author | hoelzl |
Fri, 02 Nov 2012 14:00:39 +0100 | |
changeset 49999 | dfb63b9b8908 |
parent 49784 | 5e5b2da42a69 |
child 50003 | 8c213922ed49 |
permissions | -rw-r--r-- |
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(* Title: HOL/Probability/Finite_Product_Measure.thy |
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Author: Johannes Hölzl, TU München |
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*) |
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header {*Finite product measures*} |
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theory Finite_Product_Measure |
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imports Binary_Product_Measure |
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begin |
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lemma split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)" |
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by auto |
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abbreviation |
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"Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A" |
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syntax |
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"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3PIE _:_./ _)" 10) |
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|
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syntax (xsymbols) |
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"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>\<^isub>E _\<in>_./ _)" 10) |
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|
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syntax (HTML output) |
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"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>\<^isub>E _\<in>_./ _)" 10) |
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|
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translations |
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"PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)" |
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abbreviation |
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funcset_extensional :: "['a set, 'b set] => ('a => 'b) set" |
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(infixr "->\<^isub>E" 60) where |
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"A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)" |
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notation (xsymbols) |
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funcset_extensional (infixr "\<rightarrow>\<^isub>E" 60) |
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lemma extensional_insert[intro, simp]: |
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assumes "a \<in> extensional (insert i I)" |
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shows "a(i := b) \<in> extensional (insert i I)" |
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using assms unfolding extensional_def by auto |
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lemma extensional_Int[simp]: |
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"extensional I \<inter> extensional I' = extensional (I \<inter> I')" |
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unfolding extensional_def by auto |
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definition |
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"merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)" |
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lemma merge_apply[simp]: |
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"I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i" |
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"I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i" |
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"J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i" |
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"J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i" |
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"i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I J (x, y) i = undefined" |
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unfolding merge_def by auto |
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lemma merge_commute: |
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"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) = merge J I (y, x)" |
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by (auto simp: merge_def intro!: ext) |
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lemma Pi_cancel_merge_range[simp]: |
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"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A" |
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"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A" |
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"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A" |
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"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A" |
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by (auto simp: Pi_def) |
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lemma Pi_cancel_merge[simp]: |
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"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
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"J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
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"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" |
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"J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" |
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by (auto simp: Pi_def) |
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lemma extensional_merge[simp]: "merge I J (x, y) \<in> extensional (I \<union> J)" |
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by (auto simp: extensional_def) |
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lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A" |
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by (auto simp: restrict_def Pi_def) |
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lemma restrict_merge[simp]: |
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"I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I" |
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"I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J" |
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"J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I" |
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"J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J" |
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by (auto simp: restrict_def) |
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lemma extensional_insert_undefined[intro, simp]: |
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assumes "a \<in> extensional (insert i I)" |
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shows "a(i := undefined) \<in> extensional I" |
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using assms unfolding extensional_def by auto |
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lemma extensional_insert_cancel[intro, simp]: |
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assumes "a \<in> extensional I" |
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shows "a \<in> extensional (insert i I)" |
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using assms unfolding extensional_def by auto |
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lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I {i} (x,y) = restrict (x(i := y i)) (insert i I)" |
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unfolding merge_def by (auto simp: fun_eq_iff) |
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lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)" |
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by auto |
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lemma PiE_Int: "(Pi\<^isub>E I A) \<inter> (Pi\<^isub>E I B) = Pi\<^isub>E I (\<lambda>x. A x \<inter> B x)" |
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by auto |
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lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B" |
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by (auto simp: Pi_def) |
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lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i" |
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by (auto simp: Pi_def) |
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Measurable on product space is equiv. to measurable components
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lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X" |
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by (auto simp: Pi_def) |
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lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
