author | hoelzl |
Mon, 23 May 2011 19:21:05 +0200 | |
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parent 42146 | 5b52c6a9c627 |
child 42984 | 43864e7475df |
permissions | -rw-r--r-- |
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(* Title: HOL/Probability/Binary_Product_Measure.thy |
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Author: Johannes Hölzl, TU München |
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*) |
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header {*Binary product measures*} |
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theory Binary_Product_Measure |
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imports Lebesgue_Integration |
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begin |
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lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))" |
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by auto |
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lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)" |
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by auto |
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lemma Pair_vimage_times[simp]: "\<And>A B x. Pair x -` (A \<times> B) = (if x \<in> A then B else {})" |
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by auto |
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lemma rev_Pair_vimage_times[simp]: "\<And>A B y. (\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})" |
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by auto |
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lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))" |
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by (cases x) simp |
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lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))" |
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by (auto simp: fun_eq_iff) |
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section "Binary products" |
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definition |
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"pair_measure_generator A B = |
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\<lparr> space = space A \<times> space B, |
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sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B}, |
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measure = \<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A \<rparr>" |
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|
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definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where |
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"A \<Otimes>\<^isub>M B = sigma (pair_measure_generator A B)" |
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locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2 |
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for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme" |
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|
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abbreviation (in pair_sigma_algebra) |
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"E \<equiv> pair_measure_generator M1 M2" |
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abbreviation (in pair_sigma_algebra) |
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"P \<equiv> M1 \<Otimes>\<^isub>M M2" |
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lemma sigma_algebra_pair_measure: |
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"sets M1 \<subseteq> Pow (space M1) \<Longrightarrow> sets M2 \<subseteq> Pow (space M2) \<Longrightarrow> sigma_algebra (pair_measure M1 M2)" |
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by (force simp: pair_measure_def pair_measure_generator_def intro!: sigma_algebra_sigma) |
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sublocale pair_sigma_algebra \<subseteq> sigma_algebra P |
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using M1.space_closed M2.space_closed |
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by (rule sigma_algebra_pair_measure) |
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lemma pair_measure_generatorI[intro, simp]: |
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"x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_measure_generator A B)" |
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by (auto simp add: pair_measure_generator_def) |
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lemma pair_measureI[intro, simp]: |
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"x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)" |
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by (auto simp add: pair_measure_def) |
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|
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lemma space_pair_measure: |
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"space (A \<Otimes>\<^isub>M B) = space A \<times> space B" |
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by (simp add: pair_measure_def pair_measure_generator_def) |
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lemma sets_pair_measure_generator: |
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"sets (pair_measure_generator N M) = (\<lambda>(x, y). x \<times> y) ` (sets N \<times> sets M)" |
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unfolding pair_measure_generator_def by auto |
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lemma pair_measure_generator_sets_into_space: |
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assumes "sets M \<subseteq> Pow (space M)" "sets N \<subseteq> Pow (space N)" |
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shows "sets (pair_measure_generator M N) \<subseteq> Pow (space (pair_measure_generator M N))" |
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using assms by (auto simp: pair_measure_generator_def) |
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|
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lemma pair_measure_generator_Int_snd: |
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assumes "sets S1 \<subseteq> Pow (space S1)" |
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shows "sets (pair_measure_generator S1 (algebra.restricted_space S2 A)) = |
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sets (algebra.restricted_space (pair_measure_generator S1 S2) (space S1 \<times> A))" |
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(is "?L = ?R") |
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apply (auto simp: pair_measure_generator_def image_iff) |
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using assms |
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apply (rule_tac x="a \<times> xa" in exI) |
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apply force |
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using assms |
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apply (rule_tac x="a" in exI) |
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apply (rule_tac x="b \<inter> A" in exI) |
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apply auto |
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done |
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lemma (in pair_sigma_algebra) |
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shows measurable_fst[intro!, simp]: |
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"fst \<in> measurable P M1" (is ?fst) |
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and measurable_snd[intro!, simp]: |
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"snd \<in> measurable P M2" (is ?snd) |
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Measurable on product space is equiv. to measurable components
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proof - |
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{ fix X assume "X \<in> sets M1" |
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then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2" |
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apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"]) |
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using M1.sets_into_space by force+ } |
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moreover |
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{ fix X assume "X \<in> sets M2" |
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then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2" |
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apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X]) |
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using M2.sets_into_space by force+ } |
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ultimately have "?fst \<and> ?snd" |
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by (fastsimp simp: measurable_def sets_sigma space_pair_measure |
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intro!: sigma_sets.Basic) |
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then show ?fst ?snd by auto |
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qed |
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lemma (in pair_sigma_algebra) measurable_pair_iff: |
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assumes "sigma_algebra M" |
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shows "f \<in> measurable M P \<longleftrightarrow> |
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(fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2" |
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proof - |
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interpret M: sigma_algebra M by fact |
