author | wenzelm |
Mon, 19 Jan 2015 20:39:01 +0100 | |
changeset 59409 | b7cfe12acf2e |
parent 59092 | d469103c0737 |
child 60065 | 390de6d3a7b8 |
permissions | -rw-r--r-- |
59092
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(* Title: HOL/Probability/Embed_Measure.thy |
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Author: Manuel Eberl, TU München |
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Defines measure embeddings with injective functions, i.e. lifting a measure on some values |
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to a measure on "tagged" values (e.g. embed_measure M Inl lifts a measure on values 'a to a |
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measure on the left part of the sum type 'a + 'b) |
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*) |
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section {* Embed Measure Spaces with a Function *} |
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theory Embed_Measure |
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imports Binary_Product_Measure |
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begin |
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definition embed_measure :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where |
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"embed_measure M f = measure_of (f ` space M) {f ` A |A. A \<in> sets M} |
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(\<lambda>A. emeasure M (f -` A \<inter> space M))" |
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lemma space_embed_measure: "space (embed_measure M f) = f ` space M" |
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unfolding embed_measure_def |
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by (subst space_measure_of) (auto dest: sets.sets_into_space) |
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lemma sets_embed_measure': |
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assumes inj: "inj_on f (space M)" |
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shows "sets (embed_measure M f) = {f ` A |A. A \<in> sets M}" |
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unfolding embed_measure_def |
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proof (intro sigma_algebra.sets_measure_of_eq sigma_algebra_iff2[THEN iffD2] conjI allI ballI impI) |
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fix s assume "s \<in> {f ` A |A. A \<in> sets M}" |
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then obtain s' where s'_props: "s = f ` s'" "s' \<in> sets M" by auto |
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hence "f ` space M - s = f ` (space M - s')" using inj |
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by (auto dest: inj_onD sets.sets_into_space) |
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also have "... \<in> {f ` A |A. A \<in> sets M}" using s'_props by auto |
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finally show "f ` space M - s \<in> {f ` A |A. A \<in> sets M}" . |
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next |
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fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> {f ` A |A. A \<in> sets M}" |
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then obtain A' where "\<And>i. A i = f ` A' i" "\<And>i. A' i \<in> sets M" |
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by (auto simp: subset_eq choice_iff) |
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moreover from this have "(\<Union>x. f ` A' x) = f ` (\<Union>x. A' x)" by blast |
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ultimately show "(\<Union>i. A i) \<in> {f ` A |A. A \<in> sets M}" by blast |
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qed (auto dest: sets.sets_into_space) |
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lemma the_inv_into_vimage: |
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"inj_on f X \<Longrightarrow> A \<subseteq> X \<Longrightarrow> the_inv_into X f -` A \<inter> (f`X) = f ` A" |
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by (auto simp: the_inv_into_f_f) |
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lemma sets_embed_eq_vimage_algebra: |
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assumes "inj_on f (space M)" |
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shows "sets (embed_measure M f) = sets (vimage_algebra (f`space M) (the_inv_into (space M) f) M)" |
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by (auto simp: sets_embed_measure'[OF assms] Pi_iff the_inv_into_f_f assms sets_vimage_algebra2 simple_image |
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dest: sets.sets_into_space |
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intro!: image_cong the_inv_into_vimage[symmetric]) |
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lemma sets_embed_measure: |
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assumes inj: "inj f" |
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shows "sets (embed_measure M f) = {f ` A |A. A \<in> sets M}" |
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using assms by (subst sets_embed_measure') (auto intro!: inj_onI dest: injD) |
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lemma in_sets_embed_measure: "A \<in> sets M \<Longrightarrow> f ` A \<in> sets (embed_measure M f)" |
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unfolding embed_measure_def |
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by (intro in_measure_of) (auto dest: sets.sets_into_space) |
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lemma measurable_embed_measure1: |
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assumes g: "(\<lambda>x. g (f x)) \<in> measurable M N" |
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shows "g \<in> measurable (embed_measure M f) N" |
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unfolding measurable_def |
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proof safe |
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fix A assume "A \<in> sets N" |
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with g have "(\<lambda>x. g (f x)) -` A \<inter> space M \<in> sets M" |
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by (rule measurable_sets) |
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then have "f ` ((\<lambda>x. g (f x)) -` A \<inter> space M) \<in> sets (embed_measure M f)" |
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by (rule in_sets_embed_measure) |
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also have "f ` ((\<lambda>x. g (f x)) -` A \<inter> space M) = g -` A \<inter> space (embed_measure M f)" |
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by (auto simp: space_embed_measure) |
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finally show "g -` A \<inter> space (embed_measure M f) \<in> sets (embed_measure M f)" . |
