author | hoelzl |
Mon, 19 May 2014 13:44:13 +0200 | |
changeset 56994 | 8d5e5ec1cac3 |
parent 56993 | e5366291d6aa |
child 56996 | 891e992e510f |
permissions | -rw-r--r-- |
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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 |
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imports Nonnegative_Lebesgue_Integration |
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begin |
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lemma Pair_vimage_times[simp]: "Pair x -` (A \<times> B) = (if x \<in> A then B else {})" |
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by auto |
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lemma rev_Pair_vimage_times[simp]: "(\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})" |
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by auto |
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subsection "Binary products" |
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definition pair_measure (infixr "\<Otimes>\<^sub>M" 80) where |
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"A \<Otimes>\<^sub>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>\<^sup>+x. (\<integral>\<^sup>+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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|
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lemma space_pair_measure: |
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"space (A \<Otimes>\<^sub>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>\<^sub>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>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')" |
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unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq) |
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lemma pair_measureI[intro, simp, measurable]: |
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"x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)" |
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by (auto simp: sets_pair_measure) |
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lemma measurable_pair_measureI: |
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assumes 1: "f \<in> space M \<rightarrow> space M1 \<times> space M2" |
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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>\<^sub>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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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" |
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unfolding split_beta' . |
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lemma measurable_Pair[measurable (raw)]: |
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assumes f: "f \<in> measurable M M1" and g: "g \<in> measurable M M2" |
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shows "(\<lambda>x. (f x, g x)) \<in> measurable M (M1 \<Otimes>\<^sub>M M2)" |
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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" |
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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" . |
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qed |
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lemma measurable_Pair_compose_split[measurable_dest]: |
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assumes f: "split f \<in> measurable (M1 \<Otimes>\<^sub>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 |
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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>\<^sub>M M2)" |
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using measurable_Pair[OF assms] by simp |
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lemma measurable_fst[intro!, simp, measurable]: "fst \<in> measurable (M1 \<Otimes>\<^sub>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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lemma measurable_snd[intro!, simp, measurable]: "snd \<in> measurable (M1 \<Otimes>\<^sub>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) |
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lemma |
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assumes f[measurable]: "f \<in> measurable M (N \<Otimes>\<^sub>M P)" |
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shows measurable_fst': "(\<lambda>x. fst (f x)) \<in> measurable M N" |
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and measurable_snd': "(\<lambda>x. snd (f x)) \<in> measurable M P" |
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by simp_all |
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lemma |
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assumes f[measurable]: "f \<in> measurable M N" |
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shows measurable_fst'': "(\<lambda>x. f (fst x)) \<in> measurable (M \<Otimes>\<^sub>M P) N" |
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and measurable_snd'': "(\<lambda>x. f (snd x)) \<in> measurable (P \<Otimes>\<^sub>M M) N" |
