author  wenzelm 
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parent 50244  de72bbe42190 
child 53015  a1119cf551e8 
permissions  rwrr 
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(* Title: HOL/Probability/Binary_Product_Measure.thy 
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Author: Johannes Hölzl, TU München 
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*) 

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header {*Binary product measures*} 
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theory Binary_Product_Measure 
38656  8 
imports Lebesgue_Integration 
35833  9 
begin 
10 

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lemma Pair_vimage_times[simp]: "Pair x ` (A \<times> B) = (if x \<in> A then B else {})" 
40859  12 
by auto 
13 

50104  14 
lemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) ` (A \<times> B) = (if y \<in> B then A else {})" 
40859  15 
by auto 
16 

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section "Binary products" 

18 

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definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where 
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"A \<Otimes>\<^isub>M B = measure_of (space A \<times> space B) 
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{a \<times> b  a b. a \<in> sets A \<and> b \<in> sets B} 

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(\<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A)" 

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lemma pair_measure_closed: "{a \<times> b  a b. a \<in> sets A \<and> b \<in> sets B} \<subseteq> Pow (space A \<times> space B)" 
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using sets.space_closed[of A] sets.space_closed[of B] by auto 
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lemma space_pair_measure: 
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"space (A \<Otimes>\<^isub>M B) = space A \<times> space B" 
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unfolding pair_measure_def using pair_measure_closed[of A B] 
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by (rule space_measure_of) 
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lemma sets_pair_measure: 

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"sets (A \<Otimes>\<^isub>M B) = sigma_sets (space A \<times> space B) {a \<times> b  a b. a \<in> sets A \<and> b \<in> sets B}" 

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unfolding pair_measure_def using pair_measure_closed[of A B] 
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by (rule sets_measure_of) 
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lemma sets_pair_measure_cong[cong]: 
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"sets M1 = sets M1' \<Longrightarrow> sets M2 = sets M2' \<Longrightarrow> sets (M1 \<Otimes>\<^isub>M M2) = sets (M1' \<Otimes>\<^isub>M M2')" 

39 
unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq) 

40 

50003  41 
lemma pair_measureI[intro, simp, measurable]: 
47694  42 
"x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)" 
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by (auto simp: sets_pair_measure) 

41095  44 

47694  45 
lemma measurable_pair_measureI: 
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assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2" 

47 
assumes 2: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> f ` (A \<times> B) \<inter> space M \<in> sets M" 

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shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)" 

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unfolding pair_measure_def using 1 2 

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by (intro measurable_measure_of) (auto dest: sets.sets_into_space) 
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50003  52 
lemma measurable_split_replace[measurable (raw)]: 
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"(\<lambda>x. f x (fst (g x)) (snd (g x))) \<in> measurable M N \<Longrightarrow> (\<lambda>x. split (f x) (g x)) \<in> measurable M N" 

54 
unfolding split_beta' . 

55 

56 
lemma measurable_Pair[measurable (raw)]: 

49776  57 
assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2" 
58 
shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^isub>M M2)" 

59 
proof (rule measurable_pair_measureI) 

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show "(\<lambda>x. (f x, g x)) \<in> space M \<rightarrow> space M1 \<times> space M2" 

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using f g by (auto simp: measurable_def) 

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fix A B assume *: "A \<in> sets M1" "B \<in> sets M2" 

63 
have "(\<lambda>x. (f x, g x)) ` (A \<times> B) \<inter> space M = (f ` A \<inter> space M) \<inter> (g ` B \<inter> space M)" 

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by auto 

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also have "\<dots> \<in> sets M" 

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by (rule sets.Int) (auto intro!: measurable_sets * f g) 
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finally show "(\<lambda>x. (f x, g x)) ` (A \<times> B) \<inter> space M \<in> sets M" . 
68 
qed 

69 

50003  70 
lemma measurable_Pair_compose_split[measurable_dest]: 
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assumes f: "split f \<in> measurable (M1 \<Otimes>\<^isub>M M2) N" 

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assumes g: "g \<in> measurable M M1" and h: "h \<in> measurable M M2" 

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shows "(\<lambda>x. f (g x) (h x)) \<in> measurable M N" 

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using measurable_compose[OF measurable_Pair f, OF g h] by simp 

75 

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lemma measurable_pair: 
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assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2" 

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shows "f \<in> measurable M (M1 \<Otimes>\<^isub>M M2)" 

79 
using measurable_Pair[OF assms] by simp 

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50003  81 
lemma measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^isub>M M2) M1" 
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by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times 
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measurable_def) 
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50003  85 
lemma measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^isub>M M2) M2" 
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by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space times_Int_times 
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measurable_def) 
47694  88 

50003  89 
lemma 
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assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^isub>M P)" 

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shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N" 

92 
and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P" 

93 
by simp_all 

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50003  95 
lemma 
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assumes f[measurable]: "f \<in> measurable M N" 

97 
shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^isub>M P) N" 

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and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^isub>M M) N" 

99 
by simp_all 

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101 
lemma measurable_pair_iff: 

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"f \<in> measurable M (M1 \<Otimes>\<^isub>M M2) \<longleftrightarrow> (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2" 

50003  103 
by (auto intro: measurable_pair[of f M M1 M2]) 
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49776  105 
lemma measurable_split_conv: 
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"(\<lambda>(x, y). f x y) \<in> measurable A B \<longleftrightarrow> (\<lambda>x. f (fst x) (snd x)) \<in> measurable A B" 

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by (intro arg_cong2[where f="op \<in>"]) auto 

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47694  109 
lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)" 
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by (auto intro!: measurable_Pair simp: measurable_split_conv) 
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lemma measurable_pair_swap: 

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assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M" 

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using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def) 
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lemma measurable_pair_swap_iff: 
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"f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" 

50003  118 
by (auto dest: measurable_pair_swap) 
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47694  120 
lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^isub>M M2)" 
50003  121 
by simp 
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50003  123 
lemma sets_Pair1[measurable (raw)]: 
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assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "Pair x ` A \<in> sets M2" 

40859  125 
proof  
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have "Pair x ` A = (if x \<in> space M1 then Pair x ` A \<inter> space M2 else {})" 
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using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure) 
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also have "\<dots> \<in> sets M2" 
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using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: split_if_asm) 

130 
finally show ?thesis . 