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from assms show ?thesis |
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proof (safe intro!: measurable_comp[where b=P]) |
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assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2" |
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show "f \<in> measurable M P" unfolding pair_measure_def |
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proof (rule M.measurable_sigma) |
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show "sets (pair_measure_generator M1 M2) \<subseteq> Pow (space E)" |
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unfolding pair_measure_generator_def using M1.sets_into_space M2.sets_into_space by auto |
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show "f \<in> space M \<rightarrow> space E" |
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using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma pair_measure_generator_def) |
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fix A assume "A \<in> sets E" |
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then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C" |
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unfolding pair_measure_generator_def by auto |
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moreover have "(fst \<circ> f) -` B \<inter> space M \<in> sets M" |
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using f `B \<in> sets M1` unfolding measurable_def by auto |
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moreover have "(snd \<circ> f) -` C \<inter> space M \<in> sets M" |
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using s `C \<in> sets M2` unfolding measurable_def by auto |
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moreover have "f -` A \<inter> space M = ((fst \<circ> f) -` B \<inter> space M) \<inter> ((snd \<circ> f) -` C \<inter> space M)" |
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unfolding `A = B \<times> C` by (auto simp: vimage_Times) |
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ultimately show "f -` A \<inter> space M \<in> sets M" by auto |
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qed |
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qed |
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qed |
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lemma (in pair_sigma_algebra) measurable_pair: |
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assumes "sigma_algebra M" |
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assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2" |
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shows "f \<in> measurable M P" |
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unfolding measurable_pair_iff[OF assms(1)] using assms(2,3) by simp |
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lemma pair_measure_generatorE: |
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assumes "X \<in> sets (pair_measure_generator M1 M2)" |
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obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2" |
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using assms unfolding pair_measure_generator_def by auto |
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lemma (in pair_sigma_algebra) pair_measure_generator_swap: |
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"(\<lambda>X. (\<lambda>(x,y). (y,x)) -` X \<inter> space M2 \<times> space M1) ` sets E = sets (pair_measure_generator M2 M1)" |
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156 |
proof (safe elim!: pair_measure_generatorE) |
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fix A B assume "A \<in> sets M1" "B \<in> sets M2" |
158 |
moreover then have "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A" |
|
159 |
using M1.sets_into_space M2.sets_into_space by auto |
|
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
160 |
ultimately show "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_measure_generator M2 M1)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
161 |
by (auto intro: pair_measure_generatorI) |
40859 | 162 |
next |
163 |
fix A B assume "A \<in> sets M1" "B \<in> sets M2" |
|
164 |
then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E" |
|
165 |
using M1.sets_into_space M2.sets_into_space |
|
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
166 |
by (auto intro!: image_eqI[where x="A \<times> B"] pair_measure_generatorI) |
40859 | 167 |
qed |
168 |
||
169 |
lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap: |
|
170 |
assumes Q: "Q \<in> sets P" |
|
41689
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parents:
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diff
changeset
|
171 |
shows "(\<lambda>(x,y). (y, x)) -` Q \<in> sets (M2 \<Otimes>\<^isub>M M1)" (is "_ \<in> sets ?Q") |
40859 | 172 |
proof - |
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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parents:
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changeset
|
173 |
let "?f Q" = "(\<lambda>(x,y). (y, x)) -` Q \<inter> space M2 \<times> space M1" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
174 |
have *: "(\<lambda>(x,y). (y, x)) -` Q = ?f Q" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
175 |
using sets_into_space[OF Q] by (auto simp: space_pair_measure) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
176 |
have "sets (M2 \<Otimes>\<^isub>M M1) = sets (sigma (pair_measure_generator M2 M1))" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
177 |
unfolding pair_measure_def .. |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
178 |
also have "\<dots> = sigma_sets (space M2 \<times> space M1) (?f ` sets E)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
179 |
unfolding sigma_def pair_measure_generator_swap[symmetric] |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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parents:
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diff
changeset
|
180 |
by (simp add: pair_measure_generator_def) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
181 |
also have "\<dots> = ?f ` sigma_sets (space M1 \<times> space M2) (sets E)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
182 |
using M1.sets_into_space M2.sets_into_space |
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
|
183 |
by (intro sigma_sets_vimage) (auto simp: pair_measure_generator_def) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
184 |
also have "\<dots> = ?f ` sets P" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
185 |
unfolding pair_measure_def pair_measure_generator_def sigma_def by simp |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
186 |
finally show ?thesis |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
187 |
using Q by (subst *) auto |
40859 | 188 |
qed |
189 |
||
190 |
lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable: |
|
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
191 |
shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (M2 \<Otimes>\<^isub>M M1)" |
40859 | 192 |
(is "?f \<in> measurable ?P ?Q") |
193 |
unfolding measurable_def |
|
194 |
proof (intro CollectI conjI Pi_I ballI) |
|
195 |
fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
196 |
unfolding pair_measure_generator_def pair_measure_def by auto |
40859 | 197 |
next |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
198 |
fix A assume "A \<in> sets (M2 \<Otimes>\<^isub>M M1)" |
40859 | 199 |
interpret Q: pair_sigma_algebra M2 M1 by default |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
200 |
with Q.sets_pair_sigma_algebra_swap[OF `A \<in> sets (M2 \<Otimes>\<^isub>M M1)`] |
40859 | 201 |
show "?f -` A \<inter> space ?P \<in> sets ?P" by simp |
202 |
qed |
|
203 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset
|
204 |
lemma (in pair_sigma_algebra) measurable_cut_fst[simp,intro]: |
40859 | 205 |
assumes "Q \<in> sets P" shows "Pair x -` Q \<in> sets M2" |
206 |
proof - |
|
207 |
let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x -` Q \<in> sets M2}" |
|
208 |
let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>" |
|
209 |
interpret Q: sigma_algebra ?Q |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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changeset
|
210 |
proof qed (auto simp: vimage_UN vimage_Diff space_pair_measure) |
40859 | 211 |
have "sets E \<subseteq> sets ?Q" |
212 |
using M1.sets_into_space M2.sets_into_space |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
213 |
by (auto simp: pair_measure_generator_def space_pair_measure) |
40859 | 214 |
then have "sets P \<subseteq> sets ?Q" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
215 |
apply (subst pair_measure_def, intro Q.sets_sigma_subset) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
216 |
by (simp add: pair_measure_def) |
40859 | 217 |
with assms show ?thesis by auto |
218 |
qed |
|
219 |
||
220 |
lemma (in pair_sigma_algebra) measurable_cut_snd: |
|
221 |
assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) -` Q \<in> sets M1" (is "?cut Q \<in> sets M1") |
|
222 |
proof - |
|
223 |
interpret Q: pair_sigma_algebra M2 M1 by default |
|
224 |
with Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y] |
|
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
|
225 |
show ?thesis by (simp add: vimage_compose[symmetric] comp_def) |
40859 | 226 |
qed |
227 |
||
228 |
lemma (in pair_sigma_algebra) measurable_pair_image_snd: |
|
229 |
assumes m: "f \<in> measurable P M" and "x \<in> space M1" |
|
230 |
shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M" |
|
231 |
unfolding measurable_def |
|
232 |
proof (intro CollectI conjI Pi_I ballI) |
|
233 |