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by simp_all |
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lemma measurable_pair_iff: |
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"f \<in> measurable M (M1 \<Otimes>\<^sub>M M2) \<longleftrightarrow> (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2" |
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by (auto intro: measurable_pair[of f M M1 M2]) |
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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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lemma measurable_pair_swap': "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>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>\<^sub>M M2) M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^sub>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>\<^sub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" |
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by (auto dest: measurable_pair_swap) |
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lemma measurable_Pair1': "x \<in> space M1 \<Longrightarrow> Pair x \<in> measurable M2 (M1 \<Otimes>\<^sub>M M2)" |
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by simp |
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lemma sets_Pair1[measurable (raw)]: |
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assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "Pair x -` A \<in> sets M2" |
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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) |
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finally show ?thesis . |
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qed |
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lemma measurable_Pair2': "y \<in> space M2 \<Longrightarrow> (\<lambda>x. (x, y)) \<in> measurable M1 (M1 \<Otimes>\<^sub>M M2)" |
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by (auto intro!: measurable_Pair) |
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lemma sets_Pair2: assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>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) |
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finally show ?thesis . |
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qed |
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lemma measurable_Pair2: |
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assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>M M2) M" and x: "x \<in> space M1" |
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shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M" |
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using measurable_comp[OF measurable_Pair1' f, OF x] |
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by (simp add: comp_def) |
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lemma measurable_Pair1: |
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assumes f: "f \<in> measurable (M1 \<Otimes>\<^sub>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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40859 | 156 |
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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) |
40859 | 160 |
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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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lemma (in finite_measure) finite_measure_cut_measurable: |
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assumes [measurable]: "Q \<in> sets (N \<Otimes>\<^sub>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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179 |
case (union F) |
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180 |
then have "\<And>x. emeasure M (Pair x -` (\<Union>i. F i)) = (\<Sum>i. ?s (F i) x)" |
50003 | 181 |
by (simp add: suminf_emeasure disjoint_family_vimageI subset_eq vimage_UN sets_pair_measure[symmetric]) |
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182 |
with union show ?case |
50003 | 183 |
unfolding sets_pair_measure[symmetric] by simp |
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184 |
qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If) |
49776 | 185 |
|
186 |
lemma (in sigma_finite_measure) measurable_emeasure_Pair: |
|
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187 |
assumes Q: "Q \<in> sets (N \<Otimes>\<^sub>M M)" shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N" (is "?s Q \<in> _") |
49776 | 188 |
proof - |
189 |
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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194 |
using F sets.sets_into_space by auto |
49776 | 195 |
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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204 |
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" |
|
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212 |
using F `Q \<in> sets (N \<Otimes>\<^sub>M M)` by (auto intro!: sets_Pair1) |