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by (auto simp: Pi_def) |
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lemma restrict_vimage: |
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assumes "I \<inter> J = {}" |
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shows "(\<lambda>x. (restrict x I, restrict x J)) -` (Pi\<^isub>E I E \<times> Pi\<^isub>E J F) = Pi (I \<union> J) (merge I J (E, F))" |
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using assms by (auto simp: restrict_Pi_cancel) |
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lemma merge_vimage: |
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assumes "I \<inter> J = {}" |
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shows "merge I J -` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E" |
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using assms by (auto simp: restrict_Pi_cancel) |
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lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I" |
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by (auto simp: restrict_def) |
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lemma merge_restrict[simp]: |
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"merge I J (restrict x I, y) = merge I J (x, y)" |
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"merge I J (x, restrict y J) = merge I J (x, y)" |
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unfolding merge_def by auto |
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lemma merge_x_x_eq_restrict[simp]: |
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"merge I J (x, x) = restrict x (I \<union> J)" |
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unfolding merge_def by auto |
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lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A" |
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apply auto |
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apply (drule_tac x=x in Pi_mem) |
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apply (simp_all split: split_if_asm) |
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apply (drule_tac x=i in Pi_mem) |
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apply (auto dest!: Pi_mem) |
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done |
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lemma Pi_UN: |
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fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set" |
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assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i" |
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shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)" |
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proof (intro set_eqI iffI) |
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fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)" |
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then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto |
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from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto |
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obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k" |
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using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto |
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have "f \<in> Pi I (A k)" |
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proof (intro Pi_I) |
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fix i assume "i \<in> I" |
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from mono[OF this, of "n i" k] k[OF this] n[OF this] |
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show "f i \<in> A k i" by auto |
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qed |
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then show "f \<in> (\<Union>n. Pi I (A n))" by auto |
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qed auto |
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lemma PiE_cong: |
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assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i" |
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shows "Pi\<^isub>E I A = Pi\<^isub>E I B" |
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using assms by (auto intro!: Pi_cong) |
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lemma restrict_upd[simp]: |
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"i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)" |
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by (auto simp: fun_eq_iff) |
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lemma Pi_eq_subset: |
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assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}" |
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assumes eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and "i \<in> I" |
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shows "F i \<subseteq> F' i" |
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proof |
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fix x assume "x \<in> F i" |
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with ne have "\<forall>j. \<exists>y. ((j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined))" by auto |
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from choice[OF this] guess f .. note f = this |
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then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def) |
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then have "f \<in> Pi\<^isub>E I F'" using assms by simp |
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then show "x \<in> F' i" using f `i \<in> I` by auto |
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qed |
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189 |
|
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lemma Pi_eq_iff_not_empty: |
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assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}" |
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shows "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)" |
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proof (intro iffI ballI) |
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194 |
fix i assume eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and i: "i \<in> I" |
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195 |
show "F i = F' i" |
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using Pi_eq_subset[of I F F', OF ne eq i] |
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197 |
using Pi_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i] |
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198 |
by auto |
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199 |
qed auto |
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200 |
|
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lemma Pi_eq_empty_iff: |
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202 |
"Pi\<^isub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})" |
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203 |
proof |
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204 |
assume "Pi\<^isub>E I F = {}" |
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205 |
show "\<exists>i\<in>I. F i = {}" |
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206 |
proof (rule ccontr) |
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207 |
assume "\<not> ?thesis" |
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208 |
then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto |
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parents:
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|
209 |
from choice[OF this] guess f .. |
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210 |
then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def) |
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|
211 |
with `Pi\<^isub>E I F = {}` show False by auto |
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|
212 |
qed |
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|
213 |
qed auto |
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|
214 |
|
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|
215 |
lemma Pi_eq_iff: |
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216 |
"Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))" |
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|
217 |
proof (intro iffI disjCI) |
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|
218 |
assume eq[simp]: "Pi\<^isub>E I F = Pi\<^isub>E I F'" |
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|
219 |
assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))" |
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|
220 |
then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})" |