40859  131 
qed 
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47694  133 
lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^isub>M M2)" 
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by (auto intro!: measurable_Pair) 
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47694  136 
lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>x. (x, y)) ` A \<in> sets M1" 
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proof  

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have "(\<lambda>x. (x, y)) ` A = (if y \<in> space M2 then (\<lambda>x. (x, y)) ` A \<inter> space M1 else {})" 

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using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure) 
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also have "\<dots> \<in> sets M1" 
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using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: split_if_asm) 

142 
finally show ?thesis . 

40859  143 
qed 
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47694  145 
lemma measurable_Pair2: 
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assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and x: "x \<in> space M1" 

147 
shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M" 

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using measurable_comp[OF measurable_Pair1' f, OF x] 

149 
by (simp add: comp_def) 

150 

151 
lemma measurable_Pair1: 

152 
assumes f: "f \<in> measurable (M1 \<Otimes>\<^isub>M M2) M" and y: "y \<in> space M2" 

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shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M" 
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using measurable_comp[OF measurable_Pair2' f, OF y] 
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by (simp add: comp_def) 

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47694  157 
lemma Int_stable_pair_measure_generator: "Int_stable {a \<times> b  a b. a \<in> sets A \<and> b \<in> sets B}" 
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unfolding Int_stable_def 
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by safe (auto simp add: times_Int_times) 
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50003  161 
lemma disjoint_family_vimageI: "disjoint_family F \<Longrightarrow> disjoint_family (\<lambda>i. f ` F i)" 
162 
by (auto simp: disjoint_family_on_def) 

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49776  164 
lemma (in finite_measure) finite_measure_cut_measurable: 
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assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^isub>M M)" 
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shows "(\<lambda>x. emeasure M (Pair x ` Q)) \<in> borel_measurable N" 
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(is "?s Q \<in> _") 
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using Int_stable_pair_measure_generator pair_measure_closed assms 
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unfolding sets_pair_measure 
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proof (induct rule: sigma_sets_induct_disjoint) 
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case (compl A) 
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with sets.sets_into_space have "\<And>x. emeasure M (Pair x ` ((space N \<times> space M)  A)) = 
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(if x \<in> space N then emeasure M (space M)  ?s A x else 0)" 
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unfolding sets_pair_measure[symmetric] 
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by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1) 
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with compl sets.top show ?case 
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by (auto intro!: measurable_If simp: space_pair_measure) 
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next 
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case (union F) 
50003  180 
moreover then have *: "\<And>x. emeasure M (Pair x ` (\<Union>i. F i)) = (\<Sum>i. ?s (F i) x)" 
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by (simp add: suminf_emeasure disjoint_family_vimageI subset_eq vimage_UN sets_pair_measure[symmetric]) 

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ultimately show ?case 
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qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If) 
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lemma (in sigma_finite_measure) measurable_emeasure_Pair: 

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assumes Q: "Q \<in> sets (N \<Otimes>\<^isub>M M)" shows "(\<lambda>x. emeasure M (Pair x ` Q)) \<in> borel_measurable N" (is "?s Q \<in> _") 

188 
proof  

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from sigma_finite_disjoint guess F . note F = this 

190 
then have F_sets: "\<And>i. F i \<in> sets M" by auto 

191 
let ?C = "\<lambda>x i. F i \<inter> Pair x ` Q" 

192 
{ fix i 

193 
have [simp]: "space N \<times> F i \<inter> space N \<times> space M = space N \<times> F i" 

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using F sets.sets_into_space by auto 
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let ?R = "density M (indicator (F i))" 
196 
have "finite_measure ?R" 

197 
using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq) 

198 
then have "(\<lambda>x. emeasure ?R (Pair x ` (space N \<times> space ?R \<inter> Q))) \<in> borel_measurable N" 

199 
by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q) 

200 
moreover have "\<And>x. emeasure ?R (Pair x ` (space N \<times> space ?R \<inter> Q)) 

201 
= emeasure M (F i \<inter> Pair x ` (space N \<times> space ?R \<inter> Q))" 

202 
using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1) 

203 
moreover have "\<And>x. F i \<inter> Pair x ` (space N \<times> space ?R \<inter> Q) = ?C x i" 

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using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure) 
49776  205 
ultimately have "(\<lambda>x. emeasure M (?C x i)) \<in> borel_measurable N" 
206 
by simp } 

207 
moreover 

208 
{ fix x 

209 
have "(\<Sum>i. emeasure M (?C x i)) = emeasure M (\<Union>i. ?C x i)" 

210 
proof (intro suminf_emeasure) 

211 
show "range (?C x) \<subseteq> sets M" 

212 
using F `Q \<in> sets (N \<Otimes>\<^isub>M M)` by (auto intro!: sets_Pair1) 

213 
have "disjoint_family F" using F by auto 

214 
show "disjoint_family (?C x)" 

215 
by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto 

216 
qed 

217 
also have "(\<Union>i. ?C x i) = Pair x ` Q" 

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using F sets.sets_into_space[OF `Q \<in> sets (N \<Otimes>\<^isub>M M)`] 
49776  219 
by (auto simp: space_pair_measure) 
220 
finally have "emeasure M (Pair x ` Q) = (\<Sum>i. emeasure M (?C x i))" 

221 
by simp } 

222 
ultimately show ?thesis using `Q \<in> sets (N \<Otimes>\<^isub>M M)` F_sets 

223 
by auto 

224 
qed 

225 

50003  226 
lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]: 
227 
assumes space: "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M" 

228 
assumes A: "{x\<in>space (N \<Otimes>\<^isub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^isub>M M)" 

229 
shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N" 

230 
proof  

231 
from space have "\<And>x. x \<in> space N \<Longrightarrow> Pair x ` {x \<in> space (N \<Otimes>\<^isub>M M). snd x \<in> A (fst x)} = A x" 