fix y assume "y \<in> space M2" with `f \<in> measurable P M` `x \<in> space M1` |
|
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
|
234 |
show "f (x, y) \<in> space 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
|
235 |
unfolding measurable_def pair_measure_generator_def pair_measure_def by auto |
40859 | 236 |
next |
237 |
fix A assume "A \<in> sets M" |
|
238 |
then have "Pair x -` (f -` A \<inter> space P) \<in> sets M2" (is "?C \<in> _") |
|
239 |
using `f \<in> measurable P M` |
|
240 |
by (intro measurable_cut_fst) (auto simp: measurable_def) |
|
241 |
also have "?C = (\<lambda>y. f (x, y)) -` A \<inter> space M2" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
242 |
using `x \<in> space M1` by (auto simp: pair_measure_generator_def pair_measure_def) |
40859 | 243 |
finally show "(\<lambda>y. f (x, y)) -` A \<inter> space M2 \<in> sets M2" . |
244 |
qed |
|
245 |
||
246 |
lemma (in pair_sigma_algebra) measurable_pair_image_fst: |
|
247 |
assumes m: "f \<in> measurable P M" and "y \<in> space M2" |
|
248 |
shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M" |
|
249 |
proof - |
|
250 |
interpret Q: pair_sigma_algebra M2 M1 by default |
|
251 |
from Q.measurable_pair_image_snd[OF measurable_comp `y \<in> space M2`, |
|
252 |
OF Q.pair_sigma_algebra_swap_measurable m] |
|
253 |
show ?thesis by simp |
|
254 |
qed |
|
255 |
||
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
|
256 |
lemma (in pair_sigma_algebra) Int_stable_pair_measure_generator: "Int_stable E" |
40859 | 257 |
unfolding Int_stable_def |
258 |
proof (intro ballI) |
|
259 |
fix A B assume "A \<in> sets E" "B \<in> sets E" |
|
260 |
then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2" |
|
261 |
"A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
262 |
unfolding pair_measure_generator_def by auto |
40859 | 263 |
then show "A \<inter> B \<in> sets E" |
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
|
264 |
by (auto simp add: times_Int_times pair_measure_generator_def) |
40859 | 265 |
qed |
266 |
||
267 |
lemma finite_measure_cut_measurable: |
|
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
|
268 |
fixes M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme" |
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
|
269 |
assumes "sigma_finite_measure M1" "finite_measure M2" |
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
|
270 |
assumes "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)" |
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
|
271 |
shows "(\<lambda>x. measure M2 (Pair x -` Q)) \<in> borel_measurable M1" |
40859 | 272 |
(is "?s Q \<in> _") |
273 |
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
|
274 |
interpret M1: sigma_finite_measure M1 by 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
|
275 |
interpret M2: finite_measure M2 by fact |
40859 | 276 |
interpret pair_sigma_algebra M1 M2 by default |
277 |
have [intro]: "sigma_algebra M1" by fact |
|
278 |
have [intro]: "sigma_algebra M2" by fact |
|
279 |
let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1} \<rparr>" |
|
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
|
280 |
note space_pair_measure[simp] |
40859 | 281 |
interpret dynkin_system ?D |
282 |
proof (intro dynkin_systemI) |
|
283 |
fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D" |
|
284 |
using sets_into_space by simp |
|
285 |
next |
|
286 |
from top show "space ?D \<in> sets ?D" |
|
287 |
by (auto simp add: if_distrib intro!: M1.measurable_If) |
|
288 |
next |
|
289 |
fix A assume "A \<in> sets ?D" |
|
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
|
290 |
with sets_into_space have "\<And>x. measure M2 (Pair x -` (space M1 \<times> space M2 - A)) = |
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
|
291 |
(if x \<in> space M1 then measure M2 (space M2) - ?s A x else 0)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
292 |
by (auto intro!: M2.measure_compl simp: vimage_Diff) |
40859 | 293 |
with `A \<in> sets ?D` top show "space ?D - A \<in> sets ?D" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
294 |
by (auto intro!: Diff M1.measurable_If M1.borel_measurable_extreal_diff) |
40859 | 295 |
next |
296 |
fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
297 |
moreover then have "\<And>x. measure M2 (\<Union>i. Pair x -` F i) = (\<Sum>i. ?s (F i) x)" |
40859 | 298 |
by (intro M2.measure_countably_additive[symmetric]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
299 |
(auto simp: disjoint_family_on_def) |
40859 | 300 |
ultimately show "(\<Union>i. F i) \<in> sets ?D" |
301 |
by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf) |
|
302 |
qed |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
303 |
have "sets P = sets ?D" apply (subst pair_measure_def) |
40859 | 304 |
proof (intro dynkin_lemma) |
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
|
305 |
show "Int_stable E" by (rule Int_stable_pair_measure_generator) |
40859 | 306 |
from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A" |
307 |
by auto |
|
308 |
then show "sets E \<subseteq> sets ?D" |
|
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
|
309 |
by (auto simp: pair_measure_generator_def sets_sigma if_distrib |
40859 | 310 |
intro: sigma_sets.Basic intro!: M1.measurable_If) |
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
|
311 |
qed (auto simp: pair_measure_def) |
40859 | 312 |
with `Q \<in> sets P` have "Q \<in> sets ?D" by simp |
313 |
then show "?s Q \<in> borel_measurable M1" by simp |
|
314 |
qed |
|
315 |
||
316 |
subsection {* Binary products of $\sigma$-finite measure spaces *} |
|
317 |
||
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
|
318 |
locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2 |
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
|
319 |
for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme" |
40859 | 320 |
|
321 |
sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2 |
|
322 |
by default |
|
323 |
||
42146
5b52c6a9c627
split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents:
42067
diff
changeset
|
324 |
lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B" |
5b52c6a9c627
split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents:
42067
diff
changeset
|
325 |
proof |
5b52c6a9c627
split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents:
42067
diff
changeset
|
326 |
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" |
5b52c6a9c627
split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents:
42067
diff
changeset
|
327 |
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros) |
5b52c6a9c627
split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents:
42067
diff
changeset
|
328 |
qed |
5b52c6a9c627
split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents:
42067
diff
changeset
|
329 |
|
40859 | 330 |
lemma (in pair_sigma_finite) measure_cut_measurable_fst: |
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
|
331 |
assumes "Q \<in> sets P" shows "(\<lambda>x. measure M2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _") |
40859 | 332 |
proof - |
333 |
have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+ |
|
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
|
334 |
have M1: "sigma_finite_measure M1" by default |
40859 | 335 |
from M2.disjoint_sigma_finite guess F .. note F = this |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
336 |
then have F_sets: "\<And>i. F i \<in> sets M2" by auto |
40859 | 337 |
let "?C x i" = "F i \<inter> Pair x -` Q" |
338 |
{ fix i |
|
339 |
let ?R = "M2.restricted_space (F i)" |
|
340 |
have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i" |
|
341 |
using F M2.sets_into_space 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
|
342 |
let ?R2 = "M2.restricted_space (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
|
343 |
have "(\<lambda>x. measure ?R2 (Pair x -` (space M1 \<times> space ?R2 \<inter> Q))) \<in> borel_measurable M1" |
40859 | 344 |
proof (intro finite_measure_cut_measurable[OF M1]) |
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
|
345 |
show "finite_measure ?R2" |
40859 | 346 |
using F by (intro M2.restricted_to_finite_measure) 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
|
347 |
have "(space M1 \<times> space ?R2) \<inter> Q \<in> (op \<inter> (space M1 \<times> F i)) ` sets P" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
348 |
using `Q \<in> sets P` by (auto simp: image_iff) |
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
|
349 |
also have "\<dots> = sigma_sets (space M1 \<times> F i) ((op \<inter> (space M1 \<times> F i)) ` sets E)" |
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
|
350 |
unfolding pair_measure_def pair_measure_generator_def sigma_def |
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
|
351 |
using `F i \<in> sets M2` M2.sets_into_space |
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
|
352 |
by (auto intro!: sigma_sets_Int sigma_sets.Basic) |
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
|
353 |
also have "\<dots> \<subseteq> sets (M1 \<Otimes>\<^isub>M ?R2)" |
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
|
354 |
using M1.sets_into_space |
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
|
355 |
apply (auto simp: times_Int_times pair_measure_def pair_measure_generator_def sigma_def |
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
|
356 |
intro!: sigma_sets_subseteq) |
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
|
357 |
apply (rule_tac x="a" in exI) |
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
|
358 |
apply (rule_tac x="b \<inter> F i" in exI) |
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
|
359 |
by auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
360 |
finally show "(space M1 \<times> space ?R2) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R2)" . |
