_RN_deriv_density: "density M (RN_deriv M (density M f)) = density M f" (is ?density) 

1071 
and RN_deriv_density_nonneg: "0 \<le> RN_deriv M (density M f) x" (is ?pos) 

40859  1072 
proof  
47694  1073 
let ?f = "\<lambda>x. max 0 (f x)" 
1074 
let ?P = "\<lambda>g. g \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> g x) \<and> density M g = density M f" 

1075 
from f have "?P ?f" using f by (auto intro!: density_cong simp: split: split_max) 

1076 
then have "?P (RN_deriv M (density M f))" 

1077 
unfolding RN_deriv_def by (rule someI[where P="?P"]) 

1078 
then show ?borel ?density ?pos by auto 

40859  1079 
qed 
1080 

38656  1081 
lemma (in sigma_finite_measure) RN_deriv: 
47694  1082 
assumes "absolutely_continuous M N" "sets N = sets M" 
50003  1083 
shows borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M" (is ?borel) 
47694  1084 
and density_RN_deriv: "density M (RN_deriv M N) = N" (is ?density) 
1085 
and RN_deriv_nonneg: "0 \<le> RN_deriv M N x" (is ?pos) 

38656  1086 
proof  
1087 
note Ex = Radon_Nikodym[OF assms, unfolded Bex_def] 

47694  1088 
from Ex show ?borel unfolding RN_deriv_def by (rule someI2_ex) simp 
1089 
from Ex show ?density unfolding RN_deriv_def by (rule someI2_ex) simp 

1090 
from Ex show ?pos unfolding RN_deriv_def by (rule someI2_ex) simp 

38656  1091 
qed 
1092 

40859  1093 
lemma (in sigma_finite_measure) RN_deriv_positive_integral: 
47694  1094 
assumes N: "absolutely_continuous M N" "sets N = sets M" 
40859  1095 
and f: "f \<in> borel_measurable M" 
47694  1096 
shows "integral\<^isup>P N f = (\<integral>\<^isup>+x. RN_deriv M N x * f x \<partial>M)" 
40859  1097 
proof  
47694  1098 
have "integral\<^isup>P N f = integral\<^isup>P (density M (RN_deriv M N)) f" 
1099 
using N by (simp add: density_RN_deriv) 

1100 
also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M N x * f x \<partial>M)" 

1101 
using RN_deriv(1,3)[OF N] f by (simp add: positive_integral_density) 

1102 
finally show ?thesis by simp 

40859  1103 
qed 
1104 

47694  1105 
lemma null_setsD_AE: "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N" 
1106 
using AE_iff_null_sets[of N M] by auto 

1107 

1108 
lemma (in sigma_finite_measure) RN_deriv_unique: 

1109 
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" 

1110 
and eq: "density M f = N" 

1111 
shows "AE x in M. f x = RN_deriv M N x" 

49785  1112 
unfolding eq[symmetric] 
1113 
by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv_density 

1114 
RN_deriv_density_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric]) 

1115 

1116 
lemma RN_deriv_unique_sigma_finite: 

1117 
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" 

1118 
and eq: "density M f = N" and fin: "sigma_finite_measure N" 

1119 
shows "AE x in M. f x = RN_deriv M N x" 

1120 
using fin unfolding eq[symmetric] 

1121 
by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv_density 

1122 
RN_deriv_density_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric]) 

47694  1123 

1124 
lemma (in sigma_finite_measure) RN_deriv_distr: 

1125 
fixes T :: "'a \<Rightarrow> 'b" 

1126 
assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M" 

1127 
and inv: "\<forall>x\<in>space M. T' (T x) = x" 

50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset

1128 
and ac[simp]: "absolutely_continuous (distr M M' T) (distr N M' T)" 
47694  1129 
and N: "sets N = sets M" 
1130 
shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x" 

41832  1131 
proof (rule RN_deriv_unique) 
47694  1132 
have [simp]: "sets N = sets M" by fact 
1133 
note sets_eq_imp_space_eq[OF N, simp] 

1134 
have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def) 

1135 
{ fix A assume "A \<in> sets M" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

1136 
with inv T T' sets.sets_into_space[OF this] 
47694  1137 
have "T ` T' ` A \<inter> T ` space M' \<inter> space M = A" 
1138 
by (auto simp: measurable_def) } 

1139 
note eq = this[simp] 

1140 
{ fix A assume "A \<in> sets M" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

1141 
with inv T T' sets.sets_into_space[OF this] 
47694  1142 
have "(T' \<circ> T) ` A \<inter> space M = A" 
1143 
by (auto simp: measurable_def) } 