232 
by (auto simp: space_pair_measure) 

233 
with measurable_emeasure_Pair[OF A] show ?thesis 

234 
by (auto cong: measurable_cong) 

235 
qed 

236 

49776  237 
lemma (in sigma_finite_measure) emeasure_pair_measure: 
238 
assumes "X \<in> sets (N \<Otimes>\<^isub>M M)" 

239 
shows "emeasure (N \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator X (x, y) \<partial>M \<partial>N)" (is "_ = ?\<mu> X") 

240 
proof (rule emeasure_measure_of[OF pair_measure_def]) 

241 
show "positive (sets (N \<Otimes>\<^isub>M M)) ?\<mu>" 

242 
by (auto simp: positive_def positive_integral_positive) 

243 
have eq[simp]: "\<And>A x y. indicator A (x, y) = indicator (Pair x ` A) y" 

244 
by (auto simp: indicator_def) 

245 
show "countably_additive (sets (N \<Otimes>\<^isub>M M)) ?\<mu>" 

246 
proof (rule countably_additiveI) 

247 
fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^isub>M M)" "disjoint_family F" 

248 
from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^isub>M M)" "(\<Union>i. F i) \<in> sets (N \<Otimes>\<^isub>M M)" by auto 

249 
moreover from F have "\<And>i. (\<lambda>x. emeasure M (Pair x ` F i)) \<in> borel_measurable N" 

250 
by (intro measurable_emeasure_Pair) auto 

251 
moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x ` F i)" 

252 
by (intro disjoint_family_on_bisimulation[OF F(2)]) auto 

253 
moreover have "\<And>x. range (\<lambda>i. Pair x ` F i) \<subseteq> sets M" 

254 
using F by (auto simp: sets_Pair1) 

255 
ultimately show "(\<Sum>n. ?\<mu> (F n)) = ?\<mu> (\<Union>i. F i)" 

256 
by (auto simp add: vimage_UN positive_integral_suminf[symmetric] suminf_emeasure subset_eq emeasure_nonneg sets_Pair1 

257 
intro!: positive_integral_cong positive_integral_indicator[symmetric]) 

258 
qed 

259 
show "{a \<times> b a b. a \<in> sets N \<and> b \<in> sets M} \<subseteq> Pow (space N \<times> space M)" 

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260 
using sets.space_closed[of N] sets.space_closed[of M] by auto 
49776  261 
qed fact 
262 

263 
lemma (in sigma_finite_measure) emeasure_pair_measure_alt: 

264 
assumes X: "X \<in> sets (N \<Otimes>\<^isub>M M)" 

265 
shows "emeasure (N \<Otimes>\<^isub>M M) X = (\<integral>\<^isup>+x. emeasure M (Pair x ` X) \<partial>N)" 

266 
proof  

267 
have [simp]: "\<And>x y. indicator X (x, y) = indicator (Pair x ` X) y" 

268 
by (auto simp: indicator_def) 

269 
show ?thesis 

270 
using X by (auto intro!: positive_integral_cong simp: emeasure_pair_measure sets_Pair1) 

271 
qed 

272 

273 
lemma (in sigma_finite_measure) emeasure_pair_measure_Times: 

274 
assumes A: "A \<in> sets N" and B: "B \<in> sets M" 

275 
shows "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = emeasure N A * emeasure M B" 

276 
proof  

277 
have "emeasure (N \<Otimes>\<^isub>M M) (A \<times> B) = (\<integral>\<^isup>+x. emeasure M B * indicator A x \<partial>N)" 

278 
using A B by (auto intro!: positive_integral_cong simp: emeasure_pair_measure_alt) 

279 
also have "\<dots> = emeasure M B * emeasure N A" 

280 
using A by (simp add: emeasure_nonneg positive_integral_cmult_indicator) 

281 
finally show ?thesis 

282 
by (simp add: ac_simps) 

40859  283 
qed 
284 

47694  285 
subsection {* Binary products of $\sigma$finite emeasure spaces *} 
40859  286 

47694  287 
locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2 
288 
for M1 :: "'a measure" and M2 :: "'b measure" 

40859  289 

47694  290 
lemma (in pair_sigma_finite) measurable_emeasure_Pair1: 
49776  291 
"Q \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x ` Q)) \<in> borel_measurable M1" 
292 
using M2.measurable_emeasure_Pair . 

40859  293 

47694  294 
lemma (in pair_sigma_finite) measurable_emeasure_Pair2: 
295 
assumes Q: "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) ` Q)) \<in> borel_measurable M2" 

40859  296 
proof  
47694  297 
have "(\<lambda>(x, y). (y, x)) ` Q \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)" 
298 
using Q measurable_pair_swap' by (auto intro: measurable_sets) 

49776  299 
note M1.measurable_emeasure_Pair[OF this] 
47694  300 
moreover have "\<And>y. Pair y ` ((\<lambda>(x, y). (y, x)) ` Q \<inter> space (M2 \<Otimes>\<^isub>M M1)) = (\<lambda>x. (x, y)) ` Q" 
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301 
using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure) 
47694  302 
ultimately show ?thesis by simp 
39088
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303 
qed 
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Measurable on product space is equiv. to measurable components
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304 

41689
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305 
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator: 
47694  306 
defines "E \<equiv> {A \<times> B  A B. A \<in> sets M1 \<and> B \<in> sets M2}" 
307 
shows "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> E \<and> incseq F \<and> (\<Union>i. F i) = space M1 \<times> space M2 \<and> 

308 
(\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)" 

40859  309 
proof  
47694  310 
from M1.sigma_finite_incseq guess F1 . note F1 = this 
311 
from M2.sigma_finite_incseq guess F2 . note F2 = this 

312 
from F1 F2 have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto 

40859  313 
let ?F = "\<lambda>i. F1 i \<times> F2 i" 
47694  314 
show ?thesis 
40859  315 
proof (intro exI[of _ ?F] conjI allI) 
47694  316 
show "range ?F \<subseteq> E" using F1 F2 by (auto simp: E_def) (metis range_subsetD) 
40859  317 
next 
318 
have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)" 