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|
221 |
using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto |
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|
222 |
with Pi_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto |
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|
223 |
next |
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|
224 |
assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})" |
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|
225 |
then show "Pi\<^isub>E I F = Pi\<^isub>E I F'" |
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|
226 |
using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto |
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|
227 |
qed |
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|
228 |
|
40859 | 229 |
section "Finite product spaces" |
230 |
||
231 |
section "Products" |
|
232 |
||
47694 | 233 |
definition prod_emb where |
234 |
"prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))" |
|
235 |
||
236 |
lemma prod_emb_iff: |
|
237 |
"f \<in> prod_emb I M K X \<longleftrightarrow> f \<in> extensional I \<and> (restrict f K \<in> X) \<and> (\<forall>i\<in>I. f i \<in> space (M i))" |
|
238 |
unfolding prod_emb_def by auto |
|
40859 | 239 |
|
47694 | 240 |
lemma |
241 |
shows prod_emb_empty[simp]: "prod_emb M L K {} = {}" |
|
242 |
and prod_emb_Un[simp]: "prod_emb M L K (A \<union> B) = prod_emb M L K A \<union> prod_emb M L K B" |
|
243 |
and prod_emb_Int: "prod_emb M L K (A \<inter> B) = prod_emb M L K A \<inter> prod_emb M L K B" |
|
244 |
and prod_emb_UN[simp]: "prod_emb M L K (\<Union>i\<in>I. F i) = (\<Union>i\<in>I. prod_emb M L K (F i))" |
|
245 |
and prod_emb_INT[simp]: "I \<noteq> {} \<Longrightarrow> prod_emb M L K (\<Inter>i\<in>I. F i) = (\<Inter>i\<in>I. prod_emb M L K (F i))" |
|
246 |
and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B" |
|
247 |
by (auto simp: prod_emb_def) |
|
40859 | 248 |
|
47694 | 249 |
lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> |
250 |
prod_emb I M J (\<Pi>\<^isub>E i\<in>J. E i) = (\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i))" |
|
251 |
by (force simp: prod_emb_def Pi_iff split_if_mem2) |
|
252 |
||
253 |
lemma prod_emb_PiE_same_index[simp]: "(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^isub>E I E) = Pi\<^isub>E I E" |
|
254 |
by (auto simp: prod_emb_def Pi_iff) |
|
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255 |
|
47694 | 256 |
definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where |
257 |
"PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i)) |
|
258 |
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} |
|
259 |
(\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) |
|
260 |
(\<lambda>(J, X). \<Prod>j\<in>J \<union> {i\<in>I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))" |
|
261 |
||
262 |
definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where |
|
263 |
"prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) ` |
|
264 |
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" |
|
265 |
||
266 |
abbreviation |
|
267 |
"Pi\<^isub>M I M \<equiv> PiM I M" |
|
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268 |
|
40859 | 269 |
syntax |
47694 | 270 |
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3PIM _:_./ _)" 10) |
40859 | 271 |
|
272 |
syntax (xsymbols) |
|
47694 | 273 |
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10) |
40859 | 274 |
|
275 |
syntax (HTML output) |
|
47694 | 276 |
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10) |
40859 | 277 |
|
278 |
translations |
|
47694 | 279 |
"PIM x:I. M" == "CONST PiM I (%x. M)" |
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280 |
|
47694 | 281 |
lemma prod_algebra_sets_into_space: |
282 |
"prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
283 |
using assms by (auto simp: prod_emb_def prod_algebra_def) |
|
40859 | 284 |
|
47694 | 285 |
lemma prod_algebra_eq_finite: |
286 |
assumes I: "finite I" |
|
287 |
shows "prod_algebra I M = {(\<Pi>\<^isub>E i\<in>I. X i) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R") |
|
288 |
proof (intro iffI set_eqI) |
|
289 |
fix A assume "A \<in> ?L" |
|
290 |
then obtain J E where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)" |
|
291 |
and A: "A = prod_emb I M J (PIE j:J. E j)" |
|
292 |
by (auto simp: prod_algebra_def) |
|
293 |
let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)" |
|
294 |
have A: "A = ?A" |
|
295 |
unfolding A using J by (intro prod_emb_PiE sets_into_space) auto |
|
296 |
show "A \<in> ?R" unfolding A using J top |
|
297 |
by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp |
|
298 |
next |
|
299 |
fix A assume "A \<in> ?R" |
|
300 |
then obtain X where "A = (\<Pi>\<^isub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto |
|
301 |
then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)" |
|
302 |
using sets_into_space by (force simp: prod_emb_def Pi_iff) |
|
303 |
from X I show "A \<in> ?L" unfolding A |
|
304 |
by (auto simp: prod_algebra_def) |
|
305 |
qed |
|
41095 | 306 |
|
47694 | 307 |
lemma prod_algebraI: |
308 |
"finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)) |
|
309 |
\<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M" |
|
310 |
by (auto simp: prod_algebra_def Pi_iff) |
|
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311 |
|
47694 | 312 |
lemma prod_algebraE: |
313 |
assumes A: "A \<in> prod_algebra I M" |
|
314 |
obtains J E where "A = prod_emb I M J (PIE j:J. E j)" |
|
315 |
"finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" |
|
316 |
using A by (auto simp: prod_algebra_def) |
|
42988 | 317 |
|
47694 | 318 |
lemma prod_algebraE_all: |
319 |
assumes A: "A \<in> prod_algebra I M" |
|
320 |
obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))" |
|
321 |
proof - |
|
322 |
from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)" |
|
323 |
and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))" |
|
324 |
by (auto simp: prod_algebra_def) |
|
325 |
from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)" |
|
326 |
using sets_into_space by auto |
|
327 |
then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))" |
|
328 |
using A J by (auto simp: prod_emb_PiE) |
|
329 |
moreover then have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))" |
|
330 |
using top E by auto |
|
331 |
ultimately show ?thesis using that by auto |
|
332 |
qed |
|
40859 | 333 |
|
47694 | 334 |
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)" |
335 |
proof (unfold Int_stable_def, safe) |
|
336 |
fix A assume "A \<in> prod_algebra I M" |
|
337 |
from prod_algebraE[OF this] guess J E . note A = this |
|
338 |
fix B assume "B \<in> prod_algebra I M" |
|
339 |
from prod_algebraE[OF this] guess K F . note B = this |
|
340 |
have "A \<inter> B = prod_emb I M (J \<union> K) (\<Pi>\<^isub>E i\<in>J \<union> K. (if i \<in> J then E i else space (M i)) \<inter> |
|
341 |
(if i \<in> K then F i else space (M i)))" |
|
342 |
unfolding A B using A(2,3,4) A(5)[THEN sets_into_space] B(2,3,4) B(5)[THEN sets_into_space] |
|
343 |
apply (subst (1 2 3) prod_emb_PiE) |
|
344 |
apply (simp_all add: subset_eq PiE_Int) |
|
345 |
apply blast |
|
346 |
apply (intro PiE_cong) |
|
347 |
apply auto |
|
348 |
done |
|
349 |
also have "\<dots> \<in> prod_algebra I M" |
|
350 |
using A B by (auto intro!: prod_algebraI) |
|
351 |
finally show "A \<inter> B \<in> prod_algebra I M" . |
|
352 |
qed |
|
353 |
||
354 |
lemma prod_algebra_mono: |
|
355 |
assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)" |
|
356 |
assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)" |
|
357 |
shows "prod_algebra I E \<subseteq> prod_algebra I F" |
|
358 |
proof |
|
359 |
fix A assume "A \<in> prod_algebra I E" |
|
360 |
then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" |
|
361 |
and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)" |
|
362 |
and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)" |
|
363 |
by (auto simp: prod_algebra_def) |
|
364 |
moreover |
|
365 |
from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))" |
|
366 |
by (rule PiE_cong) |
|
367 |
with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)" |
|
368 |
by (simp add: prod_emb_def) |
|
369 |
moreover |
|
370 |
from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)" |
|
371 |
by auto |
|
372 |
ultimately show "A \<in> prod_algebra I F" |
|
373 |
apply (simp add: prod_algebra_def image_iff) |
|
374 |
apply (intro exI[of _ J] exI[of _ G] conjI) |
|
375 |
apply auto |
|
376 |
done |
|
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|
377 |
qed |
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|
378 |
|
47694 | 379 |
lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))" |
380 |
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp |
|
381 |
||
382 |
lemma sets_PiM: "sets (\<Pi>\<^isub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) (prod_algebra I M)" |
|
383 |
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp |
|
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|