49776 | 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" |
|
53015
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|
218 |
using F sets.sets_into_space[OF `Q \<in> sets (N \<Otimes>\<^sub>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 } |
|
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222 |
ultimately show ?thesis using `Q \<in> sets (N \<Otimes>\<^sub>M M)` F_sets |
49776 | 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" |
|
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228 |
assumes A: "{x\<in>space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M)" |
50003 | 229 |
shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N" |
230 |
proof - |
|
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|
231 |
from space have "\<And>x. x \<in> space N \<Longrightarrow> Pair x -` {x \<in> space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} = A x" |
50003 | 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: |
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238 |
assumes "X \<in> sets (N \<Otimes>\<^sub>M M)" |
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|
239 |
shows "emeasure (N \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>M \<partial>N)" (is "_ = ?\<mu> X") |
49776 | 240 |
proof (rule emeasure_measure_of[OF pair_measure_def]) |
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|
241 |
show "positive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>" |
49776 | 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) |
|
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|
245 |
show "countably_additive (sets (N \<Otimes>\<^sub>M M)) ?\<mu>" |
49776 | 246 |
proof (rule countably_additiveI) |
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|
247 |
fix F :: "nat \<Rightarrow> ('b \<times> 'a) set" assume F: "range F \<subseteq> sets (N \<Otimes>\<^sub>M M)" "disjoint_family F" |
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|
248 |
from F have *: "\<And>i. F i \<in> sets (N \<Otimes>\<^sub>M M)" "(\<Union>i. F i) \<in> sets (N \<Otimes>\<^sub>M M)" by auto |
49776 | 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: |
|
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|
264 |
assumes X: "X \<in> sets (N \<Otimes>\<^sub>M M)" |
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|
265 |
shows "emeasure (N \<Otimes>\<^sub>M M) X = (\<integral>\<^sup>+x. emeasure M (Pair x -` X) \<partial>N)" |
49776 | 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" |
|
53015
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|
275 |
shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B" |
49776 | 276 |
proof - |
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|
277 |
have "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = (\<integral>\<^sup>+x. emeasure M B * indicator A x \<partial>N)" |
49776 | 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: |
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|
291 |
"Q \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> (\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable M1" |
49776 | 292 |
using M2.measurable_emeasure_Pair . |
40859 | 293 |
|
47694 | 294 |
lemma (in pair_sigma_finite) measurable_emeasure_Pair2: |
53015
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|
295 |
assumes Q: "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "(\<lambda>y. emeasure M1 ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2" |
40859 | 296 |
proof - |
53015
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|
297 |
have "(\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>M M1)" |
47694 | 298 |
using Q measurable_pair_swap' by (auto intro: measurable_sets) |
49776 | 299 |
note M1.measurable_emeasure_Pair[OF this] |
53015
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|
300 |
moreover have "\<And>y. Pair y -` ((\<lambda>(x, y). (y, x)) -` Q \<inter> space (M2 \<Otimes>\<^sub>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 |
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|
303 |
qed |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
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changeset
|
304 |
|
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3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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41661
diff
changeset
|
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> |
|
53015
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changeset
|
308 |
(\<forall>i. emeasure (M1 \<Otimes>\<^sub>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
cdf7693bbe08
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diff
changeset
|
324 |
using `incseq F1` `incseq F2` unfolding incseq_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset
|
325 |
by (force split: split_max)+ |
40859 | 326 |
then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)" |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
53374
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changeset
|
327 |