319 
proof (intro subsetI) 

320 
fix x assume "x \<in> space M1 \<times> space M2" 

321 
then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j" 

322 
by (auto simp: space) 

323 
then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)" 

41981
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324 
using `incseq F1` `incseq F2` unfolding incseq_def 
cdf7693bbe08
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325 
by (force split: split_max)+ 
40859  326 
then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)" 
327 
by (intro SigmaI) (auto simp add: min_max.sup_commute) 

328 
then show "x \<in> (\<Union>i. ?F i)" by auto 

329 
qed 

47694  330 
then show "(\<Union>i. ?F i) = space M1 \<times> space M2" 
331 
using space by (auto simp: space) 

40859  332 
next 
41981
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333 
fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)" 
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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changeset

334 
using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto 
40859  335 
next 
336 
fix i 

337 
from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto 

47694  338 
with F1 F2 emeasure_nonneg[of M1 "F1 i"] emeasure_nonneg[of M2 "F2 i"] 
339 
show "emeasure (M1 \<Otimes>\<^isub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>" 

340 
by (auto simp add: emeasure_pair_measure_Times) 

341 
qed 

342 
qed 

343 

49800  344 
sublocale pair_sigma_finite \<subseteq> P: sigma_finite_measure "M1 \<Otimes>\<^isub>M M2" 
47694  345 
proof 
346 
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this 

347 
show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2) \<and> (\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2) \<and> (\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)" 

348 
proof (rule exI[of _ F], intro conjI) 

349 
show "range F \<subseteq> sets (M1 \<Otimes>\<^isub>M M2)" using F by (auto simp: pair_measure_def) 

350 
show "(\<Union>i. F i) = space (M1 \<Otimes>\<^isub>M M2)" 

351 
using F by (auto simp: space_pair_measure) 

352 
show "\<forall>i. emeasure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>" using F by auto 

40859  353 
qed 
354 
qed 

355 

47694  356 
lemma sigma_finite_pair_measure: 
357 
assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B" 

358 
shows "sigma_finite_measure (A \<Otimes>\<^isub>M B)" 

359 
proof  

360 
interpret A: sigma_finite_measure A by fact 

361 
interpret B: sigma_finite_measure B by fact 

362 
interpret AB: pair_sigma_finite A B .. 

363 
show ?thesis .. 

40859  364 
qed 
39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset

365 

47694  366 
lemma sets_pair_swap: 
367 
assumes "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" 

41689
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hoelzl
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diff
changeset

368 
shows "(\<lambda>(x, y). (y, x)) ` A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)" 
47694  369 
using measurable_pair_swap' assms by (rule measurable_sets) 
41661  370 

47694  371 
lemma (in pair_sigma_finite) distr_pair_swap: 
372 
"M1 \<Otimes>\<^isub>M M2 = distr (M2 \<Otimes>\<^isub>M M1) (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D") 

40859  373 
proof  
41689
3e39b0e730d6
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hoelzl
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41661
diff
changeset

374 
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this 
47694  375 
let ?E = "{a \<times> b a b. a \<in> sets M1 \<and> b \<in> sets M2}" 
376 
show ?thesis 

377 
proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) 

378 
show "?E \<subseteq> Pow (space ?P)" 

50244
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changeset

379 
using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure) 
47694  380 
show "sets ?P = sigma_sets (space ?P) ?E" 
381 
by (simp add: sets_pair_measure space_pair_measure) 

382 
then show "sets ?D = sigma_sets (space ?P) ?E" 

383 
by simp 

384 
next 

49784
5e5b2da42a69
remove incseq assumption from measure_eqI_generator_eq
hoelzl
parents:
49776
diff
changeset

385 
show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>" 
47694  386 
using F by (auto simp: space_pair_measure) 
387 
next 

388 
fix X assume "X \<in> ?E" 

389 
then obtain A B where X[simp]: "X = A \<times> B" and A: "A \<in> sets M1" and B: "B \<in> sets M2" by auto 

390 
have "(\<lambda>(y, x). (x, y)) ` X \<inter> space (M2 \<Otimes>\<^isub>M M1) = B \<times> A" 

50244
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parents:
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diff
changeset

391 
using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure) 
47694  392 
with A B show "emeasure (M1 \<Otimes>\<^isub>M M2) X = emeasure ?D X" 
49776  393 
by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr 
47694  394 
measurable_pair_swap' ac_simps) 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

395 
qed 
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

396 
qed 
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

397 

47694  398 
lemma (in pair_sigma_finite) emeasure_pair_measure_alt2: 
399 
assumes A: "A \<in> sets (M1 \<Otimes>\<^isub>M M2)" 

400 
shows "emeasure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+y. emeasure M1 ((\<lambda>x. (x, y)) ` A) \<partial>M2)" 

401 
(is "_ = ?\<nu> A") 

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

402 
proof  
47694  403 
have [simp]: "\<And>y. (Pair y ` ((\<lambda>(x, y). (y, x)) ` A \<inter> space (M2 \<Otimes>\<^isub>M M1))) = (\<lambda>x. (x, y)) ` A" 
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset

404 
using sets.sets_into_space[OF A] by (auto simp: space_pair_measure) 
47694  405 
show ?thesis using A 
406 
by (subst distr_pair_swap) 

407 
(simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap'] 

49776  408 
M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A]) 
409 
qed 

410 

411 
lemma (in pair_sigma_finite) AE_pair: 

412 
assumes "AE x in (M1 \<Otimes>\<^isub>M M2). Q x" 

413 
shows "AE x in M1. (AE y in M2. Q (x, y))" 

414 
proof  

415 
obtain N where N: "N \<in> sets (M1 \<Otimes>\<^isub>M M2)" "emeasure (M1 \<Otimes>\<^isub>M M2) N = 0" "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> Q x} \<subseteq> N" 

416 
using assms unfolding eventually_ae_filter by auto 

417 
show ?thesis 

418 
proof (rule AE_I) 