384 |
|
47694 | 385 |
lemma sets_PiM_single: "sets (PiM I M) = |
386 |
sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}" |
|
387 |
(is "_ = sigma_sets ?\<Omega> ?R") |
|
388 |
unfolding sets_PiM |
|
389 |
proof (rule sigma_sets_eqI) |
|
390 |
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto |
|
391 |
fix A assume "A \<in> prod_algebra I M" |
|
392 |
from prod_algebraE[OF this] guess J X . note X = this |
|
393 |
show "A \<in> sigma_sets ?\<Omega> ?R" |
|
394 |
proof cases |
|
395 |
assume "I = {}" |
|
396 |
with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def) |
|
397 |
with `I = {}` show ?thesis by (auto intro!: sigma_sets_top) |
|
398 |
next |
|
399 |
assume "I \<noteq> {}" |
|
400 |
with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^isub>E i\<in>I. space (M i)). f j \<in> X j})" |
|
401 |
using sets_into_space[OF X(5)] |
|
402 |
by (auto simp: prod_emb_PiE[OF _ sets_into_space] Pi_iff split: split_if_asm) blast |
|
403 |
also have "\<dots> \<in> sigma_sets ?\<Omega> ?R" |
|
404 |
using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto |
|
405 |
finally show "A \<in> sigma_sets ?\<Omega> ?R" . |
|
406 |
qed |
|
407 |
next |
|
408 |
fix A assume "A \<in> ?R" |
|
409 |
then obtain i B where A: "A = {f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)" |
|
410 |
by auto |
|
411 |
then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)" |
|
412 |
using sets_into_space[OF A(3)] |
|
413 |
apply (subst prod_emb_PiE) |
|
414 |
apply (auto simp: Pi_iff split: split_if_asm) |
|
415 |
apply blast |
|
416 |
done |
|
417 |
also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)" |
|
418 |
using A by (intro sigma_sets.Basic prod_algebraI) auto |
|
419 |
finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" . |
|
420 |
qed |
|
421 |
||
422 |
lemma sets_PiM_I: |
|
423 |
assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)" |
|
424 |
shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)" |
|
425 |
proof cases |
|
426 |
assume "J = {}" |
|
427 |
then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))" |
|
428 |
by (auto simp: prod_emb_def) |
|
429 |
then show ?thesis |
|
430 |
by (auto simp add: sets_PiM intro!: sigma_sets_top) |
|
431 |
next |
|
432 |
assume "J \<noteq> {}" with assms show ?thesis |
|
433 |
by (auto simp add: sets_PiM prod_algebra_def intro!: sigma_sets.Basic) |
|
40859 | 434 |
qed |
435 |
||
47694 | 436 |
lemma measurable_PiM: |
437 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
438 |
assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow> |
|
439 |
f -` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N" |
|
440 |
shows "f \<in> measurable N (PiM I M)" |
|
441 |
using sets_PiM prod_algebra_sets_into_space space |
|
442 |
proof (rule measurable_sigma_sets) |
|
443 |
fix A assume "A \<in> prod_algebra I M" |
|
444 |
from prod_algebraE[OF this] guess J X . |
|
445 |
with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto |
|
446 |
qed |
|
447 |
||
448 |
lemma measurable_PiM_Collect: |
|
449 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
450 |
assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow> |
|
451 |
{\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" |
|
452 |
shows "f \<in> measurable N (PiM I M)" |
|
453 |
using sets_PiM prod_algebra_sets_into_space space |
|
454 |
proof (rule measurable_sigma_sets) |
|
455 |
fix A assume "A \<in> prod_algebra I M" |
|
456 |
from prod_algebraE[OF this] guess J X . note X = this |
|
457 |
have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}" |
|
458 |
using sets_into_space[OF X(5)] X(2-) space unfolding X(1) |
|
459 |
by (subst prod_emb_PiE) (auto simp: Pi_iff split: split_if_asm) |
|
460 |
also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets) |
|
461 |
finally show "f -` A \<inter> space N \<in> sets N" . |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
462 |
qed |
41095 | 463 |
|
47694 | 464 |
lemma measurable_PiM_single: |
465 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
466 |
assumes sets: "\<And>A i. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N" |
|
467 |
shows "f \<in> measurable N (PiM I M)" |
|
468 |
using sets_PiM_single |
|
469 |
proof (rule measurable_sigma_sets) |
|
470 |
fix A assume "A \<in> {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}" |
|
471 |
then obtain B i where "A = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)" |
|
472 |
by auto |
|
473 |
with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto |
|
474 |
also have "\<dots> \<in> sets N" using B by (rule sets) |
|
475 |
finally show "f -` A \<inter> space N \<in> sets N" . |
|
476 |
qed (auto simp: space) |
|
40859 | 477 |
|
47694 | 478 |
lemma sets_PiM_I_finite[simp, intro]: |
479 |
assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))" |
|
480 |
shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)" |
|
481 |
using sets_PiM_I[of I I E M] sets_into_space[OF sets] `finite I` sets by auto |
|
482 |
||
483 |
lemma measurable_component_update: |
|
484 |
assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I" |
|
485 |
shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _") |
|
486 |
proof (intro measurable_PiM_single) |
|
487 |
fix j A assume "j \<in> insert i I" "A \<in> sets (M j)" |
|
488 |
moreover have "{\<omega> \<in> space (M i). (x(i := \<omega>)) j \<in> A} = |
|
489 |
(if i = j then space (M i) \<inter> A else if x j \<in> A then space (M i) else {})" |
|
490 |
by auto |
|
491 |
ultimately show "{\<omega> \<in> space (M i). (x(i := \<omega>)) j \<in> A} \<in> sets (M i)" |
|
492 |
by auto |
|
493 |
qed (insert sets_into_space assms, auto simp: space_PiM) |
|
494 |
||
495 |
lemma measurable_component_singleton: |
|
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
|
496 |
assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)" |
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
|
497 |
proof (unfold measurable_def, intro CollectI conjI ballI) |
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
|
498 |
fix A assume "A \<in> sets (M i)" |
47694 | 499 |
then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) = prod_emb I M {i} (\<Pi>\<^isub>E j\<in>{i}. A)" |
500 |
using sets_into_space `i \<in> I` |
|
501 |
by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm) |
|
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
|
502 |
then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)" |
47694 | 503 |
using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I) |
504 |
qed (insert `i \<in> I`, auto simp: space_PiM) |
|
505 |
||
506 |
lemma measurable_add_dim: |
|
49776 | 507 |
"(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)" |
47694 | 508 |
(is "?f \<in> measurable ?P ?I") |
509 |
proof (rule measurable_PiM_single) |
|
510 |
fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)" |
|
511 |
have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} = |
|
512 |
(if j = i then space (Pi\<^isub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)" |
|
513 |
using sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM) |
|
514 |
also have "\<dots> \<in> sets ?P" |
|
515 |
using A j |
|
516 |
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) |
|
517 |
finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" . |
|
518 |
qed (auto simp: space_pair_measure space_PiM) |
|
41661 | 519 |
|
47694 | 520 |
lemma measurable_merge: |
49780 | 521 |
"merge I J \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)" |
47694 | 522 |
(is "?f \<in> measurable ?P ?U") |
523 |
proof (rule measurable_PiM_single) |
|
524 |
fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J" |
|
49780 | 525 |
then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} = |
47694 | 526 |
(if i \<in> I then ((\<lambda>x. x i) \<circ> fst) -` A \<inter> space ?P else ((\<lambda>x. x i) \<circ> snd) -` A \<inter> space ?P)" |
49776 | 527 |
by (auto simp: merge_def) |
47694 | 528 |
also have "\<dots> \<in> sets ?P" |
529 |
using A |
|
530 |
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) |
|
49780 | 531 |
finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" . |
49776 | 532 |
qed (auto simp: space_pair_measure space_PiM Pi_iff merge_def extensional_def) |
42988 | 533 |
|
47694 | 534 |
lemma measurable_restrict: |
535 |
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)" |
|
536 |
shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)" |
|
537 |
proof (rule measurable_PiM_single) |
|
538 |
fix A i assume A: "i \<in> I" "A \<in> sets (M i)" |
|
539 |
then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N" |
|
540 |
by auto |
|
541 |
then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N" |
|
542 |
using A X by (auto intro!: measurable_sets) |
|
543 |
qed (insert X, auto dest: measurable_space) |
|
544 |
||
545 |
locale product_sigma_finite = |
|
546 |
fixes M :: "'i \<Rightarrow> 'a measure" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
547 |
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)" |
40859 | 548 |
|
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