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qed (insert measurable_space[OF assms], auto simp: space_embed_measure) |
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lemma measurable_embed_measure2': |
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assumes "inj_on f (space M)" |
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shows "f \<in> measurable M (embed_measure M f)" |
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proof- |
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{ |
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fix A assume A: "A \<in> sets M" |
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also from this have "A = A \<inter> space M" by auto |
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also have "... = f -` f ` A \<inter> space M" using A assms |
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by (auto dest: inj_onD sets.sets_into_space) |
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finally have "f -` f ` A \<inter> space M \<in> sets M" . |
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} |
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thus ?thesis using assms unfolding embed_measure_def |
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by (intro measurable_measure_of) (auto dest: sets.sets_into_space) |
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qed |
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|
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lemma measurable_embed_measure2: |
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assumes [simp]: "inj f" shows "f \<in> measurable M (embed_measure M f)" |
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by (auto simp: inj_vimage_image_eq embed_measure_def |
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intro!: measurable_measure_of dest: sets.sets_into_space) |
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|
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lemma embed_measure_eq_distr': |
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assumes "inj_on f (space M)" |
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shows "embed_measure M f = distr M (embed_measure M f) f" |
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proof- |
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have "distr M (embed_measure M f) f = |
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measure_of (f ` space M) {f ` A |A. A \<in> sets M} |
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(\<lambda>A. emeasure M (f -` A \<inter> space M))" unfolding distr_def |
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104 |
by (simp add: space_embed_measure sets_embed_measure'[OF assms]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
105 |
also have "... = embed_measure M f" unfolding embed_measure_def .. |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
106 |
finally show ?thesis .. |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
107 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
108 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
109 |
lemma embed_measure_eq_distr: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
110 |
"inj f \<Longrightarrow> embed_measure M f = distr M (embed_measure M f) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
111 |
by (rule embed_measure_eq_distr') (auto intro!: inj_onI dest: injD) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
112 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
113 |
lemma nn_integral_embed_measure: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
114 |
"inj f \<Longrightarrow> g \<in> borel_measurable (embed_measure M f) \<Longrightarrow> |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
115 |
nn_integral (embed_measure M f) g = nn_integral M (\<lambda>x. g (f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
116 |
apply (subst embed_measure_eq_distr, simp) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
117 |
apply (subst nn_integral_distr) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
118 |
apply (simp_all add: measurable_embed_measure2) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
119 |
done |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
120 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
121 |
lemma emeasure_embed_measure': |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
122 |
assumes "inj_on f (space M)" "A \<in> sets (embed_measure M f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
123 |
shows "emeasure (embed_measure M f) A = emeasure M (f -` A \<inter> space M)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
124 |
by (subst embed_measure_eq_distr'[OF assms(1)]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
125 |
(simp add: emeasure_distr[OF measurable_embed_measure2'[OF assms(1)] assms(2)]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
126 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
127 |
lemma emeasure_embed_measure: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
128 |
assumes "inj f" "A \<in> sets (embed_measure M f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
129 |
shows "emeasure (embed_measure M f) A = emeasure M (f -` A \<inter> space M)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
130 |
using assms by (intro emeasure_embed_measure') (auto intro!: inj_onI dest: injD) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
131 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
132 |
lemma embed_measure_comp: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
133 |
assumes [simp]: "inj f" "inj g" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
134 |
shows "embed_measure (embed_measure M f) g = embed_measure M (g \<circ> f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
135 |
proof- |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
136 |
have [simp]: "inj (\<lambda>x. g (f x))" by (subst o_def[symmetric]) (auto intro: inj_comp) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
137 |
note measurable_embed_measure2[measurable] |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
138 |
have "embed_measure (embed_measure M f) g = |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
139 |
distr M (embed_measure (embed_measure M f) g) (g \<circ> f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
140 |
by (subst (1 2) embed_measure_eq_distr) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
141 |
(simp_all add: distr_distr sets_embed_measure cong: distr_cong) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
142 |
also have "... = embed_measure M (g \<circ> f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
143 |
by (subst (3) embed_measure_eq_distr, simp add: o_def, rule distr_cong) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
144 |
(auto simp: sets_embed_measure o_def image_image[symmetric] |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