419 
from N measurable_emeasure_Pair1[OF `N \<in> sets (M1 \<Otimes>\<^isub>M M2)`] 

420 
show "emeasure M1 {x\<in>space M1. emeasure M2 (Pair x ` N) \<noteq> 0} = 0" 

421 
by (auto simp: M2.emeasure_pair_measure_alt positive_integral_0_iff emeasure_nonneg) 

422 
show "{x \<in> space M1. emeasure M2 (Pair x ` N) \<noteq> 0} \<in> sets M1" 

423 
by (intro borel_measurable_ereal_neq_const measurable_emeasure_Pair1 N) 

424 
{ fix x assume "x \<in> space M1" "emeasure M2 (Pair x ` N) = 0" 

425 
have "AE y in M2. Q (x, y)" 

426 
proof (rule AE_I) 

427 
show "emeasure M2 (Pair x ` N) = 0" by fact 

428 
show "Pair x ` N \<in> sets M2" using N(1) by (rule sets_Pair1) 

429 
show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x ` N" 

430 
using N `x \<in> space M1` unfolding space_pair_measure by auto 

431 
qed } 

432 
then show "{x \<in> space M1. \<not> (AE y in M2. Q (x, y))} \<subseteq> {x \<in> space M1. emeasure M2 (Pair x ` N) \<noteq> 0}" 

433 
by auto 

434 
qed 

435 
qed 

436 

437 
lemma (in pair_sigma_finite) AE_pair_measure: 

438 
assumes "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)" 

439 
assumes ae: "AE x in M1. AE y in M2. P (x, y)" 

440 
shows "AE x in M1 \<Otimes>\<^isub>M M2. P x" 

441 
proof (subst AE_iff_measurable[OF _ refl]) 

442 
show "{x\<in>space (M1 \<Otimes>\<^isub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^isub>M M2)" 

50244
de72bbe42190
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immler
parents:
50104
diff
changeset

443 
by (rule sets.sets_Collect) fact 
49776  444 
then have "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} = 
445 
(\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)" 

446 
by (simp add: M2.emeasure_pair_measure) 

447 
also have "\<dots> = (\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. 0 \<partial>M2 \<partial>M1)" 

448 
using ae 

449 
apply (safe intro!: positive_integral_cong_AE) 

450 
apply (intro AE_I2) 

451 
apply (safe intro!: positive_integral_cong_AE) 

452 
apply auto 

453 
done 

454 
finally show "emeasure (M1 \<Otimes>\<^isub>M M2) {x \<in> space (M1 \<Otimes>\<^isub>M M2). \<not> P x} = 0" by simp 

455 
qed 

456 

457 
lemma (in pair_sigma_finite) AE_pair_iff: 

458 
"{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2) \<Longrightarrow> 

459 
(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE x in (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x))" 

460 
using AE_pair[of "\<lambda>x. P (fst x) (snd x)"] AE_pair_measure[of "\<lambda>x. P (fst x) (snd x)"] by auto 

461 

462 
lemma (in pair_sigma_finite) AE_commute: 

463 
assumes P: "{x\<in>space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^isub>M M2)" 

464 
shows "(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE y in M2. AE x in M1. P x y)" 

465 
proof  

466 
interpret Q: pair_sigma_finite M2 M1 .. 

467 
have [simp]: "\<And>x. (fst (case x of (x, y) \<Rightarrow> (y, x))) = snd x" "\<And>x. (snd (case x of (x, y) \<Rightarrow> (y, x))) = fst x" 

468 
by auto 

469 
have "{x \<in> space (M2 \<Otimes>\<^isub>M M1). P (snd x) (fst x)} = 

470 
(\<lambda>(x, y). (y, x)) ` {x \<in> space (M1 \<Otimes>\<^isub>M M2). P (fst x) (snd x)} \<inter> space (M2 \<Otimes>\<^isub>M M1)" 

471 
by (auto simp: space_pair_measure) 

472 
also have "\<dots> \<in> sets (M2 \<Otimes>\<^isub>M M1)" 

473 
by (intro sets_pair_swap P) 

474 
finally show ?thesis 

475 
apply (subst AE_pair_iff[OF P]) 

476 
apply (subst distr_pair_swap) 

477 
apply (subst AE_distr_iff[OF measurable_pair_swap' P]) 

478 
apply (subst Q.AE_pair_iff) 

479 
apply simp_all 

480 
done 

40859  481 
qed 
482 

483 
section "Fubinis theorem" 

484 

49800  485 
lemma measurable_compose_Pair1: 
486 
"x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^isub>M M2) L \<Longrightarrow> (\<lambda>y. g (x, y)) \<in> measurable M2 L" 

50003  487 
by simp 
49800  488 

49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

489 
lemma (in sigma_finite_measure) borel_measurable_positive_integral_fst': 
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

490 
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)" "\<And>x. 0 \<le> f x" 
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

491 
shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1" 
49800  492 
using f proof induct 
493 
case (cong u v) 

49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

494 
then have "\<And>w x. w \<in> space M1 \<Longrightarrow> x \<in> space M \<Longrightarrow> u (w, x) = v (w, x)" 
49800  495 
by (auto simp: space_pair_measure) 
496 
show ?case 

497 
apply (subst measurable_cong) 

498 
apply (rule positive_integral_cong) 

499 
apply fact+ 

500 
done 

501 
next 

502 
case (set Q) 

503 
have [simp]: "\<And>x y. indicator Q (x, y) = indicator (Pair x ` Q) y" 

504 
by (auto simp: indicator_def) 

49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

505 
have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x ` Q) = \<integral>\<^isup>+ y. indicator Q (x, y) \<partial>M" 
49800  506 
by (simp add: sets_Pair1[OF set]) 
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

507 
from this measurable_emeasure_Pair[OF set] show ?case 
49800  508 
by (rule measurable_cong[THEN iffD1]) 
509 
qed (simp_all add: positive_integral_add positive_integral_cmult measurable_compose_Pair1 

510 
positive_integral_monotone_convergence_SUP incseq_def le_fun_def 

511 
cong: measurable_cong) 

512 

49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

513 
lemma (in sigma_finite_measure) positive_integral_fst: 
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