|
549 |
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i |
40859 | 550 |
by (rule sigma_finite_measures) |
551 |
||
47694 | 552 |
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" + |
553 |
fixes I :: "'i set" |
|
554 |
assumes finite_index: "finite I" |
|
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
|
555 |
|
40859 | 556 |
lemma (in finite_product_sigma_finite) sigma_finite_pairs: |
557 |
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set. |
|
558 |
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and> |
|
47694 | 559 |
(\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and> |
560 |
(\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)" |
|
40859 | 561 |
proof - |
47694 | 562 |
have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. emeasure (M i) (F k) \<noteq> \<infinity>)" |
563 |
using M.sigma_finite_incseq by metis |
|
40859 | 564 |
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. |
47694 | 565 |
then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. emeasure (M i) (F i k) \<noteq> \<infinity>" |
40859 | 566 |
by auto |
567 |
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k" |
|
47694 | 568 |
note space_PiM[simp] |
40859 | 569 |
show ?thesis |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
570 |
proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI) |
40859 | 571 |
fix i show "range (F i) \<subseteq> sets (M i)" by fact |
572 |
next |
|
47694 | 573 |
fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact |
40859 | 574 |
next |
47694 | 575 |
fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space (PiM I M)" |
576 |
using `\<And>i. range (F i) \<subseteq> sets (M i)` sets_into_space |
|
577 |
by auto blast |
|
40859 | 578 |
next |
47694 | 579 |
fix f assume "f \<in> space (PiM I M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
580 |
with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
581 |
show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def) |
40859 | 582 |
next |
583 |
fix i show "?F i \<subseteq> ?F (Suc i)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
584 |
using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto |
40859 | 585 |
qed |
586 |
qed |
|
587 |
||
49780 | 588 |
lemma |
589 |
shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}" |
|
590 |
and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }" |
|
591 |
by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq) |
|
592 |
||
593 |
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1" |
|
594 |
proof - |
|
595 |
let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)" |
|
596 |
have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1" |
|
597 |
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) |
|
598 |
show "positive (PiM {} M) ?\<mu>" |
|
599 |
by (auto simp: positive_def) |
|
600 |
show "countably_additive (PiM {} M) ?\<mu>" |
|
601 |
by (rule countably_additiveI_finite) |
|
602 |
(auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: ) |
|
603 |
qed (auto simp: prod_emb_def) |
|
604 |
also have "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}" |
|
605 |
by (auto simp: prod_emb_def) |
|
606 |
finally show ?thesis |
|
607 |
by simp |
|
608 |
qed |
|
609 |
||
610 |
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}" |
|
611 |
by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def) |
|
612 |
||
49776 | 613 |
lemma (in product_sigma_finite) emeasure_PiM: |
614 |
"finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" |
|
615 |
proof (induct I arbitrary: A rule: finite_induct) |
|
40859 | 616 |
case (insert i I) |
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
|
617 |
interpret finite_product_sigma_finite M I by default fact |
40859 | 618 |
have "finite (insert i I)" using `finite I` by auto |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
619 |
interpret I': finite_product_sigma_finite M "insert i I" by default fact |
41661 | 620 |
let ?h = "(\<lambda>(f, y). f(i := y))" |
47694 | 621 |
|
622 |
let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h" |
|
623 |
let ?\<mu> = "emeasure ?P" |
|
624 |
let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}" |
|
625 |
let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))" |
|
626 |
||
49776 | 627 |
have "emeasure (Pi\<^isub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^isub>E (insert i I) A)) = |
628 |
(\<Prod>i\<in>insert i I. emeasure (M i) (A i))" |
|
629 |
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) |
|
630 |
fix J E assume "(J \<noteq> {} \<or> insert i I = {}) \<and> finite J \<and> J \<subseteq> insert i I \<and> E \<in> (\<Pi> j\<in>J. sets (M j))" |
|
631 |
then have J: "J \<noteq> {}" "finite J" "J \<subseteq> insert i I" and E: "\<forall>j\<in>J. E j \<in> sets (M j)" by auto |
|
632 |
let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)" |
|
633 |
let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^isub>E j\<in>J-{i}. E j)" |
|
634 |
have "?\<mu> ?p = |
|
635 |
emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i))" |
|
636 |
by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+ |
|
637 |
also have "?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))" |
|
638 |
using J E[rule_format, THEN sets_into_space] |
|
639 |
by (force simp: space_pair_measure space_PiM Pi_iff prod_emb_iff split: split_if_asm) |
|
640 |
also have "emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) = |
|
641 |
emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))" |
|
642 |
using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto |
|
643 |
also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))" |
|
644 |
using J E[rule_format, THEN sets_into_space] |
|
645 |
by (auto simp: prod_emb_iff Pi_iff split: split_if_asm) blast+ |
|
646 |
also have "emeasure (Pi\<^isub>M I M) (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j)) = |
|
647 |
(\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))" |
|
648 |
using E by (subst insert) (auto intro!: setprod_cong) |
|
649 |
also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) * |
|
650 |
emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)" |
|
651 |
using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong) |
|
652 |
also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)" |
|
653 |
using insert(1,2) J E by (intro setprod_mono_one_right) auto |
|
654 |
finally show "?\<mu> ?p = \<dots>" . |
|
47694 | 655 |
|
49776 | 656 |
show "prod_emb (insert i I) M J (Pi\<^isub>E J E) \<in> Pow (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" |
657 |
using J E[rule_format, THEN sets_into_space] by (auto simp: prod_emb_iff) |
|
658 |
next |
|
659 |
show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" |
|
660 |
using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all |
|
661 |
next |
|
662 |
show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and> |
|
663 |
insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))" |
|
664 |
using insert by auto |
|
665 |
qed (auto intro!: setprod_cong) |
|
666 |
with insert show ?case |
|
667 |
by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets_into_space) |
|
49780 | 668 |
qed (simp add: emeasure_PiM_empty) |
47694 | 669 |
|
49776 | 670 |
lemma (in product_sigma_finite) sigma_finite: |
671 |
assumes "finite I" |
|
672 |
shows "sigma_finite_measure (PiM I M)" |
|
673 |
proof - |
|
674 |
interpret finite_product_sigma_finite M I by default fact |
|
675 |
||
676 |
from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. |
|
677 |
then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)" |
|
678 |
"incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)" |
|
679 |
"(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)" |
|
680 |
"\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>" |
|
47694 | 681 |
by blast+ |
49776 | 682 |
let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k" |
47694 | 683 |
|
49776 | 684 |
show ?thesis |
47694 | 685 |
proof (unfold_locales, intro exI[of _ ?F] conjI allI) |
49776 | 686 |
show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto |
47694 | 687 |
next |
49776 | 688 |
from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp |
47694 | 689 |
next |
690 |
fix j |
|
49776 | 691 |
from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M] |
692 |
show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>" |
|
693 |
by (subst emeasure_PiM) auto |
|
40859 | 694 |
qed |
695 |
qed |
|
696 |
||
47694 | 697 |
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M" |
698 |
using sigma_finite[OF finite_index] . |
|
40859 | 699 |
|
700 |
lemma (in finite_product_sigma_finite) measure_times: |
|
47694 | 701 |
"(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^isub>M I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" |
702 |
using emeasure_PiM[OF finite_index] by auto |
|
41096 | 703 |
|
40859 | 704 |
lemma (in product_sigma_finite) positive_integral_empty: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
705 |
assumes pos: "0 \<le> f (\<lambda>k. undefined)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