1144 
note eq2 = this[simp] 

1145 
let ?M' = "distr M M' T" and ?N' = "distr N M' T" 

1146 
interpret M': sigma_finite_measure ?M' 

41832  1147 
proof 
1148 
from sigma_finite guess F .. note F = this 

47694  1149 
show "\<exists>A::nat \<Rightarrow> 'b set. range A \<subseteq> sets ?M' \<and> (\<Union>i. A i) = space ?M' \<and> (\<forall>i. emeasure ?M' (A i) \<noteq> \<infinity>)" 
41832  1150 
proof (intro exI conjI allI) 
47694  1151 
show *: "range (\<lambda>i. T' ` F i \<inter> space ?M') \<subseteq> sets ?M'" 
1152 
using F T' by (auto simp: measurable_def) 

1153 
show "(\<Union>i. T' ` F i \<inter> space ?M') = space ?M'" 

1154 
using F T' by (force simp: measurable_def) 

41832  1155 
fix i 
1156 
have "T' ` F i \<inter> space M' \<in> sets M'" using * by auto 

1157 
moreover 

1158 
have Fi: "F i \<in> sets M" using F by auto 

47694  1159 
ultimately show "emeasure ?M' (T' ` F i \<inter> space ?M') \<noteq> \<infinity>" 
1160 
using F T T' by (simp add: emeasure_distr) 

41832  1161 
qed 
1162 
qed 

47694  1163 
have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M" 
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset

1164 
using T ac by measurable 
47694  1165 
then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M" 
41832  1166 
by (simp add: comp_def) 
47694  1167 
show "AE x in M. 0 \<le> RN_deriv ?M' ?N' (T x)" using M'.RN_deriv_nonneg[OF ac] by auto 
1168 

1169 
have "N = distr N M (T' \<circ> T)" 

1170 
by (subst measure_of_of_measure[of N, symmetric]) 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset

1171 
(auto simp add: distr_def sets.sigma_sets_eq intro!: measure_of_eq sets.space_closed) 
47694  1172 
also have "\<dots> = distr (distr N M' T) M T'" 
1173 
using T T' by (simp add: distr_distr) 

1174 
also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'" 

1175 
using ac by (simp add: M'.density_RN_deriv) 

1176 
also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)" 

1177 
using M'.borel_measurable_RN_deriv[OF ac] by (simp add: distr_density_distr[OF T T', OF inv]) 

1178 
finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N" 

1179 
by (simp add: comp_def) 

41832  1180 
qed 
1181 

40859  1182 
lemma (in sigma_finite_measure) RN_deriv_finite: 
47694  1183 
assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M" 
1184 
shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>" 

40859  1185 
proof  
47694  1186 
interpret N: sigma_finite_measure N by fact 
1187 
from N show ?thesis 

1188 
using sigma_finite_iff_density_finite[OF RN_deriv(1)[OF ac]] RN_deriv(2,3)[OF ac] by simp 

40859  1189 
qed 
1190 

1191 
lemma (in sigma_finite_measure) 

47694  1192 
assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M" 
40859  1193 
and f: "f \<in> borel_measurable M" 
47694  1194 
shows RN_deriv_integrable: "integrable N f \<longleftrightarrow> 
1195 
integrable M (\<lambda>x. real (RN_deriv M N x) * f x)" (is ?integrable) 

1196 
and RN_deriv_integral: "integral\<^isup>L N f = 

1197 
(\<integral>x. real (RN_deriv M N x) * f x \<partial>M)" (is ?integral) 

40859  1198 
proof  
47694  1199 
note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp] 
1200 
interpret N: sigma_finite_measure N by fact 

43920  1201 
have minus_cong: "\<And>A B A' B'::ereal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A  real B = real A'  real B'" by simp 
40859  1202 
have f': "(\<lambda>x.  f x) \<in> borel_measurable M" using f by auto 
47694  1203 
have Nf: "f \<in> borel_measurable N" using f by (simp add: measurable_def) 
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

1204 
{ fix f :: "'a \<Rightarrow> real" 
47694  1205 
{ fix x assume *: "RN_deriv M N x \<noteq> \<infinity>" 
1206 
have "ereal (real (RN_deriv M N x)) * ereal (f x) = ereal (real (RN_deriv M N x) * f x)" 

40859  1207 
by (simp add: mult_le_0_iff) 
47694  1208 
then have "RN_deriv M N x * ereal (f x) = ereal (real (RN_deriv M N x) * f x)" 
1209 
using RN_deriv(3)[OF ac] * by (auto simp add: ereal_real split: split_if_asm) } 