514 
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)" "\<And>x. 0 \<le> f x" 
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

515 
shows "(\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. f (x, y) \<partial>M \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M) f" (is "?I f = _") 
49800  516 
using f proof induct 
517 
case (cong u v) 

518 
moreover then have "?I u = ?I v" 

519 
by (intro positive_integral_cong) (auto simp: space_pair_measure) 

520 
ultimately show ?case 

521 
by (simp cong: positive_integral_cong) 

49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

522 
qed (simp_all add: emeasure_pair_measure positive_integral_cmult positive_integral_add 
49800  523 
positive_integral_monotone_convergence_SUP 
524 
measurable_compose_Pair1 positive_integral_positive 

49825
bb5db3d1d6dd
cleanup borel_measurable_positive_integral_(fstsnd)
hoelzl
parents:
49800
diff
changeset

525 
borel_measurable_positive_integral_fst' positive_integral_mono incseq_def le_fun_def 
49800  526 
cong: positive_integral_cong) 
40859  527 

49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

528 
lemma (in sigma_finite_measure) positive_integral_fst_measurable: 
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

529 
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M)" 
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

530 
shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M) \<in> borel_measurable M1" 
40859  531 
(is "?C f \<in> borel_measurable M1") 
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

532 
and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M) \<partial>M1) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M) f" 
49800  533 
using f 
49825
bb5db3d1d6dd
cleanup borel_measurable_positive_integral_(fstsnd)
hoelzl
parents:
49800
diff
changeset

534 
borel_measurable_positive_integral_fst'[of "\<lambda>x. max 0 (f x)"] 
49800  535 
positive_integral_fst[of "\<lambda>x. max 0 (f x)"] 
536 
unfolding positive_integral_max_0 by auto 

40859  537 

50003  538 
lemma (in sigma_finite_measure) borel_measurable_positive_integral[measurable (raw)]: 
539 
"split f \<in> borel_measurable (N \<Otimes>\<^isub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^isup>+ y. f x y \<partial>M) \<in> borel_measurable N" 

540 
using positive_integral_fst_measurable(1)[of "split f" N] by simp 

541 

542 
lemma (in sigma_finite_measure) borel_measurable_lebesgue_integral[measurable (raw)]: 

543 
"split f \<in> borel_measurable (N \<Otimes>\<^isub>M M) \<Longrightarrow> (\<lambda>x. \<integral> y. f x y \<partial>M) \<in> borel_measurable N" 

544 
by (simp add: lebesgue_integral_def) 

49825
bb5db3d1d6dd
cleanup borel_measurable_positive_integral_(fstsnd)
hoelzl
parents:
49800
diff
changeset

545 

47694  546 
lemma (in pair_sigma_finite) positive_integral_snd_measurable: 
547 
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" 

548 
shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f" 

41661  549 
proof  
47694  550 
note measurable_pair_swap[OF f] 
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

551 
from M1.positive_integral_fst_measurable[OF this] 
47694  552 
have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1))" 
40859  553 
by simp 
47694  554 
also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) f" 
555 
by (subst distr_pair_swap) 

556 
(auto simp: positive_integral_distr[OF measurable_pair_swap' f] intro!: positive_integral_cong) 

40859  557 
finally show ?thesis . 
558 
qed 

559 

560 
lemma (in pair_sigma_finite) Fubini: 

47694  561 
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" 
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

562 
shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1)" 
40859  563 
unfolding positive_integral_snd_measurable[OF assms] 
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

564 
unfolding M2.positive_integral_fst_measurable[OF assms] .. 
40859  565 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

566 
lemma (in pair_sigma_finite) integrable_product_swap: 
47694  567 
assumes "integrable (M1 \<Otimes>\<^isub>M M2) f" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

568 
shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))" 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

569 
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

570 
interpret Q: pair_sigma_finite M2 M1 by default 
41661  571 
have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff) 
572 
show ?thesis unfolding * 

47694  573 
by (rule integrable_distr[OF measurable_pair_swap']) 
574 
(simp add: distr_pair_swap[symmetric] assms) 

41661  575 
qed 
576 

577 
lemma (in pair_sigma_finite) integrable_product_swap_iff: 

47694  578 
"integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable (M1 \<Otimes>\<^isub>M M2) f" 
41661  579 
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

580 
interpret Q: pair_sigma_finite M2 M1 by default 
41661  581 
from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f] 
582 
show ?thesis by auto 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

583 
qed 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

584 

bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

585 
lemma (in pair_sigma_finite) integral_product_swap: 
47694  586 
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" 
587 
shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

588 
proof  
41661  589 
have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff) 
47694  590 
show ?thesis unfolding * 
591 
by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric]) 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

592 
qed 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

593 

bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

594 
lemma (in pair_sigma_finite) integrable_fst_measurable: 
47694  595 
assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f" 
596 
shows "AE x in M1. integrable M2 (\<lambda> y. f (x, y))" (is "?AE") 

597 
and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT") 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

598 
proof  
47694  599 
have f_borel: "f \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" 
600 
using f by auto 

46731  601 
let ?pf = "\<lambda>x. ereal (f x)" and ?nf = "\<lambda>x. ereal ( f x)" 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

602 
have 
47694  603 
borel: "?nf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)""?pf \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" and 
604 
int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?nf \<noteq> \<infinity>" "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) ?pf \<noteq> \<infinity>" 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

605 
using assms by auto 
43920  606 
have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>" 
607 
"(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal ( f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>" 

49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

608 
using borel[THEN M2.positive_integral_fst_measurable(1)] int 
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

609 
unfolding borel[THEN M2.positive_integral_fst_measurable(2)] by simp_all 
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

610 
with borel[THEN M2.positive_integral_fst_measurable(1)] 
43920  611 
have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>" 
612 
"AE x in M1. (\<integral>\<^isup>+y. ereal ( f (x, y)) \<partial>M2) \<noteq> \<infinity>" 

47694  613 
by (auto intro!: positive_integral_PInf_AE ) 
43920  614 
then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>" 
615 
"AE x in M1. \<bar>\<integral>\<^isup>+y. ereal ( f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>" 