145 |
intro: inj_comp cong: distr_cong) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
146 |
finally show ?thesis . |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
147 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
148 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
149 |
lemma sigma_finite_embed_measure: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
150 |
assumes "sigma_finite_measure M" and inj: "inj f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
151 |
shows "sigma_finite_measure (embed_measure M f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
152 |
proof |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
153 |
case goal1 |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
154 |
from assms(1) interpret sigma_finite_measure M . |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
155 |
from sigma_finite_countable obtain A where |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
156 |
A_props: "countable A" "A \<subseteq> sets M" "\<Union>A = space M" "\<And>X. X\<in>A \<Longrightarrow> emeasure M X \<noteq> \<infinity>" by blast |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
157 |
show ?case |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
158 |
proof (intro exI[of _ "op ` f`A"] conjI allI) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
159 |
from A_props show "countable (op ` f`A)" by auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
160 |
from inj and A_props show "op ` f`A \<subseteq> sets (embed_measure M f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
161 |
by (auto simp: sets_embed_measure) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
162 |
from A_props and inj show "\<Union>(op ` f`A) = space (embed_measure M f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
163 |
by (auto simp: space_embed_measure intro!: imageI) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
164 |
from A_props and inj show "\<forall>a\<in>op ` f ` A. emeasure (embed_measure M f) a \<noteq> \<infinity>" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
165 |
by (intro ballI, subst emeasure_embed_measure) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
166 |
(auto simp: inj_vimage_image_eq intro: in_sets_embed_measure) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
167 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
168 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
169 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
170 |
lemma embed_measure_count_space: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
171 |
"inj f \<Longrightarrow> embed_measure (count_space A) f = count_space (f`A)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
172 |
apply (subst distr_bij_count_space[of f A "f`A", symmetric]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
173 |
apply (simp add: inj_on_def bij_betw_def) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
174 |
apply (subst embed_measure_eq_distr) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
175 |
apply simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
176 |
apply (auto intro!: measure_eqI imageI simp: sets_embed_measure subset_image_iff) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
177 |
apply blast |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
178 |
apply (subst (1 2) emeasure_distr) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
179 |
apply (auto simp: space_embed_measure sets_embed_measure) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
180 |
done |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
181 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
182 |
lemma sets_embed_measure_alt: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
183 |
"inj f \<Longrightarrow> sets (embed_measure M f) = (op`f) ` sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
184 |
by (auto simp: sets_embed_measure) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
185 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
186 |
lemma emeasure_embed_measure_image': |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
187 |
assumes "inj_on f (space M)" "X \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
188 |
shows "emeasure (embed_measure M f) (f`X) = emeasure M X" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
189 |
proof- |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
190 |
from assms have "emeasure (embed_measure M f) (f`X) = emeasure M (f -` f ` X \<inter> space M)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
191 |
by (subst emeasure_embed_measure') (auto simp: sets_embed_measure') |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
192 |
also from assms have "f -` f ` X \<inter> space M = X" by (auto dest: inj_onD sets.sets_into_space) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
193 |
finally show ?thesis . |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
194 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
195 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
196 |
lemma emeasure_embed_measure_image: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
197 |
"inj f \<Longrightarrow> X \<in> sets M \<Longrightarrow> emeasure (embed_measure M f) (f`X) = emeasure M X" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
198 |
by (simp_all add: emeasure_embed_measure in_sets_embed_measure inj_vimage_image_eq) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
199 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
200 |
lemma embed_measure_eq_iff: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
201 |
assumes "inj f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
202 |
shows "embed_measure A f = embed_measure B f \<longleftrightarrow> A = B" (is "?M = ?N \<longleftrightarrow> _") |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
203 |
proof |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
204 |
from assms have I: "inj (op` f)" by (auto intro: injI dest: injD) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
205 |
assume asm: "?M = ?N" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
206 |
hence "sets (embed_measure A f) = sets (embed_measure B f)" by simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
207 |
with assms have "sets A = sets B" by (simp only: I inj_image_eq_iff sets_embed_measure_alt) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
208 |
moreover { |
d469103c0737
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parents:
diff
changeset
|
209 |
fix X assume "X \<in> sets A" |
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diff
changeset
|
210 |
from asm have "emeasure ?M (f`X) = emeasure ?N (f`X)" by simp |
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diff
changeset
|
211 |
with `X \<in> sets A` and `sets A = sets B` and assms |
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diff
changeset
|
212 |
have "emeasure A X = emeasure B X" by (simp add: emeasure_embed_measure_image) |