706 |
shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)" |
40859 | 707 |
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
|
708 |
interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI) |
47694 | 709 |
have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1" |
40859 | 710 |
using assms by (subst measure_times) auto |
711 |
then show ?thesis |
|
47694 | 712 |
unfolding positive_integral_def simple_function_def simple_integral_def[abs_def] |
713 |
proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
714 |
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
715 |
by (intro SUP_upper) (auto simp: le_fun_def split: split_max) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
716 |
show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
717 |
by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm) |
40859 | 718 |
qed |
719 |
qed |
|
720 |
||
47694 | 721 |
lemma (in product_sigma_finite) distr_merge: |
40859 | 722 |
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J" |
49780 | 723 |
shows "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M) (merge I J) = Pi\<^isub>M (I \<union> J) M" |
47694 | 724 |
(is "?D = ?P") |
40859 | 725 |
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
|
726 |
interpret I: finite_product_sigma_finite M I by default fact |
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
|
727 |
interpret J: finite_product_sigma_finite M J by default fact |
40859 | 728 |
have "finite (I \<union> J)" using fin by auto |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
729 |
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact |
47694 | 730 |
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default |
49780 | 731 |
let ?g = "merge I J" |
47694 | 732 |
|
41661 | 733 |
from IJ.sigma_finite_pairs obtain F where |
734 |
F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
735 |
"incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)" |
47694 | 736 |
"(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P" |
737 |
"\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>" |
|
41661 | 738 |
by auto |
739 |
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k" |
|
47694 | 740 |
|
741 |
show ?thesis |
|
742 |
proof (rule measure_eqI_generator_eq[symmetric]) |
|
743 |
show "Int_stable (prod_algebra (I \<union> J) M)" |
|
744 |
by (rule Int_stable_prod_algebra) |
|
745 |
show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))" |
|
746 |
by (rule prod_algebra_sets_into_space) |
|
747 |
show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)" |
|
748 |
by (rule sets_PiM) |
|
749 |
then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)" |
|
750 |
by simp |
|
751 |
||
752 |
show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F |
|
753 |
using fin by (auto simp: prod_algebra_eq_finite) |
|
754 |
show "(\<Union>i. \<Pi>\<^isub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i))" |
|
755 |
using F(3) by (simp add: space_PiM) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
756 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
757 |
fix k |
47694 | 758 |
from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M] |
759 |
show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto |
|
41661 | 760 |
next |
47694 | 761 |
fix A assume A: "A \<in> prod_algebra (I \<union> J) M" |
762 |
with fin obtain F where A_eq: "A = (Pi\<^isub>E (I \<union> J) F)" and F: "\<forall>i\<in>I \<union> J. F i \<in> sets (M i)" |
|
763 |
by (auto simp add: prod_algebra_eq_finite) |
|
764 |
let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M" |
|
765 |
let ?X = "?g -` A \<inter> space ?B" |
|
766 |
have "Pi\<^isub>E I F \<subseteq> space (Pi\<^isub>M I M)" "Pi\<^isub>E J F \<subseteq> space (Pi\<^isub>M J M)" |
|
767 |
using F[rule_format, THEN sets_into_space] by (auto simp: space_PiM) |
|
768 |
then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)" |
|
769 |
unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM) |
|
770 |
have "emeasure ?D A = emeasure ?B ?X" |
|
771 |
using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM) |
|
772 |
also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))" |
|
773 |
using `finite J` `finite I` F X |
|
49776 | 774 |
by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times Pi_iff) |
47694 | 775 |
also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))" |
41661 | 776 |
using `finite J` `finite I` `I \<inter> J = {}` by (simp add: setprod_Un_one) |
47694 | 777 |
also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)" |
41661 | 778 |
using `finite J` `finite I` F unfolding A |
779 |
by (intro IJ.measure_times[symmetric]) auto |
|
47694 | 780 |
finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp |
781 |
qed |
|
41661 | 782 |
qed |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
783 |
|
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
784 |
lemma (in product_sigma_finite) product_positive_integral_fold: |
47694 | 785 |
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
786 |
and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
787 |
shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f = |
49780 | 788 |
(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
789 |
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
|
790 |
interpret I: finite_product_sigma_finite M I by default fact |
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
|
791 |
interpret J: finite_product_sigma_finite M J by default fact |
41831 | 792 |
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default |
49780 | 793 |
have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" |
49776 | 794 |
using measurable_comp[OF measurable_merge f] by (simp add: comp_def) |
41661 | 795 |
show ?thesis |
47694 | 796 |
apply (subst distr_merge[OF IJ, symmetric]) |
49776 | 797 |
apply (subst positive_integral_distr[OF measurable_merge f]) |
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigma-finite
hoelzl
parents:
49784
diff
changeset
|
798 |
apply (subst J.positive_integral_fst_measurable(2)[symmetric, OF P_borel]) |
47694 | 799 |
apply simp |
800 |
done |
|
40859 | 801 |
qed |
802 |
||
47694 | 803 |
lemma (in product_sigma_finite) distr_singleton: |
804 |
"distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _") |
|
805 |
proof (intro measure_eqI[symmetric]) |
|
41831 | 806 |
interpret I: finite_product_sigma_finite M "{i}" by default simp |
47694 | 807 |
fix A assume A: "A \<in> sets (M i)" |
808 |
moreover then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)" |
|
809 |
using sets_into_space by (auto simp: space_PiM) |
|
810 |
ultimately show "emeasure (M i) A = emeasure ?D A" |
|
811 |
using A I.measure_times[of "\<lambda>_. A"] |
|
812 |
by (simp add: emeasure_distr measurable_component_singleton) |
|
813 |
qed simp |
|
41831 | 814 |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
815 |
lemma (in product_sigma_finite) product_positive_integral_singleton: |
40859 | 816 |
assumes f: "f \<in> borel_measurable (M i)" |
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
|
817 |
shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f" |
40859 | 818 |
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
|
819 |
interpret I: finite_product_sigma_finite M "{i}" by default simp |
47694 | 820 |
from f show ?thesis |
821 |
apply (subst distr_singleton[symmetric]) |
|
822 |
apply (subst positive_integral_distr[OF measurable_component_singleton]) |
|
823 |
apply simp_all |
|
824 |
done |
|
40859 | 825 |
qed |
826 |
||
41096 | 827 |
lemma (in product_sigma_finite) product_positive_integral_insert: |
49780 | 828 |
assumes I[simp]: "finite I" "i \<notin> I" |
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
|
829 |
and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
830 |
shows "integral\<^isup>P (Pi\<^isub>M (insert i I) M) f = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^isub>M I M))" |
41096 | 831 |
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
|
832 |
interpret I: finite_product_sigma_finite M I by default auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
833 |
interpret i: finite_product_sigma_finite M "{i}" by default auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
834 |
have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I" |
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
|
835 |
using f by auto |
41096 | 836 |
show ?thesis |
49780 | 837 |
unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f] |
838 |
proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric]) |
|
47694 | 839 |
fix x assume x: "x \<in> space (Pi\<^isub>M I M)" |
49780 | 840 |
let ?f = "\<lambda>y. f (x(i := y))" |
841 |
show "?f \<in> borel_measurable (M i)" |
|
47694 | 842 |
using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`] |
843 |
unfolding comp_def . |
|
49780 | 844 |
show "(\<integral>\<^isup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^isub>M {i} M) = (\<integral>\<^isup>+ y. f (x(i := y i)) \<partial>Pi\<^isub>M {i} M)" |
845 |
using x |
|
846 |
by (auto intro!: positive_integral_cong arg_cong[where f=f] |