40859 | 361 |
qed |
362 |
moreover have "\<And>x. Pair x -` (space M1 \<times> F i \<inter> Q) = ?C x 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
|
363 |
using `Q \<in> sets P` sets_into_space by (auto simp: space_pair_measure) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
364 |
ultimately have "(\<lambda>x. measure M2 (?C x i)) \<in> borel_measurable M1" |
40859 | 365 |
by simp } |
366 |
moreover |
|
367 |
{ fix x |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
368 |
have "(\<Sum>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)" |
40859 | 369 |
proof (intro M2.measure_countably_additive) |
370 |
show "range (?C x) \<subseteq> sets M2" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
371 |
using F `Q \<in> sets P` by (auto intro!: M2.Int) |
40859 | 372 |
have "disjoint_family F" using F by auto |
373 |
show "disjoint_family (?C x)" |
|
374 |
by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto |
|
375 |
qed |
|
376 |
also have "(\<Union>i. ?C x i) = Pair x -` Q" |
|
377 |
using F sets_into_space `Q \<in> sets P` |
|
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
|
378 |
by (auto simp: space_pair_measure) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
379 |
finally have "measure M2 (Pair x -` Q) = (\<Sum>i. measure M2 (?C x i))" |
40859 | 380 |
by simp } |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
381 |
ultimately show ?thesis using `Q \<in> sets P` F_sets |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
382 |
by (auto intro!: M1.borel_measurable_psuminf M2.Int) |
40859 | 383 |
qed |
384 |
||
385 |
lemma (in pair_sigma_finite) measure_cut_measurable_snd: |
|
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
|
386 |
assumes "Q \<in> sets P" shows "(\<lambda>y. M1.\<mu> ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2" |
40859 | 387 |
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
|
388 |
interpret Q: pair_sigma_finite M2 M1 by default |
40859 | 389 |
note sets_pair_sigma_algebra_swap[OF assms] |
390 |
from Q.measure_cut_measurable_fst[OF this] |
|
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
|
391 |
show ?thesis by (simp add: vimage_compose[symmetric] comp_def) |
40859 | 392 |
qed |
393 |
||
394 |
lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable: |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
395 |
assumes "f \<in> measurable P M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M" |
40859 | 396 |
proof - |
397 |
interpret Q: pair_sigma_algebra M2 M1 by default |
|
398 |
have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff) |
|
399 |
show ?thesis |
|
400 |
using Q.pair_sigma_algebra_swap_measurable assms |
|
401 |
unfolding * by (rule measurable_comp) |
|
39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
402 |
qed |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
403 |
|
40859 | 404 |
lemma (in pair_sigma_finite) pair_measure_alt: |
405 |
assumes "A \<in> sets P" |
|
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
|
406 |
shows "measure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+ x. measure M2 (Pair x -` A) \<partial>M1)" |
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
|
407 |
apply (simp add: pair_measure_def pair_measure_generator_def) |
40859 | 408 |
proof (rule M1.positive_integral_cong) |
409 |
fix x assume "x \<in> space M1" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
410 |
have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: extreal)" |
40859 | 411 |
unfolding indicator_def 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
|
412 |
show "(\<integral>\<^isup>+ y. indicator A (x, y) \<partial>M2) = measure M2 (Pair x -` A)" |
40859 | 413 |
unfolding * |
414 |
apply (subst M2.positive_integral_indicator) |
|
415 |
apply (rule measurable_cut_fst[OF assms]) |
|
416 |
by simp |
|
417 |
qed |
|
418 |
||
419 |
lemma (in pair_sigma_finite) pair_measure_times: |
|
420 |
assumes A: "A \<in> sets M1" and "B \<in> sets M2" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
421 |
shows "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = M1.\<mu> A * measure M2 B" |
40859 | 422 |
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
|
423 |
have "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+ x. measure M2 B * indicator A x \<partial>M1)" |
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
|
424 |
using assms by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt) |
40859 | 425 |
with assms show ?thesis |
426 |
by (simp add: M1.positive_integral_cmult_indicator ac_simps) |
|
427 |
qed |
|
428 |
||
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
429 |
lemma (in measure_space) measure_not_negative[simp,intro]: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
430 |
assumes A: "A \<in> sets M" shows "\<mu> A \<noteq> - \<infinity>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
431 |
using positive_measure[OF A] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
432 |
|
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
|
433 |
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
434 |
"\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> incseq F \<and> (\<Union>i. F i) = space E \<and> |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
435 |
(\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)" |
40859 | 436 |
proof - |
437 |
obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
438 |
F1: "range F1 \<subseteq> sets M1" "incseq F1" "(\<Union>i. F1 i) = space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<infinity>" and |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
439 |
F2: "range F2 \<subseteq> sets M2" "incseq F2" "(\<Union>i. F2 i) = space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<infinity>" |
40859 | 440 |
using M1.sigma_finite_up M2.sigma_finite_up by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
441 |
then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto |
40859 | 442 |
let ?F = "\<lambda>i. F1 i \<times> F2 i" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
443 |
show ?thesis unfolding space_pair_measure |
40859 | 444 |
proof (intro exI[of _ ?F] conjI allI) |
445 |
show "range ?F \<subseteq> sets E" using F1 F2 |
|
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
|
446 |
by (fastsimp intro!: pair_measure_generatorI) |
40859 | 447 |
next |
448 |
have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)" |
|
449 |
proof (intro subsetI) |
|
450 |
fix x assume "x \<in> space M1 \<times> space M2" |
|
451 |
then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j" |
|
452 |
by (auto simp: space) |
|
453 |
then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
454 |
using `incseq F1` `incseq F2` unfolding incseq_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
455 |
by (force split: split_max)+ |
40859 | 456 |
then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)" |
457 |
by (intro SigmaI) (auto simp add: min_max.sup_commute) |
|
458 |
then show "x \<in> (\<Union>i. ?F i)" by auto |
|
459 |
qed |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
460 |
then show "(\<Union>i. ?F i) = space E" |
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
|
461 |
using space by (auto simp: space pair_measure_generator_def) |
40859 | 462 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
463 |
fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
464 |
using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto |
40859 | 465 |
next |
466 |
fix i |
|
467 |
from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
468 |
with F1 F2 M1.positive_measure[OF this(1)] M2.positive_measure[OF this(2)] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
469 |
show "measure P (F1 i \<times> F2 i) \<noteq> \<infinity>" |
40859 | 470 |
by (simp add: pair_measure_times) |
471 |
qed |
|
472 |
qed |
|
473 |
||
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
|
474 |
sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P |
40859 | 475 |
proof |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
476 |
show "positive P (measure P)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
477 |
unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
478 |
by (auto intro: M1.positive_integral_positive M2.positive_integral_positive) |
40859 | 479 |
|
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
|
480 |
show "countably_additive P (measure P)" |
40859 | 481 |
unfolding countably_additive_def |
482 |
proof (intro allI impI) |
|
483 |
fix F :: "nat \<Rightarrow> ('a \<times> 'b) set" |
|
484 |
assume F: "range F \<subseteq> sets P" "disjoint_family F" |
|
485 |
from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" 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
|
486 |
moreover from F have "\<And>i. (\<lambda>x. measure M2 (Pair x -` F i)) \<in> borel_measurable M1" |
40859 | 487 |
by (intro measure_cut_measurable_fst) auto |
488 |
moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)" |
|
489 |
by (intro disjoint_family_on_bisimulation[OF F(2)]) auto |
|
490 |
moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
491 |
using F by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
492 |
ultimately show "(\<Sum>n. measure P (F n)) = measure P (\<Union>i. F i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
493 |
by (simp add: pair_measure_alt vimage_UN M1.positive_integral_suminf[symmetric] |
40859 | 494 |
M2.measure_countably_additive |
495 |
cong: M1.positive_integral_cong) |
|
496 |
qed |
|
497 |
||
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
|
498 |
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
499 |
show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. measure P (F i) \<noteq> \<infinity>)" |
40859 | 500 |
proof (rule exI[of _ F], intro conjI) |
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
|
501 |