by (intro SigmaI) (auto simp add: max.commute) |
40859 | 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
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
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
hoelzl
parents:
41831
diff
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"] |
53015
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changeset
|
339 |
show "emeasure (M1 \<Otimes>\<^sub>M M2) (F1 i \<times> F2 i) \<noteq> \<infinity>" |
47694 | 340 |
by (auto simp add: emeasure_pair_measure_Times) |
341 |
qed |
|
342 |
qed |
|
343 |
||
53015
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changeset
|
344 |
sublocale pair_sigma_finite \<subseteq> P: sigma_finite_measure "M1 \<Otimes>\<^sub>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 |
|
53015
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changeset
|
347 |
show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets (M1 \<Otimes>\<^sub>M M2) \<and> (\<Union>i. F i) = space (M1 \<Otimes>\<^sub>M M2) \<and> (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)" |
47694 | 348 |
proof (rule exI[of _ F], intro conjI) |
53015
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changeset
|
349 |
show "range F \<subseteq> sets (M1 \<Otimes>\<^sub>M M2)" using F by (auto simp: pair_measure_def) |
a1119cf551e8
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diff
changeset
|
350 |
show "(\<Union>i. F i) = space (M1 \<Otimes>\<^sub>M M2)" |
47694 | 351 |
using F by (auto simp: space_pair_measure) |
53015
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changeset
|
352 |
show "\<forall>i. emeasure (M1 \<Otimes>\<^sub>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" |
|
53015
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changeset
|
358 |
shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)" |
47694 | 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: |
53015
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wenzelm
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changeset
|
367 |
assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
368 |
shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>M M1) \<in> sets (M2 \<Otimes>\<^sub>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: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
372 |
"M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))" (is "?P = ?D") |
40859 | 373 |
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
|
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
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
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 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
390 |
have "(\<lambda>(y, x). (x, y)) -` X \<inter> space (M2 \<Otimes>\<^sub>M M1) = B \<times> A" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
391 |
using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
392 |
with A B show "emeasure (M1 \<Otimes>\<^sub>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: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
399 |
assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
400 |
shows "emeasure (M1 \<Otimes>\<^sub>M M2) A = (\<integral>\<^sup>+y. emeasure M1 ((\<lambda>x. (x, y)) -` A) \<partial>M2)" |
47694 | 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 - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
403 |
have [simp]: "\<And>y. (Pair y -` ((\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^sub>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: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
412 |
assumes "AE x in (M1 \<Otimes>\<^sub>M M2). Q x" |
49776 | 413 |
shows "AE x in M1. (AE y in M2. Q (x, y))" |
414 |
proof - |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
415 |
obtain N where N: "N \<in> sets (M1 \<Otimes>\<^sub>M M2)" "emeasure (M1 \<Otimes>\<^sub>M M2) N = 0" "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> Q x} \<subseteq> N" |
49776 | 416 |
using assms unfolding eventually_ae_filter by auto |
417 |
show ?thesis |
|
418 |
proof (rule AE_I) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
419 |
from N measurable_emeasure_Pair1[OF `N \<in> sets (M1 \<Otimes>\<^sub>M M2)`] |
49776 | 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: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
438 |
assumes "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
49776 | 439 |
assumes ae: "AE x in M1. AE y in M2. P (x, y)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
440 |
shows "AE x in M1 \<Otimes>\<^sub>M M2. P x" |
49776 | 441 |
proof (subst AE_iff_measurable[OF _ refl]) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
442 |
show "{x\<in>space (M1 \<Otimes>\<^sub>M M2). \<not> P x} \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
443 |
by (rule sets.sets_Collect) fact |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
444 |
then have "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
445 |
(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} (x, y) \<partial>M2 \<partial>M1)" |
49776 | 446 |