47694  616 
by (auto simp: positive_integral_positive) 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

617 
from AE_pos show ?AE using assms 
47694  618 
by (simp add: measurable_Pair2[OF f_borel] integrable_def) 
43920  619 
{ fix f have "(\<integral>\<^isup>+ x.  \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)" 
47694  620 
using positive_integral_positive 
621 
by (intro positive_integral_cong_pos) (auto simp: ereal_uminus_le_reorder) 

43920  622 
then have "(\<integral>\<^isup>+ x.  \<integral>\<^isup>+ y. ereal (f x y) \<partial>M2 \<partial>M1) = 0" by simp } 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

623 
note this[simp] 
47694  624 
{ fix f assume borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M1 \<Otimes>\<^isub>M M2)" 
625 
and int: "integral\<^isup>P (M1 \<Otimes>\<^isub>M M2) (\<lambda>x. ereal (f x)) \<noteq> \<infinity>" 

626 
and AE: "AE x in M1. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<noteq> \<infinity>" 

43920  627 
have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f") 
41705  628 
proof (intro integrable_def[THEN iffD2] conjI) 
629 
show "?f \<in> borel_measurable M1" 

49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

630 
using borel by (auto intro!: M2.positive_integral_fst_measurable) 
43920  631 
have "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. ereal (f (x, y)) \<partial>M2) \<partial>M1)" 
47694  632 
using AE positive_integral_positive[of M2] 
633 
by (auto intro!: positive_integral_cong_AE simp: ereal_real) 

43920  634 
then show "(\<integral>\<^isup>+x. ereal (?f x) \<partial>M1) \<noteq> \<infinity>" 
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

635 
using M2.positive_integral_fst_measurable[OF borel] int by simp 
43920  636 
have "(\<integral>\<^isup>+x. ereal ( ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)" 
47694  637 
by (intro positive_integral_cong_pos) 
638 
(simp add: positive_integral_positive real_of_ereal_pos) 

43920  639 
then show "(\<integral>\<^isup>+x. ereal ( ?f x) \<partial>M1) \<noteq> \<infinity>" by simp 
41705  640 
qed } 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

641 
with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)] 
41705  642 
show ?INT 
47694  643 
unfolding lebesgue_integral_def[of "M1 \<Otimes>\<^isub>M M2"] lebesgue_integral_def[of M2] 
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigmafinite
hoelzl
parents:
49825
diff
changeset

644 
borel[THEN M2.positive_integral_fst_measurable(2), symmetric] 
47694  645 
using AE[THEN integral_real] 
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset

646 
by simp 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

647 
qed 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

648 

bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

649 
lemma (in pair_sigma_finite) integrable_snd_measurable: 
47694  650 
assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f" 
651 
shows "AE y in M2. integrable M1 (\<lambda>x. f (x, y))" (is "?AE") 

652 
and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L (M1 \<Otimes>\<^isub>M M2) f" (is "?INT") 

41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

653 
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

654 
interpret Q: pair_sigma_finite M2 M1 by default 
47694  655 
have Q_int: "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x, y). f (y, x))" 
41661  656 
using f unfolding integrable_product_swap_iff . 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

657 
show ?INT 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

658 
using Q.integrable_fst_measurable(2)[OF Q_int] 
47694  659 
using integral_product_swap[of f] f by auto 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

660 
show ?AE 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

661 
using Q.integrable_fst_measurable(1)[OF Q_int] 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

662 
by simp 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

663 
qed 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

664 

bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

665 
lemma (in pair_sigma_finite) Fubini_integral: 
47694  666 
assumes f: "integrable (M1 \<Otimes>\<^isub>M M2) f" 
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset

667 
shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)" 
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

668 
unfolding integrable_snd_measurable[OF assms] 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

669 
unfolding integrable_fst_measurable[OF assms] .. 
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset

670 

47694  671 
section {* Products on counting spaces, densities and distributions *} 
40859  672 

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

673 
lemma sigma_sets_pair_measure_generator_finite: 
38656  674 
assumes "finite A" and "finite B" 
47694  675 
shows "sigma_sets (A \<times> B) { a \<times> b  a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)" 
40859  676 
(is "sigma_sets ?prod ?sets = _") 
38656  677 
proof safe 
678 
have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product) 

679 
fix x assume subset: "x \<subseteq> A \<times> B" 

680 
hence "finite x" using fin by (rule finite_subset) 

40859  681 
from this subset show "x \<in> sigma_sets ?prod ?sets" 
38656  682 
proof (induct x) 
683 
case empty show ?case by (rule sigma_sets.Empty) 

684 
next 

685 
case (insert a x) 

47694  686 
hence "{a} \<in> sigma_sets ?prod ?sets" by auto 
38656  687 
moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto 
688 
ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un) 

689 
qed 

690 
next 

691 
fix x a b 

40859  692 
assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x" 
38656  693 
from sigma_sets_into_sp[OF _ this(1)] this(2) 
40859  694 
show "a \<in> A" and "b \<in> B" by auto 
35833  695 
qed 
696 

47694  697 
lemma pair_measure_count_space: 
698 
assumes A: "finite A" and B: "finite B" 

699 
shows "count_space A \<Otimes>\<^isub>M count_space B = count_space (A \<times> B)" (is "?P = ?C") 

700 
proof (rule measure_eqI) 

701 
interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact 

702 
interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact 

703 
interpret P: pair_sigma_finite "count_space A" "count_space B" by default 

704 
show eq: "sets ?P = sets ?C" 

705 
by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B) 

706 
fix X assume X: "X \<in> sets ?P" 

707 
with eq have X_subset: "X \<subseteq> A \<times> B" by simp 

708 
with A B have fin_Pair: "\<And>x. finite (Pair x ` X)" 

709 
by (intro finite_subset[OF _ B]) auto 

710 
have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B) 

711 
show "emeasure ?P X = emeasure ?C X" 