show "range F \<subseteq> sets P" using F by (auto simp: pair_measure_def) |
40859 | 502 |
show "(\<Union>i. F i) = space P" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
503 |
using F by (auto simp: pair_measure_def pair_measure_generator_def) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
504 |
show "\<forall>i. measure P (F i) \<noteq> \<infinity>" using F by auto |
40859 | 505 |
qed |
506 |
qed |
|
39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
507 |
|
41661 | 508 |
lemma (in pair_sigma_algebra) sets_swap: |
509 |
assumes "A \<in> sets P" |
|
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
|
510 |
shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)" |
41661 | 511 |
(is "_ -` A \<inter> space ?Q \<in> sets ?Q") |
512 |
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
|
513 |
have *: "(\<lambda>(x, y). (y, x)) -` A \<inter> space ?Q = (\<lambda>(x, y). (y, x)) -` A" |
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
|
514 |
using `A \<in> sets P` sets_into_space by (auto simp: space_pair_measure) |
41661 | 515 |
show ?thesis |
516 |
unfolding * using assms by (rule sets_pair_sigma_algebra_swap) |
|
517 |
qed |
|
518 |
||
40859 | 519 |
lemma (in pair_sigma_finite) pair_measure_alt2: |
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
520 |
assumes A: "A \<in> sets P" |
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
|
521 |
shows "\<mu> A = (\<integral>\<^isup>+y. M1.\<mu> ((\<lambda>x. (x, y)) -` A) \<partial>M2)" |
40859 | 522 |
(is "_ = ?\<nu> A") |
523 |
proof - |
|
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
524 |
interpret Q: pair_sigma_finite M2 M1 by default |
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
|
525 |
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this |
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
|
526 |
have [simp]: "\<And>m. \<lparr> space = space E, sets = sets (sigma E), measure = m \<rparr> = P\<lparr> measure := m \<rparr>" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
527 |
unfolding pair_measure_def by simp |
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
528 |
|
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
529 |
have "\<mu> A = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` A \<inter> space Q.P)" |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
530 |
proof (rule measure_unique_Int_stable_vimage[OF Int_stable_pair_measure_generator]) |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
531 |
show "measure_space P" "measure_space Q.P" by default |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
532 |
show "(\<lambda>(y, x). (x, y)) \<in> measurable Q.P P" by (rule Q.pair_sigma_algebra_swap_measurable) |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
533 |
show "sets (sigma E) = sets P" "space E = space P" "A \<in> sets (sigma E)" |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
534 |
using assms unfolding pair_measure_def by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
535 |
show "range F \<subseteq> sets E" "incseq F" "(\<Union>i. F i) = space E" "\<And>i. \<mu> (F i) \<noteq> \<infinity>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
536 |
using F `A \<in> sets P` by (auto simp: pair_measure_def) |
40859 | 537 |
fix X assume "X \<in> sets E" |
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
538 |
then obtain A B where X[simp]: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
539 |
unfolding pair_measure_def pair_measure_generator_def by auto |
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
540 |
then have "(\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P = B \<times> A" |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
541 |
using M1.sets_into_space M2.sets_into_space by (auto simp: space_pair_measure) |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
542 |
then show "\<mu> X = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P)" |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
543 |
using AB by (simp add: pair_measure_times Q.pair_measure_times ac_simps) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
544 |
qed |
41706
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
545 |
then show ?thesis |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
546 |
using sets_into_space[OF A] Q.pair_measure_alt[OF sets_swap[OF A]] |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
547 |
by (auto simp add: Q.pair_measure_alt space_pair_measure |
a207a858d1f6
prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents:
41705
diff
changeset
|
548 |
intro!: M2.positive_integral_cong arg_cong[where f="M1.\<mu>"]) |
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 |
qed |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
550 |
|
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
|
551 |
lemma pair_sigma_algebra_sigma: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
552 |
assumes 1: "incseq S1" "(\<Union>i. S1 i) = space E1" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
553 |
assumes 2: "decseq S2" "(\<Union>i. S2 i) = space E2" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)" |
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
|
554 |
shows "sets (sigma (pair_measure_generator (sigma E1) (sigma E2))) = sets (sigma (pair_measure_generator E1 E2))" |
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 |
(is "sets ?S = sets ?E") |
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
|
556 |
proof - |
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
|
557 |
interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma) |
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
|
558 |
interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma) |
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
|
559 |
have P: "sets (pair_measure_generator E1 E2) \<subseteq> Pow (space E1 \<times> space E2)" |
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
|
560 |
using E1 E2 by (auto simp add: pair_measure_generator_def) |
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
|
561 |
interpret E: sigma_algebra ?E unfolding pair_measure_generator_def |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
562 |
using E1 E2 by (intro sigma_algebra_sigma) 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
|
563 |
{ fix A assume "A \<in> sets E1" |
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
|
564 |
then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
565 |
using E1 2 unfolding pair_measure_generator_def 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
|
566 |
also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
567 |
also have "\<dots> \<in> sets ?E" unfolding pair_measure_generator_def sets_sigma |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
568 |
using 2 `A \<in> sets E1` |
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
|
569 |
by (intro sigma_sets.Union) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
570 |
(force simp: image_subset_iff intro!: sigma_sets.Basic) |
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
|
571 |
finally have "fst -` A \<inter> space ?E \<in> sets ?E" . } |
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
|
572 |
moreover |
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
|
573 |
{ fix B assume "B \<in> sets E2" |
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
|
574 |
then have "snd -` B \<inter> space ?E = (\<Union>i. S1 i) \<times> B" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
575 |
using E2 1 unfolding pair_measure_generator_def 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
|
576 |
also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
577 |
also have "\<dots> \<in> sets ?E" |
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
|
578 |
using 1 `B \<in> sets E2` unfolding pair_measure_generator_def sets_sigma |
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
|
579 |
by (intro sigma_sets.Union) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
580 |
(force simp: image_subset_iff intro!: sigma_sets.Basic) |
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
|
581 |
finally have "snd -` B \<inter> space ?E \<in> sets ?E" . } |
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
|
582 |
ultimately have proj: |
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
|
583 |
"fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)" |
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
|
584 |
using E1 E2 by (subst (1 2) E.measurable_iff_sigma) |
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
|
585 |
(auto simp: pair_measure_generator_def sets_sigma) |
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
|
586 |
{ fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)" |
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
|
587 |
with proj have "fst -` A \<inter> space ?E \<in> sets ?E" "snd -` B \<inter> space ?E \<in> sets ?E" |
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
|
588 |
unfolding measurable_def by simp_all |
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
|
589 |
moreover have "A \<times> B = (fst -` A \<inter> space ?E) \<inter> (snd -` B \<inter> space ?E)" |
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
|
590 |
using A B M1.sets_into_space M2.sets_into_space |
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
|
591 |
by (auto simp: pair_measure_generator_def) |
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
|
592 |
ultimately have "A \<times> B \<in> sets ?E" by auto } |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
593 |
then have "sigma_sets (space ?E) (sets (pair_measure_generator (sigma E1) (sigma E2))) \<subseteq> sets ?E" |
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
|
594 |
by (intro E.sigma_sets_subset) (auto simp add: pair_measure_generator_def sets_sigma) |
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