by (simp add: M2.emeasure_pair_measure) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
447 |
also have "\<dots> = (\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. 0 \<partial>M2 \<partial>M1)" |
49776 | 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 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
454 |
finally show "emeasure (M1 \<Otimes>\<^sub>M M2) {x \<in> space (M1 \<Otimes>\<^sub>M M2). \<not> P x} = 0" by simp |
49776 | 455 |
qed |
456 |
||
457 |
lemma (in pair_sigma_finite) AE_pair_iff: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
458 |
"{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2) \<Longrightarrow> |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
459 |
(AE x in M1. AE y in M2. P x y) \<longleftrightarrow> (AE x in (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x))" |
49776 | 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: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
463 |
assumes P: "{x\<in>space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)" |
49776 | 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 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
469 |
have "{x \<in> space (M2 \<Otimes>\<^sub>M M1). P (snd x) (fst x)} = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
470 |
(\<lambda>(x, y). (y, x)) -` {x \<in> space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<inter> space (M2 \<Otimes>\<^sub>M M1)" |
49776 | 471 |
by (auto simp: space_pair_measure) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
472 |
also have "\<dots> \<in> sets (M2 \<Otimes>\<^sub>M M1)" |
49776 | 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 |
||
56994 | 483 |
subsection "Fubinis theorem" |
40859 | 484 |
|
49800 | 485 |
lemma measurable_compose_Pair1: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
486 |
"x \<in> space M1 \<Longrightarrow> g \<in> measurable (M1 \<Otimes>\<^sub>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 sigma-finite
hoelzl
parents:
49825
diff
changeset
|
489 |
lemma (in sigma_finite_measure) borel_measurable_positive_integral_fst': |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
490 |
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" "\<And>x. 0 \<le> f x" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
491 |
shows "(\<lambda>x. \<integral>\<^sup>+ 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 sigma-finite
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) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
505 |
have "\<And>x. x \<in> space M1 \<Longrightarrow> emeasure M (Pair x -` Q) = \<integral>\<^sup>+ 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 sigma-finite
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 |
||
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54863
diff
changeset
|
513 |
lemma (in sigma_finite_measure) positive_integral_fst': |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
514 |
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" "\<And>x. 0 \<le> f x" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
515 |
shows "(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M \<partial>M1) = integral\<^sup>P (M1 \<Otimes>\<^sub>M M) f" (is "?I f = _") |
49800 | 516 |
using f proof induct |
517 |
case (cong u v) |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
518 |
then have "?I u = ?I v" |
49800 | 519 |
by (intro positive_integral_cong) (auto simp: space_pair_measure) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
520 |
with cong show ?case |
49800 | 521 |
by (simp cong: positive_integral_cong) |
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigma-finite
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_(fst|snd)
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 |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54863
diff
changeset
|
528 |
lemma (in sigma_finite_measure) positive_integral_fst: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
529 |
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54863
diff
changeset
|
530 |
shows "(\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M) \<partial>M1) = integral\<^sup>P (M1 \<Otimes>\<^sub>M M) f" |
49800 | 531 |
using f |
49825
bb5db3d1d6dd
cleanup borel_measurable_positive_integral_(fst|snd)
hoelzl
parents:
49800
diff
changeset
|
532 |
borel_measurable_positive_integral_fst'[of "\<lambda>x. max 0 (f x)"] |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54863
diff
changeset
|
533 |
positive_integral_fst'[of "\<lambda>x. max 0 (f x)"] |
49800 | 534 |
unfolding positive_integral_max_0 by auto |
40859 | 535 |
|
50003 | 536 |
lemma (in sigma_finite_measure) borel_measurable_positive_integral[measurable (raw)]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
537 |
"split f \<in> borel_measurable (N \<Otimes>\<^sub>M M) \<Longrightarrow> (\<lambda>x. \<integral>\<^sup>+ y. f x y \<partial>M) \<in> borel_measurable N" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54863
diff
changeset
|
538 |
using borel_measurable_positive_integral_fst'[of "\<lambda>x. max 0 (split f x)" N] |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54863