49776  712 
apply (subst B.emeasure_pair_measure_alt[OF X]) 
47694  713 
apply (subst emeasure_count_space) 
714 
using X_subset apply auto [] 

715 
apply (simp add: fin_Pair emeasure_count_space X_subset fin_X) 

716 
apply (subst positive_integral_count_space) 

717 
using A apply simp 

718 
apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric]) 

719 
apply (subst card_gt_0_iff) 

720 
apply (simp add: fin_Pair) 

721 
apply (subst card_SigmaI[symmetric]) 

722 
using A apply simp 

723 
using fin_Pair apply simp 

724 
using X_subset apply (auto intro!: arg_cong[where f=card]) 

725 
done 

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:
44890
diff
changeset

726 
qed 
35833  727 

47694  728 
lemma pair_measure_density: 
729 
assumes f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x" 

730 
assumes g: "g \<in> borel_measurable M2" "AE x in M2. 0 \<le> g x" 

50003  731 
assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)" 
47694  732 
shows "density M1 f \<Otimes>\<^isub>M density M2 g = density (M1 \<Otimes>\<^isub>M M2) (\<lambda>(x,y). f x * g y)" (is "?L = ?R") 
733 
proof (rule measure_eqI) 

734 
interpret M2: sigma_finite_measure M2 by fact 

735 
interpret D2: sigma_finite_measure "density M2 g" by fact 

736 

737 
fix A assume A: "A \<in> sets ?L" 

50003  738 
with f g have "(\<integral>\<^isup>+ x. f x * \<integral>\<^isup>+ y. g y * indicator A (x, y) \<partial>M2 \<partial>M1) = 
739 
(\<integral>\<^isup>+ x. \<integral>\<^isup>+ y. f x * g y * indicator A (x, y) \<partial>M2 \<partial>M1)" 

740 
by (intro positive_integral_cong_AE) 

741 
(auto simp add: positive_integral_cmult[symmetric] ac_simps) 

742 
with A f g show "emeasure ?L A = emeasure ?R A" 

743 
by (simp add: D2.emeasure_pair_measure emeasure_density positive_integral_density 

744 
M2.positive_integral_fst_measurable(2)[symmetric] 

745 
cong: positive_integral_cong) 

47694  746 
qed simp 
747 

748 
lemma sigma_finite_measure_distr: 

749 
assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N" 

750 
shows "sigma_finite_measure M" 

40859  751 
proof  
47694  752 
interpret sigma_finite_measure "distr M N f" by fact 
753 
from sigma_finite_disjoint guess A . note A = this 

754 
show ?thesis 

755 
proof (unfold_locales, intro conjI exI allI) 

756 
show "range (\<lambda>i. f ` A i \<inter> space M) \<subseteq> sets M" 

50003  757 
using A f by auto 
47694  758 
show "(\<Union>i. f ` A i \<inter> space M) = space M" 
759 
using A(1) A(2)[symmetric] f by (auto simp: measurable_def Pi_def) 

760 
fix i show "emeasure M (f ` A i \<inter> space M) \<noteq> \<infinity>" 

761 
using f A(1,2) A(3)[of i] by (simp add: emeasure_distr subset_eq) 

762 
qed 

38656  763 
qed 
764 

47694  765 
lemma pair_measure_distr: 
766 
assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T" 

50003  767 
assumes "sigma_finite_measure (distr N T g)" 
47694  768 
shows "distr M S f \<Otimes>\<^isub>M distr N T g = distr (M \<Otimes>\<^isub>M N) (S \<Otimes>\<^isub>M T) (\<lambda>(x, y). (f x, g y))" (is "?P = ?D") 
769 
proof (rule measure_eqI) 

770 
interpret T: sigma_finite_measure "distr N T g" by fact 

771 
interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+ 

50003  772 

47694  773 
fix A assume A: "A \<in> sets ?P" 
50003  774 
with f g show "emeasure ?P A = emeasure ?D A" 
775 
by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr 

776 
T.emeasure_pair_measure_alt positive_integral_distr 

777 
intro!: positive_integral_cong arg_cong[where f="emeasure N"]) 

778 
qed simp 

39097  779 

50104  780 
lemma pair_measure_eqI: 
781 
assumes "sigma_finite_measure M1" "sigma_finite_measure M2" 

782 
assumes sets: "sets (M1 \<Otimes>\<^isub>M M2) = sets M" 

783 
assumes emeasure: "\<And>A B. A \<in> sets M1 \<Longrightarrow> B \<in> sets M2 \<Longrightarrow> emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)" 

784 
shows "M1 \<Otimes>\<^isub>M M2 = M" 

785 
proof  

786 
interpret M1: sigma_finite_measure M1 by fact 

787 
interpret M2: sigma_finite_measure M2 by fact 

788 
interpret pair_sigma_finite M1 M2 by default 

789 
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this 

790 
let ?E = "{a \<times> b a b. a \<in> sets M1 \<and> b \<in> sets M2}" 

791 
let ?P = "M1 \<Otimes>\<^isub>M M2" 

792 
show ?thesis 

793 
proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) 

794 
show "?E \<subseteq> Pow (space ?P)" 

50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset

795 
using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure) 
50104  796 
show "sets ?P = sigma_sets (space ?P) ?E" 
797 
by (simp add: sets_pair_measure space_pair_measure) 

798 
then show "sets M = sigma_sets (space ?P) ?E" 

799 
using sets[symmetric] by simp 

800 
next 

801 
show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>" 

802 
using F by (auto simp: space_pair_measure) 

803 
next 

804 
fix X assume "X \<in> ?E" 

805 
then obtain A B where X[simp]: "X = A \<times> B" and A: "A \<in> sets M1" and B: "B \<in> sets M2" by auto 

806 
then have "emeasure ?P X = emeasure M1 A * emeasure M2 B" 

807 
by (simp add: M2.emeasure_pair_measure_Times) 

808 
also have "\<dots> = emeasure M (A \<times> B)" 

809 
using A B emeasure by auto 

810 
finally show "emeasure ?P X = emeasure M X" 

811 
by simp 

812 
qed 

813 
qed 

814 

40859  815 
end 