|
595 |
then have subset: "sets ?S \<subseteq> sets ?E" |
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
|
596 |
by (simp add: sets_sigma pair_measure_generator_def) |
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
|
597 |
show "sets ?S = sets ?E" |
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
|
598 |
proof (intro set_eqI iffI) |
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
|
599 |
fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S" |
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
|
600 |
unfolding sets_sigma |
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
|
601 |
proof induct |
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
|
602 |
case (Basic A) then show ?case |
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
|
603 |
by (auto simp: pair_measure_generator_def sets_sigma intro: sigma_sets.Basic) |
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
|
604 |
qed (auto intro: sigma_sets.intros simp: pair_measure_generator_def) |
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
|
605 |
next |
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
|
606 |
fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
607 |
qed |
40859 | 608 |
qed |
609 |
||
610 |
section "Fubinis theorem" |
|
611 |
||
612 |
lemma (in pair_sigma_finite) simple_function_cut: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
613 |
assumes f: "simple_function P f" "\<And>x. 0 \<le> f x" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
614 |
shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1" |
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
|
615 |
and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" |
40859 | 616 |
proof - |
617 |
have f_borel: "f \<in> borel_measurable P" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
618 |
using f(1) by (rule borel_measurable_simple_function) |
40859 | 619 |
let "?F z" = "f -` {z} \<inter> space P" |
620 |
let "?F' x z" = "Pair x -` ?F z" |
|
621 |
{ fix x assume "x \<in> space M1" |
|
622 |
have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y" |
|
623 |
by (auto simp: indicator_def) |
|
624 |
have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using `x \<in> space M1` |
|
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
|
625 |
by (simp add: space_pair_measure) |
40859 | 626 |
moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel |
627 |
by (intro borel_measurable_vimage measurable_cut_fst) |
|
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
|
628 |
ultimately have "simple_function M2 (\<lambda> y. f (x, y))" |
40859 | 629 |
apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
630 |
apply (rule simple_function_indicator_representation[OF f(1)]) |
40859 | 631 |
using `x \<in> space M1` by (auto simp del: space_sigma) } |
632 |
note M2_sf = this |
|
633 |
{ fix x assume x: "x \<in> space M1" |
|
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
|
634 |
then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f ` space P. z * M2.\<mu> (?F' x z))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
635 |
unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)] |
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
|
636 |
unfolding simple_integral_def |
40859 | 637 |
proof (safe intro!: setsum_mono_zero_cong_left) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
638 |
from f(1) show "finite (f ` space P)" by (rule simple_functionD) |
40859 | 639 |
next |
640 |
fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space P" |
|
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
|
641 |
using `x \<in> space M1` by (auto simp: space_pair_measure) |
40859 | 642 |
next |
643 |
fix x' y assume "(x', y) \<in> space P" |
|
644 |
"f (x', y) \<notin> (\<lambda>y. f (x, y)) ` space M2" |
|
645 |
then have *: "?F' x (f (x', y)) = {}" |
|
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
|
646 |
by (force simp: space_pair_measure) |
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
|
647 |
show "f (x', y) * M2.\<mu> (?F' x (f (x', y))) = 0" |
40859 | 648 |
unfolding * by simp |
649 |
qed (simp add: vimage_compose[symmetric] 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
|
650 |
space_pair_measure) } |
40859 | 651 |
note eq = this |
652 |
moreover have "\<And>z. ?F z \<in> sets P" |
|
653 |
by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma) |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
654 |
moreover then have "\<And>z. (\<lambda>x. M2.\<mu> (?F' x z)) \<in> borel_measurable M1" |
40859 | 655 |
by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
656 |
moreover have *: "\<And>i x. 0 \<le> M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
657 |
using f(1)[THEN simple_functionD(2)] f(2) by (intro M2.positive_measure measurable_cut_fst) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
658 |
moreover { fix i assume "i \<in> f`space P" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
659 |
with * have "\<And>x. 0 \<le> i * M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
660 |
using f(2) by auto } |
40859 | 661 |
ultimately |
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
|
662 |
show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
663 |
and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" using f(2) |
40859 | 664 |
by (auto simp del: vimage_Int cong: measurable_cong |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
665 |
intro!: M1.borel_measurable_extreal_setsum setsum_cong |
40859 | 666 |
simp add: M1.positive_integral_setsum simple_integral_def |
667 |
M1.positive_integral_cmult |
|
668 |
M1.positive_integral_cong[OF eq] |
|
669 |
positive_integral_eq_simple_integral[OF f] |
|
670 |
pair_measure_alt[symmetric]) |
|
671 |
qed |
|
672 |
||
673 |
lemma (in pair_sigma_finite) positive_integral_fst_measurable: |
|
674 |
assumes f: "f \<in> borel_measurable P" |
|
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
|
675 |
shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1" |
40859 | 676 |
(is "?C f \<in> borel_measurable M1") |
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
|
677 |
and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" |
40859 | 678 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
679 |
from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this |
40859 | 680 |
then have F_borel: "\<And>i. F i \<in> borel_measurable P" |
681 |
by (auto intro: borel_measurable_simple_function) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
682 |
note sf = simple_function_cut[OF F(1,5)] |
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41096
diff
changeset
|
683 |
then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1" |
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41096
diff
changeset
|
684 |
using F(1) by auto |
40859 | 685 |
moreover |
686 |
{ fix x assume "x \<in> space M1" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
687 |
from F measurable_pair_image_snd[OF F_borel`x \<in> space M1`] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
688 |
have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
689 |
by (intro M2.positive_integral_monotone_convergence_SUP) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
690 |
(auto simp: incseq_Suc_iff le_fun_def) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
691 |
then have "(SUP i. ?C (F i) x) = ?C f x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
692 |
unfolding F(4) positive_integral_max_0 by simp } |
40859 | 693 |
note SUPR_C = this |
694 |
ultimately show "?C f \<in> borel_measurable M1" |
|
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41096
diff
changeset
|
695 |
by (simp cong: measurable_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
|
696 |
have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>P) = (SUP i. integral\<^isup>P P (F i))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
697 |
using F_borel F |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
698 |
by (intro positive_integral_monotone_convergence_SUP) 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
|
699 |
also have "(SUP i. integral\<^isup>P P (F i)) = (SUP i. \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1)" |
40859 | 700 |
unfolding sf(2) by simp |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
701 |
also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
702 |
by (intro M1.positive_integral_monotone_convergence_SUP[symmetric]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
703 |
(auto intro!: M2.positive_integral_mono M2.positive_integral_positive |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
704 |
simp: incseq_Suc_iff le_fun_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
|
705 |
also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
706 |
using F_borel F(2,5) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
707 |
by (auto intro!: M1.positive_integral_cong M2.positive_integral_monotone_convergence_SUP[symmetric] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
708 |
simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd) |
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
|
709 |
finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
710 |
using F by (simp add: positive_integral_max_0) |
40859 | 711 |
qed |
712 |
||
41831 | 713 |
lemma (in pair_sigma_finite) measure_preserving_swap: |
714 |
"(\<lambda>(x,y). (y, x)) \<in> measure_preserving (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)" |
|
715 |
proof |
|
716 |
interpret Q: pair_sigma_finite M2 M1 by default |
|
717 |
show *: "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)" |
|
718 |
using pair_sigma_algebra_swap_measurable . |
|
719 |
fix X assume "X \<in> sets (M2 \<Otimes>\<^isub>M M1)" |
|
720 |
from measurable_sets[OF * this] this Q.sets_into_space[OF this] |
|
721 |