diff
changeset
|
539 |
by (simp add: positive_integral_max_0) |
50003 | 540 |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54863
diff
changeset
|
541 |
lemma (in pair_sigma_finite) positive_integral_snd: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
542 |
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
543 |
shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^sup>P (M1 \<Otimes>\<^sub>M M2) f" |
41661 | 544 |
proof - |
47694 | 545 |
note measurable_pair_swap[OF f] |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54863
diff
changeset
|
546 |
from M1.positive_integral_fst[OF this] |
53015
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standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
547 |
have "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1))" |
40859 | 548 |
by simp |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
549 |
also have "(\<integral>\<^sup>+ (x, y). f (y, x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>P (M1 \<Otimes>\<^sub>M M2) f" |
47694 | 550 |
by (subst distr_pair_swap) |
551 |
(auto simp: positive_integral_distr[OF measurable_pair_swap' f] intro!: positive_integral_cong) |
|
40859 | 552 |
finally show ?thesis . |
553 |
qed |
|
554 |
||
555 |
lemma (in pair_sigma_finite) Fubini: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
556 |
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
557 |
shows "(\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x, y) \<partial>M2) \<partial>M1)" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54863
diff
changeset
|
558 |
unfolding positive_integral_snd[OF assms] M2.positive_integral_fst[OF assms] .. |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
559 |
|
56994 | 560 |
subsection {* Products on counting spaces, densities and distributions *} |
40859 | 561 |
|
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 |
lemma sigma_sets_pair_measure_generator_finite: |
38656 | 563 |
assumes "finite A" and "finite B" |
47694 | 564 |
shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)" |
40859 | 565 |
(is "sigma_sets ?prod ?sets = _") |
38656 | 566 |
proof safe |
567 |
have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product) |
|
568 |
fix x assume subset: "x \<subseteq> A \<times> B" |
|
569 |
hence "finite x" using fin by (rule finite_subset) |
|
40859 | 570 |
from this subset show "x \<in> sigma_sets ?prod ?sets" |
38656 | 571 |
proof (induct x) |
572 |
case empty show ?case by (rule sigma_sets.Empty) |
|
573 |
next |
|
574 |
case (insert a x) |
|
47694 | 575 |
hence "{a} \<in> sigma_sets ?prod ?sets" by auto |
38656 | 576 |
moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto |
577 |
ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un) |
|
578 |
qed |
|
579 |
next |
|
580 |
fix x a b |
|
40859 | 581 |
assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x" |
38656 | 582 |
from sigma_sets_into_sp[OF _ this(1)] this(2) |
40859 | 583 |
show "a \<in> A" and "b \<in> B" by auto |
35833 | 584 |
qed |
585 |
||
47694 | 586 |
lemma pair_measure_count_space: |
587 |
assumes A: "finite A" and B: "finite B" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
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changeset
|
588 |
shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)" (is "?P = ?C") |
47694 | 589 |
proof (rule measure_eqI) |
590 |
interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact |
|
591 |
interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact |
|
592 |
interpret P: pair_sigma_finite "count_space A" "count_space B" by default |
|
593 |
show eq: "sets ?P = sets ?C" |
|
594 |
by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B) |
|
595 |
fix X assume X: "X \<in> sets ?P" |
|
596 |
with eq have X_subset: "X \<subseteq> A \<times> B" by simp |
|
597 |
with A B have fin_Pair: "\<And>x. finite (Pair x -` X)" |
|
598 |
by (intro finite_subset[OF _ B]) auto |
|
599 |
have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B) |
|
600 |
show "emeasure ?P X = emeasure ?C X" |
|
49776 | 601 |
apply (subst B.emeasure_pair_measure_alt[OF X]) |
47694 | 602 |
apply (subst emeasure_count_space) |
603 |
using X_subset apply auto [] |
|
604 |
apply (simp add: fin_Pair emeasure_count_space X_subset fin_X) |
|
605 |
apply (subst positive_integral_count_space) |
|
606 |
using A apply simp |
|
607 |
apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric]) |
|
608 |
apply (subst card_gt_0_iff) |
|
609 |
apply (simp add: fin_Pair) |
|
610 |
apply (subst card_SigmaI[symmetric]) |
|
611 |
using A apply simp |
|
612 |
using fin_Pair apply simp |
|
613 |
using X_subset apply (auto intro!: arg_cong[where f=card]) |
|
614 |
done |
|
45777
c36637603821
remove unnecessary sublocale instantiations in HOL-Probability (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
|
615 |
qed |
35833 | 616 |
|
47694 | 617 |
lemma pair_measure_density: |
618 |
assumes f: "f \<in> borel_measurable M1" "AE x in M1. 0 \<le> f x" |
|
619 |
assumes g: "g \<in> borel_measurable M2" "AE x in M2. 0 \<le> g x" |