|
847 |
simp add: space_PiM extensional_def) |
|
41096 | 848 |
qed |
849 |
qed |
|
850 |
||
851 |
lemma (in product_sigma_finite) product_positive_integral_setprod: |
|
43920 | 852 |
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal" |
41096 | 853 |
assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
854 |
and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
855 |
shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))" |
41096 | 856 |
using assms proof induct |
857 |
case (insert i I) |
|
858 |
note `finite I`[intro, simp] |
|
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
|
859 |
interpret I: finite_product_sigma_finite M I by default auto |
41096 | 860 |
have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))" |
861 |
using insert by (auto intro!: setprod_cong) |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
862 |
have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^isub>M J M)" |
41096 | 863 |
using sets_into_space insert |
47694 | 864 |
by (intro borel_measurable_ereal_setprod |
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
|
865 |
measurable_comp[OF measurable_component_singleton, unfolded comp_def]) |
41096 | 866 |
auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
867 |
then show ?case |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
868 |
apply (simp add: product_positive_integral_insert[OF insert(1,2) prod]) |
47694 | 869 |
apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc) |
870 |
apply (subst positive_integral_cmult) |
|
871 |
apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
872 |
done |
47694 | 873 |
qed (simp add: space_PiM) |
41096 | 874 |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
875 |
lemma (in product_sigma_finite) product_integral_singleton: |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
876 |
assumes f: "f \<in> borel_measurable (M i)" |
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
|
877 |
shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
878 |
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
|
879 |
interpret I: finite_product_sigma_finite M "{i}" by default simp |
43920 | 880 |
have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)" |
881 |
"(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)" |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
882 |
using assms by auto |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
883 |
show ?thesis |
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
|
884 |
unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] .. |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
885 |
qed |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
886 |
|
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
887 |
lemma (in product_sigma_finite) product_integral_fold: |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
888 |
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
889 |
and f: "integrable (Pi\<^isub>M (I \<union> J) M) f" |
49780 | 890 |
shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I J (x, y)) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
891 |
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
|
892 |
interpret I: finite_product_sigma_finite M I by default fact |
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
|
893 |
interpret J: finite_product_sigma_finite M J by default fact |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
894 |
have "finite (I \<union> J)" using fin by auto |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
895 |
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact |
47694 | 896 |
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default |
49780 | 897 |
let ?M = "merge I J" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
898 |
let ?f = "\<lambda>x. f (?M x)" |
47694 | 899 |
from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)" |
900 |
by auto |
|
49780 | 901 |
have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" |
49776 | 902 |
using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def) |
47694 | 903 |
have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f" |
49776 | 904 |
by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f) |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
905 |
show ?thesis |
47694 | 906 |
apply (subst distr_merge[symmetric, OF IJ fin]) |
49776 | 907 |
apply (subst integral_distr[OF measurable_merge f_borel]) |
47694 | 908 |
apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int]) |
909 |
apply simp |
|
910 |
done |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
911 |
qed |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
912 |
|
49776 | 913 |
lemma (in product_sigma_finite) |
914 |
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)" |
|
915 |
shows emeasure_fold_integral: |
|
49780 | 916 |
"emeasure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I) |
49776 | 917 |
and emeasure_fold_measurable: |
49780 | 918 |
"(\<lambda>x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B) |
49776 | 919 |
proof - |
920 |
interpret I: finite_product_sigma_finite M I by default fact |
|
921 |
interpret J: finite_product_sigma_finite M J by default fact |
|
922 |
interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" .. |
|
49780 | 923 |
have merge: "merge I J -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" |
49776 | 924 |
by (intro measurable_sets[OF _ A] measurable_merge assms) |
925 |
||
926 |
show ?I |
|
927 |
apply (subst distr_merge[symmetric, OF IJ]) |
|
928 |
apply (subst emeasure_distr[OF measurable_merge A]) |
|
929 |
apply (subst J.emeasure_pair_measure_alt[OF merge]) |
|
930 |
apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure) |
|
931 |
done |
|
932 |
||
933 |
show ?B |
|
934 |
using IJ.measurable_emeasure_Pair1[OF merge] |
|
935 |
by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong) |
|
936 |
qed |
|
937 |
||
41096 | 938 |
lemma (in product_sigma_finite) product_integral_insert: |
47694 | 939 |
assumes I: "finite I" "i \<notin> I" |
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
|
940 |
and f: "integrable (Pi\<^isub>M (insert i I) M) f" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
941 |
shows "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^isub>M I M)" |
41096 | 942 |
proof - |
47694 | 943 |
have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f" |
944 |
by simp |
|
49780 | 945 |
also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) \<partial>Pi\<^isub>M I M)" |
47694 | 946 |
using f I by (intro product_integral_fold) auto |
947 |
also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)" |
|
948 |
proof (rule integral_cong, subst product_integral_singleton[symmetric]) |
|
949 |
fix x assume x: "x \<in> space (Pi\<^isub>M I M)" |
|
950 |
have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)" |
|
951 |
using f by auto |
|
952 |
show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)" |
|
953 |
using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`] |
|
954 |
unfolding comp_def . |
|
49780 | 955 |
from x I show "(\<integral> y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^isub>M {i} M)" |
47694 | 956 |
by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def) |
41096 | 957 |
qed |
47694 | 958 |
finally show ?thesis . |
41096 | 959 |
qed |
960 |
||
961 |
lemma (in product_sigma_finite) product_integrable_setprod: |
|
962 |
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
963 |
assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)" |
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
|
964 |
shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f") |
41096 | 965 |
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
|
966 |
interpret finite_product_sigma_finite M I by default fact |
41096 | 967 |
have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)" |
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
|
968 |
using integrable unfolding integrable_def by auto |
47694 | 969 |
have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)" |
970 |
using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def) |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
971 |
moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)" |
41096 | 972 |
proof (unfold integrable_def, intro conjI) |
47694 | 973 |
show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)" |
41096 | 974 |
using borel by auto |
47694 | 975 |
have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>Pi\<^isub>M I M) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>Pi\<^isub>M I M)" |
43920 | 976 |
by (simp add: setprod_ereal abs_setprod) |
977 |
also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))" |
|
41096 | 978 |
using f by (subst product_positive_integral_setprod) auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
979 |
also have "\<dots> < \<infinity>" |
47694 | 980 |
using integrable[THEN integrable_abs] |
981 |
by (simp add: setprod_PInf integrable_def positive_integral_positive) |
|
982 |
finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto |
|
983 |
have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
984 |
by (intro positive_integral_cong_pos) auto |
47694 | 985 |
then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp |
41096 | 986 |
qed |
987 |
ultimately show ?thesis |
|
988 |
by (rule integrable_abs_iff[THEN iffD1]) |
|
989 |
qed |
|
990 |
||
991 |
lemma (in product_sigma_finite) product_integral_setprod: |
|
992 |
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real" |
|
49780 | 993 |
assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)" |
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
|
994 |
shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>L (M i) (f i))" |
49780 | 995 |
using assms proof induct |
996 |
case empty |
|
997 |
interpret finite_measure "Pi\<^isub>M {} M" |
|
998 |
by rule (simp add: space_PiM) |
|
999 |
show ?case by (simp add: space_PiM measure_def) |
|
41096 | 1000 |
next |
1001 |
case (insert i I) |
|
1002 |
then have iI: "finite (insert i I)" by auto |
|
1003 |
then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> |
|
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
|
1004 |
integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))" |
49780 | 1005 |
by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset) |
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
|
1006 |
interpret I: finite_product_sigma_finite M I by default fact |
41096 | 1007 |
have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))" |
1008 |
using `i \<notin> I` by (auto intro!: setprod_cong) |
|
1009 |
show ?case |
|
49780 | 1010 |
unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]] |
47694 | 1011 |
by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI) |
41096 | 1012 |
qed |
1013 |
||
49776 | 1014 |
lemma sets_Collect_single: |
1015 |
"i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^isub>M I M). x i \<in> A } \<in> sets (Pi\<^isub>M I M)" |
|
1016 |
unfolding sets_PiM_single |
|
1017 |
by (auto intro!: sigma_sets.Basic exI[of _ i] exI[of _ A]) (auto simp: space_PiM) |
|
1018 |
||
1019 |
lemma sigma_prod_algebra_sigma_eq_infinite: |
|
1020 |
fixes E :: "'i \<Rightarrow> 'a set set" |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1021 |
assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" |
49776 | 1022 |
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" |
1023 |
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" |
|
1024 |
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" |
|
1025 |
defines "P == {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> E i}" |
|
1026 |
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P" |
|
1027 |
proof |
|
1028 |
let ?P = "sigma (space (Pi\<^isub>M I M)) P" |
|
1029 |
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))" |
|
1030 |
using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq) |
|
1031 |
then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
1032 |
by (simp add: space_PiM) |
|
1033 |
have "sets (PiM I M) = |
|
1034 |
sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}" |
|
1035 |
using sets_PiM_single[of I M] by (simp add: space_P) |
|
1036 |
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)" |
|
1037 |
proof (safe intro!: sigma_sets_subset) |
|
1038 |
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" |
|
1039 |
then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))" |
|
1040 |
apply (subst measurable_iff_measure_of) |
|
1041 |
apply (simp_all add: P_closed) |
|
1042 |
using E_closed |
|
1043 |
apply (force simp: subset_eq space_PiM) |
|
1044 |
apply (force simp: subset_eq space_PiM) |
|
1045 |
apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i]) |
|
1046 |
apply (rule_tac x=Aa in exI) |
|
1047 |
apply (auto simp: space_PiM) |
|
1048 |
done |
|
1049 |
from measurable_sets[OF this, of A] A `i \<in> I` E_closed |
|
1050 |
have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
1051 |
by (simp add: E_generates) |
|
1052 |
also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}" |
|
1053 |
using P_closed by (auto simp: space_PiM) |
|
1054 |
finally show "\<dots> \<in> sets ?P" . |
|
1055 |
qed |
|
1056 |
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P" |
|
1057 |
by (simp add: P_closed) |
|
1058 |
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)" |
|
1059 |
unfolding P_def space_PiM[symmetric] |
|
1060 |
by (intro sigma_sets_subset) (auto simp: E_generates sets_Collect_single) |
|
1061 |
qed |
|
1062 |
||
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1063 |
lemma bchoice_iff: "(\<forall>a\<in>A. \<exists>b. P a b) \<longleftrightarrow> (\<exists>f. \<forall>a\<in>A. P a (f a))" |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1064 |
by metis |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1065 |
|
47694 | 1066 |
lemma sigma_prod_algebra_sigma_eq: |
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1067 |
fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" |
47694 | 1068 |
assumes "finite I" |
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1069 |
assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" |
47694 | 1070 |
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" |
1071 |
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" |
|
1072 |
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" |
|
1073 |
defines "P == { Pi\<^isub>E I F | F. \<forall>i\<in>I. F i \<in> E i }" |
|
1074 |
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P" |
|
1075 |
proof |
|
1076 |
let ?P = "sigma (space (Pi\<^isub>M I M)) P" |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1077 |
from `finite I`[THEN ex_bij_betw_finite_nat] guess T .. |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1078 |
then have T: "\<And>i. i \<in> I \<Longrightarrow> T i < card I" "\<And>i. i\<in>I \<Longrightarrow> the_inv_into I T (T i) = i" |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1079 |
by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f) |
47694 | 1080 |
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))" |
1081 |
using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq) |
|
1082 |
then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
1083 |
by (simp add: space_PiM) |
|
1084 |
have "sets (PiM I M) = |
|
1085 |
sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}" |
|
1086 |
using sets_PiM_single[of I M] by (simp add: space_P) |
|
1087 |
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)" |
|
1088 |
proof (safe intro!: sigma_sets_subset) |
|
1089 |
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" |
|
1090 |
have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))" |
|
1091 |
proof (subst measurable_iff_measure_of) |
|
1092 |
show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact |
|
1093 |
from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" |
|
1094 |
by (auto simp: Pi_iff) |
|
1095 |
show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
1096 |
proof |
|
1097 |
fix A assume A: "A \<in> E i" |
|
1098 |
then have "(\<lambda>x. x i) -` A \<inter> space ?P = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))" |
|
1099 |
using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm) |
|
1100 |
also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)" |
|
1101 |
by (intro PiE_cong) (simp add: S_union) |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1102 |
also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j))" |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1103 |
using T |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1104 |
apply (auto simp: Pi_iff bchoice_iff) |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1105 |
apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI) |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1106 |
apply (auto simp: bij_betw_def) |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1107 |
done |
47694 | 1108 |
also have "\<dots> \<in> sets ?P" |
1109 |
proof (safe intro!: countable_UN) |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1110 |
fix xs show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P" |
47694 | 1111 |
using A S_in_E |
1112 |
by (simp add: P_closed) |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1113 |
(auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"]) |
47694 | 1114 |
qed |
1115 |
finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
1116 |
using P_closed by simp |
|
1117 |
qed |
|
1118 |
qed |
|
1119 |
from measurable_sets[OF this, of A] A `i \<in> I` E_closed |
|
1120 |
have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
1121 |
by (simp add: E_generates) |
|
1122 |
also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}" |
|
1123 |
using P_closed by (auto simp: space_PiM) |
|
1124 |
finally show "\<dots> \<in> sets ?P" . |
|
1125 |
qed |
|
1126 |
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P" |
|
1127 |
by (simp add: P_closed) |
|
1128 |
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)" |
|
1129 |
using `finite I` |
|
1130 |
by (auto intro!: sigma_sets_subset simp: E_generates P_def) |
|
1131 |
qed |
|
1132 |
||
1133 |
end |