show "measure (M1 \<Otimes>\<^isub>M M2) ((\<lambda>(x, y). (y, x)) -` X \<inter> space P) = measure (M2 \<Otimes>\<^isub>M M1) X" |
|
722 |
by (auto intro!: M1.positive_integral_cong arg_cong[where f="M2.\<mu>"] |
|
723 |
simp: pair_measure_alt Q.pair_measure_alt2 space_pair_measure) |
|
724 |
qed |
|
725 |
||
41661 | 726 |
lemma (in pair_sigma_finite) positive_integral_product_swap: |
727 |
assumes f: "f \<in> borel_measurable P" |
|
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
|
728 |
shows "(\<integral>\<^isup>+x. f (case x of (x,y)\<Rightarrow>(y,x)) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P P f" |
41661 | 729 |
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
|
730 |
interpret Q: pair_sigma_finite M2 M1 by default |
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
|
731 |
have "sigma_algebra P" by default |
41831 | 732 |
with f show ?thesis |
733 |
by (subst Q.positive_integral_vimage[OF _ Q.measure_preserving_swap]) auto |
|
41661 | 734 |
qed |
735 |
||
40859 | 736 |
lemma (in pair_sigma_finite) positive_integral_snd_measurable: |
737 |
assumes f: "f \<in> borel_measurable P" |
|
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
|
738 |
shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P P f" |
40859 | 739 |
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
|
740 |
interpret Q: pair_sigma_finite M2 M1 by default |
40859 | 741 |
note pair_sigma_algebra_measurable[OF f] |
742 |
from Q.positive_integral_fst_measurable[OF this] |
|
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
|
743 |
have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P)" |
40859 | 744 |
by 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
|
745 |
also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P) = integral\<^isup>P P f" |
41661 | 746 |
unfolding positive_integral_product_swap[OF f, symmetric] |
747 |
by (auto intro!: Q.positive_integral_cong) |
|
40859 | 748 |
finally show ?thesis . |
749 |
qed |
|
750 |
||
751 |
lemma (in pair_sigma_finite) Fubini: |
|
752 |
assumes f: "f \<in> borel_measurable P" |
|
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
|
753 |
shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1)" |
40859 | 754 |
unfolding positive_integral_snd_measurable[OF assms] |
755 |
unfolding positive_integral_fst_measurable[OF assms] .. |
|
756 |
||
757 |
lemma (in pair_sigma_finite) AE_pair: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
758 |
assumes "AE x in P. Q x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
759 |
shows "AE x in M1. (AE y in M2. Q (x, y))" |
40859 | 760 |
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
|
761 |
obtain N where N: "N \<in> sets P" "\<mu> N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N" |
40859 | 762 |
using assms unfolding almost_everywhere_def by auto |
763 |
show ?thesis |
|
764 |
proof (rule M1.AE_I) |
|
765 |
from N measure_cut_measurable_fst[OF `N \<in> sets P`] |
|
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
|
766 |
show "M1.\<mu> {x\<in>space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} = 0" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
767 |
by (auto simp: pair_measure_alt M1.positive_integral_0_iff) |
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
|
768 |
show "{x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} \<in> sets M1" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
769 |
by (intro M1.borel_measurable_extreal_neq_const measure_cut_measurable_fst 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
|
770 |
{ fix x assume "x \<in> space M1" "M2.\<mu> (Pair x -` N) = 0" |
40859 | 771 |
have "M2.almost_everywhere (\<lambda>y. Q (x, y))" |
772 |
proof (rule M2.AE_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
|
773 |
show "M2.\<mu> (Pair x -` N) = 0" by fact |
40859 | 774 |
show "Pair x -` N \<in> sets M2" by (intro measurable_cut_fst N) |
775 |
show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` 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
|
776 |
using N `x \<in> space M1` unfolding space_sigma space_pair_measure by auto |
40859 | 777 |
qed } |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
778 |
then show "{x \<in> space M1. \<not> M2.almost_everywhere (\<lambda>y. Q (x, y))} \<subseteq> {x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0}" |
40859 | 779 |
by auto |
39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
780 |
qed |
ca17017c10e6
Measurable on product space is equiv. to measurable components
hoelzl
parents:
39082
diff
changeset
|
781 |
qed |
35833 | 782 |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
783 |
lemma (in pair_sigma_algebra) measurable_product_swap: |
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
|
784 |
"f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
785 |
proof - |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
786 |
interpret Q: pair_sigma_algebra M2 M1 by default |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
787 |
show ?thesis |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
788 |
using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"] |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
789 |
by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI) |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
790 |
qed |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
791 |
|
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
792 |
lemma (in pair_sigma_finite) integrable_product_swap: |
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
|
793 |
assumes "integrable P 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
|
794 |
shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
795 |
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
|
796 |
interpret Q: pair_sigma_finite M2 M1 by default |
41661 | 797 |
have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff) |
798 |
show ?thesis unfolding * |
|
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
|
799 |
using assms unfolding integrable_def |
41661 | 800 |
apply (subst (1 2) positive_integral_product_swap) |
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
|
801 |
using `integrable P f` unfolding integrable_def |
41661 | 802 |
by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric]) |
803 |
qed |
|
804 |
||
805 |
lemma (in pair_sigma_finite) integrable_product_swap_iff: |
|
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
|
806 |
"integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable P f" |
41661 | 807 |
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
|
808 |
interpret Q: pair_sigma_finite M2 M1 by default |
41661 | 809 |
from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f] |
810 |
show ?thesis by auto |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
811 |
qed |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
812 |
|
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
813 |
lemma (in pair_sigma_finite) integral_product_swap: |
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
|
814 |
assumes "integrable P 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
|
815 |
shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L P f" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
816 |
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
|
817 |
interpret Q: pair_sigma_finite M2 M1 by default |
41661 | 818 |
have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff) |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
819 |
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
|
820 |
unfolding lebesgue_integral_def * |
41661 | 821 |
apply (subst (1 2) positive_integral_product_swap) |
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
|
822 |
using `integrable P f` unfolding integrable_def |
41661 | 823 |
by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric]) |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
824 |
qed |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
825 |
|
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
826 |
lemma (in pair_sigma_finite) integrable_fst_measurable: |
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
|
827 |
assumes f: "integrable P 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
|
828 |
shows "M1.almost_everywhere (\<lambda>x. integrable M2 (\<lambda> y. f (x, y)))" (is "?AE") |
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 "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L P f" (is "?INT") |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
830 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
831 |
let "?pf x" = "extreal (f x)" and "?nf x" = "extreal (- f x)" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
832 |
have |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
833 |
borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
834 |
int: "integral\<^isup>P P ?nf \<noteq> \<infinity>" "integral\<^isup>P P ?pf \<noteq> \<infinity>" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
835 |
using assms by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
836 |
have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
837 |
"(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
838 |
using borel[THEN positive_integral_fst_measurable(1)] int |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
839 |
unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
840 |
with borel[THEN positive_integral_fst_measurable(1)] |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
841 |
have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
842 |
"AE x in M1. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
843 |
by (auto intro!: M1.positive_integral_PInf_AE ) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
844 |
then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
845 |
"AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
846 |