|
50003 | 620 |
assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
621 |
shows "density M1 f \<Otimes>\<^sub>M density M2 g = density (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x,y). f x * g y)" (is "?L = ?R") |
47694 | 622 |
proof (rule measure_eqI) |
623 |
interpret M2: sigma_finite_measure M2 by fact |
|
624 |
interpret D2: sigma_finite_measure "density M2 g" by fact |
|
625 |
||
626 |
fix A assume A: "A \<in> sets ?L" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
627 |
with f g have "(\<integral>\<^sup>+ x. f x * \<integral>\<^sup>+ y. g y * indicator A (x, y) \<partial>M2 \<partial>M1) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
628 |
(\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f x * g y * indicator A (x, y) \<partial>M2 \<partial>M1)" |
50003 | 629 |
by (intro positive_integral_cong_AE) |
630 |
(auto simp add: positive_integral_cmult[symmetric] ac_simps) |
|
631 |
with A f g show "emeasure ?L A = emeasure ?R A" |
|
632 |
by (simp add: D2.emeasure_pair_measure emeasure_density positive_integral_density |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
54863
diff
changeset
|
633 |
M2.positive_integral_fst[symmetric] |
50003 | 634 |
cong: positive_integral_cong) |
47694 | 635 |
qed simp |
636 |
||
637 |
lemma sigma_finite_measure_distr: |
|
638 |
assumes "sigma_finite_measure (distr M N f)" and f: "f \<in> measurable M N" |
|
639 |
shows "sigma_finite_measure M" |
|
40859 | 640 |
proof - |
47694 | 641 |
interpret sigma_finite_measure "distr M N f" by fact |
642 |
from sigma_finite_disjoint guess A . note A = this |
|
643 |
show ?thesis |
|
644 |
proof (unfold_locales, intro conjI exI allI) |
|
645 |
show "range (\<lambda>i. f -` A i \<inter> space M) \<subseteq> sets M" |
|
50003 | 646 |
using A f by auto |
47694 | 647 |
show "(\<Union>i. f -` A i \<inter> space M) = space M" |
648 |
using A(1) A(2)[symmetric] f by (auto simp: measurable_def Pi_def) |
|
649 |
fix i show "emeasure M (f -` A i \<inter> space M) \<noteq> \<infinity>" |
|
650 |
using f A(1,2) A(3)[of i] by (simp add: emeasure_distr subset_eq) |
|
651 |
qed |
|
38656 | 652 |
qed |
653 |
||
47694 | 654 |
lemma pair_measure_distr: |
655 |
assumes f: "f \<in> measurable M S" and g: "g \<in> measurable N T" |
|
50003 | 656 |
assumes "sigma_finite_measure (distr N T g)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
657 |
shows "distr M S f \<Otimes>\<^sub>M distr N T g = distr (M \<Otimes>\<^sub>M N) (S \<Otimes>\<^sub>M T) (\<lambda>(x, y). (f x, g y))" (is "?P = ?D") |
47694 | 658 |
proof (rule measure_eqI) |
659 |
interpret T: sigma_finite_measure "distr N T g" by fact |
|
660 |
interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+ |
|
50003 | 661 |
|
47694 | 662 |
fix A assume A: "A \<in> sets ?P" |
50003 | 663 |
with f g show "emeasure ?P A = emeasure ?D A" |
664 |
by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr |
|
665 |
T.emeasure_pair_measure_alt positive_integral_distr |
|
666 |
intro!: positive_integral_cong arg_cong[where f="emeasure N"]) |
|
667 |
qed simp |
|
39097 | 668 |
|
50104 | 669 |
lemma pair_measure_eqI: |
670 |
assumes "sigma_finite_measure M1" "sigma_finite_measure M2" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
671 |
assumes sets: "sets (M1 \<Otimes>\<^sub>M M2) = sets M" |
50104 | 672 |
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)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
673 |
shows "M1 \<Otimes>\<^sub>M M2 = M" |
50104 | 674 |
proof - |
675 |
interpret M1: sigma_finite_measure M1 by fact |
|
676 |
interpret M2: sigma_finite_measure M2 by fact |
|
677 |
interpret pair_sigma_finite M1 M2 by default |
|
678 |
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this |
|
679 |
let ?E = "{a \<times> b |a b. a \<in> sets M1 \<and> b \<in> sets M2}" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50244
diff
changeset
|
680 |
let ?P = "M1 \<Otimes>\<^sub>M M2" |
50104 | 681 |
show ?thesis |
682 |
proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]]) |
|
683 |
show "?E \<subseteq> Pow (space ?P)" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
684 |
using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure) |
50104 | 685 |
show "sets ?P = sigma_sets (space ?P) ?E" |
686 |
by (simp add: sets_pair_measure space_pair_measure) |
|
687 |
then show "sets M = sigma_sets (space ?P) ?E" |
|
688 |
using sets[symmetric] by simp |
|
689 |
next |
|
690 |
show "range F \<subseteq> ?E" "(\<Union>i. F i) = space ?P" "\<And>i. emeasure ?P (F i) \<noteq> \<infinity>" |
|
691 |
using F by (auto simp: space_pair_measure) |
|
692 |
next |
|
693 |
fix X assume "X \<in> ?E" |
|
694 |
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 |
|
695 |
then have "emeasure ?P X = emeasure M1 A * emeasure M2 B" |
|
696 |
by (simp add: M2.emeasure_pair_measure_Times) |
|
697 |
also have "\<dots> = emeasure M (A \<times> B)" |
|
698 |
using A B emeasure by auto |
|
699 |
finally show "emeasure ?P X = emeasure M X" |
|
700 |
by simp |
|
701 |
qed |
|
702 |
qed |
|
703 |
||
40859 | 704 |
end |