by (auto simp: M2.positive_integral_positive) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
847 |
from AE_pos show ?AE using assms |
41705 | 848 |
by (simp add: measurable_pair_image_snd integrable_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
849 |
{ fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
850 |
using M2.positive_integral_positive |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
851 |
by (intro M1.positive_integral_cong_pos) (auto simp: extreal_uminus_le_reorder) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
852 |
then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = 0" by simp } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
853 |
note this[simp] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
854 |
{ fix f assume borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable P" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
855 |
and int: "integral\<^isup>P P (\<lambda>x. extreal (f x)) \<noteq> \<infinity>" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
856 |
and AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
857 |
have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f") |
41705 | 858 |
proof (intro integrable_def[THEN iffD2] conjI) |
859 |
show "?f \<in> borel_measurable M1" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
860 |
using borel by (auto intro!: M1.borel_measurable_real_of_extreal positive_integral_fst_measurable) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
861 |
have "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<partial>M1)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
862 |
using AE M2.positive_integral_positive |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
863 |
by (auto intro!: M1.positive_integral_cong_AE simp: extreal_real) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
864 |
then show "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) \<noteq> \<infinity>" |
41705 | 865 |
using positive_integral_fst_measurable[OF borel] int by simp |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
866 |
have "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
867 |
by (intro M1.positive_integral_cong_pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
868 |
(simp add: M2.positive_integral_positive real_of_extreal_pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
869 |
then show "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp |
41705 | 870 |
qed } |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
871 |
with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)] |
41705 | 872 |
show ?INT |
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
|
873 |
unfolding lebesgue_integral_def[of P] lebesgue_integral_def[of M2] |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
874 |
borel[THEN positive_integral_fst_measurable(2), symmetric] |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
875 |
using AE[THEN M1.integral_real] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
876 |
by simp |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
877 |
qed |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
878 |
|
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
879 |
lemma (in pair_sigma_finite) integrable_snd_measurable: |
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
|
880 |
assumes f: "integrable P 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
|
881 |
shows "M2.almost_everywhere (\<lambda>y. integrable M1 (\<lambda>x. f (x, y)))" (is "?AE") |
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
|
882 |
and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L P f" (is "?INT") |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
883 |
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
|
884 |
interpret Q: pair_sigma_finite M2 M1 by default |
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
|
885 |
have Q_int: "integrable Q.P (\<lambda>(x, y). f (y, x))" |
41661 | 886 |
using f unfolding integrable_product_swap_iff . |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
887 |
show ?INT |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
888 |
using Q.integrable_fst_measurable(2)[OF Q_int] |
41661 | 889 |
using integral_product_swap[OF f] by simp |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
890 |
show ?AE |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
891 |
using Q.integrable_fst_measurable(1)[OF Q_int] |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
892 |
by simp |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
893 |
qed |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
894 |
|
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
895 |
lemma (in pair_sigma_finite) Fubini_integral: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
896 |
assumes f: "integrable P f" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
897 |
shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
898 |
unfolding integrable_snd_measurable[OF assms] |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
899 |
unfolding integrable_fst_measurable[OF assms] .. |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
900 |
|
40859 | 901 |
section "Products on finite spaces" |
902 |
||
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
|
903 |
lemma sigma_sets_pair_measure_generator_finite: |
38656 | 904 |
assumes "finite A" and "finite B" |
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
|
905 |
shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<in> Pow A \<and> b \<in> Pow B} = Pow (A \<times> B)" |
40859 | 906 |
(is "sigma_sets ?prod ?sets = _") |
38656 | 907 |
proof safe |
908 |
have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product) |
|
909 |
fix x assume subset: "x \<subseteq> A \<times> B" |
|
910 |
hence "finite x" using fin by (rule finite_subset) |
|
40859 | 911 |
from this subset show "x \<in> sigma_sets ?prod ?sets" |
38656 | 912 |
proof (induct x) |
913 |
case empty show ?case by (rule sigma_sets.Empty) |
|
914 |
next |
|
915 |
case (insert a x) |
|
40859 | 916 |
hence "{a} \<in> sigma_sets ?prod ?sets" |
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
|
917 |
by (auto simp: pair_measure_generator_def intro!: sigma_sets.Basic) |
38656 | 918 |
moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto |
919 |
ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un) |
|
920 |
qed |
|
921 |
next |
|
922 |
fix x a b |
|
40859 | 923 |
assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x" |
38656 | 924 |
from sigma_sets_into_sp[OF _ this(1)] this(2) |
40859 | 925 |
show "a \<in> A" and "b \<in> B" by auto |
35833 | 926 |
qed |
927 |
||
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
|
928 |
locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2 |
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
|
929 |
for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme" |
40859 | 930 |
|
931 |
sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default |
|
932 |
||
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
|
933 |
lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra: |
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
|
934 |
shows "P = \<lparr> space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2), \<dots> = algebra.more P \<rparr>" |
35977 | 935 |
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
|
936 |
show ?thesis |
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
|
937 |
using sigma_sets_pair_measure_generator_finite[OF M1.finite_space M2.finite_space] |
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
|
938 |
by (intro algebra.equality) (simp_all add: pair_measure_def pair_measure_generator_def sigma_def) |
40859 | 939 |
qed |
940 |
||
941 |
sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P |
|
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
|
942 |
apply default |
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
|
943 |
using M1.finite_space M2.finite_space |
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
|
944 |
apply (subst finite_pair_sigma_algebra) apply simp |
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
|
945 |
apply (subst (1 2) finite_pair_sigma_algebra) apply simp |
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
|
946 |
done |
35833 | 947 |
|
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
|
948 |
locale pair_finite_space = M1: finite_measure_space M1 + M2: finite_measure_space M2 |
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
|
949 |
for M1 M2 |
40859 | 950 |
|
951 |
sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra |
|
952 |
by default |
|
35833 | 953 |
|
40859 | 954 |
sublocale pair_finite_space \<subseteq> pair_sigma_finite |
955 |
by default |
|
38656 | 956 |
|
40859 | 957 |
lemma (in pair_finite_space) pair_measure_Pair[simp]: |
958 |
assumes "a \<in> space M1" "b \<in> space M2" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
959 |
shows "\<mu> {(a, b)} = M1.\<mu> {a} * M2.\<mu> {b}" |
40859 | 960 |
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
|
961 |
have "\<mu> ({a}\<times>{b}) = M1.\<mu> {a} * M2.\<mu> {b}" |
40859 | 962 |
using M1.sets_eq_Pow M2.sets_eq_Pow assms |
963 |
by (subst pair_measure_times) auto |
|
964 |
then show ?thesis by simp |
|
38656 | 965 |
qed |
966 |
||
40859 | 967 |
lemma (in pair_finite_space) pair_measure_singleton[simp]: |
968 |
assumes "x \<in> space M1 \<times> space M2" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
969 |
shows "\<mu> {x} = M1.\<mu> {fst x} * M2.\<mu> {snd x}" |
40859 | 970 |
using pair_measure_Pair assms by (cases x) auto |
38656 | 971 |
|
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
|
972 |
sublocale pair_finite_space \<subseteq> finite_measure_space P |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
973 |
by default (auto simp: space_pair_measure) |
39097 | 974 |
|
40859 | 975 |
end |