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diff
changeset
|
213 |
} |
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diff
changeset
|
214 |
ultimately show "A = B" by (rule measure_eqI) |
d469103c0737
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diff
changeset
|
215 |
qed simp |
d469103c0737
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diff
changeset
|
216 |
|
d469103c0737
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diff
changeset
|
217 |
lemma the_inv_into_in_Pi: "inj_on f A \<Longrightarrow> the_inv_into A f \<in> f ` A \<rightarrow> A" |
d469103c0737
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diff
changeset
|
218 |
by (auto simp: the_inv_into_f_f) |
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diff
changeset
|
219 |
|
d469103c0737
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diff
changeset
|
220 |
lemma map_prod_image: "map_prod f g ` (A \<times> B) = (f`A) \<times> (g`B)" |
d469103c0737
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diff
changeset
|
221 |
using map_prod_surj_on[OF refl refl] . |
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parents:
diff
changeset
|
222 |
|
d469103c0737
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diff
changeset
|
223 |
lemma map_prod_vimage: "map_prod f g -` (A \<times> B) = (f-`A) \<times> (g-`B)" |
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diff
changeset
|
224 |
by auto |
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hoelzl
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diff
changeset
|
225 |
|
d469103c0737
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diff
changeset
|
226 |
lemma embed_measure_prod: |
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diff
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|
227 |
assumes f: "inj f" and g: "inj g" and [simp]: "sigma_finite_measure M" "sigma_finite_measure N" |
d469103c0737
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diff
changeset
|
228 |
shows "embed_measure M f \<Otimes>\<^sub>M embed_measure N g = embed_measure (M \<Otimes>\<^sub>M N) (\<lambda>(x, y). (f x, g y))" |
d469103c0737
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diff
changeset
|
229 |
(is "?L = _") |
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diff
changeset
|
230 |
unfolding map_prod_def[symmetric] |
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diff
changeset
|
231 |
proof (rule pair_measure_eqI) |
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diff
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|
232 |
have fg[simp]: "\<And>A. inj_on (map_prod f g) A" "\<And>A. inj_on f A" "\<And>A. inj_on g A" |
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diff
changeset
|
233 |
using f g by (auto simp: inj_on_def) |
d469103c0737
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diff
changeset
|
234 |
|
d469103c0737
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diff
changeset
|
235 |
show sets: "sets ?L = sets (embed_measure (M \<Otimes>\<^sub>M N) (map_prod f g))" |
d469103c0737
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diff
changeset
|
236 |
unfolding map_prod_def[symmetric] |
d469103c0737
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diff
changeset
|
237 |
apply (simp add: sets_pair_eq_sets_fst_snd sets_embed_eq_vimage_algebra |
d469103c0737
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diff
changeset
|
238 |
cong: vimage_algebra_cong) |
d469103c0737
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diff
changeset
|
239 |
apply (subst vimage_algebra_Sup_sigma) |
d469103c0737
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diff
changeset
|
240 |
apply (simp_all add: space_pair_measure[symmetric]) |
d469103c0737
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diff
changeset
|
241 |
apply (auto simp add: the_inv_into_f_f |
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diff
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|
242 |
simp del: map_prod_simp |
d469103c0737
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diff
changeset
|
243 |
del: prod_fun_imageE) [] |
d469103c0737
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diff
changeset
|
244 |
apply (subst (1 2 3 4 ) vimage_algebra_vimage_algebra_eq) |
d469103c0737
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diff
changeset
|
245 |
apply (simp_all add: the_inv_into_in_Pi Pi_iff[of snd] Pi_iff[of fst] space_pair_measure) |
d469103c0737
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diff
changeset
|
246 |
apply (simp_all add: Pi_iff[of snd] Pi_iff[of fst] the_inv_into_in_Pi vimage_algebra_vimage_algebra_eq |
d469103c0737
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diff
changeset
|
247 |
space_pair_measure[symmetric] map_prod_image[symmetric]) |
d469103c0737
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diff
changeset
|
248 |
apply (intro arg_cong[where f=sets] arg_cong[where f=Sup_sigma] arg_cong2[where f=insert] vimage_algebra_cong) |
d469103c0737
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diff
changeset
|
249 |
apply (auto simp: map_prod_image the_inv_into_f_f |
d469103c0737
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diff
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|
250 |
simp del: map_prod_simp del: prod_fun_imageE) |
d469103c0737
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diff
changeset
|
251 |
apply (simp_all add: the_inv_into_f_f space_pair_measure) |
d469103c0737
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diff
changeset
|
252 |
done |
d469103c0737
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diff
changeset
|
253 |
|
d469103c0737
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diff
changeset
|
254 |
note measurable_embed_measure2[measurable] |
d469103c0737
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diff
changeset
|
255 |
fix A B assume AB: "A \<in> sets (embed_measure M f)" "B \<in> sets (embed_measure N g)" |
d469103c0737
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diff
changeset
|
256 |
moreover have "f -` A \<times> g -` B \<inter> space (M \<Otimes>\<^sub>M N) = (f -` A \<inter> space M) \<times> (g -` B \<inter> space N)" |
d469103c0737
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diff
changeset
|
257 |
by (auto simp: space_pair_measure) |
d469103c0737
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diff
changeset
|
258 |
ultimately show "emeasure (embed_measure M f) A * emeasure (embed_measure N g) B = |
d469103c0737
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diff
changeset
|
259 |
emeasure (embed_measure (M \<Otimes>\<^sub>M N) (map_prod f g)) (A \<times> B)" |
d469103c0737
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diff
changeset
|
260 |
by (simp add: map_prod_vimage sets[symmetric] emeasure_embed_measure |
d469103c0737
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diff
changeset
|
261 |
sigma_finite_measure.emeasure_pair_measure_Times) |
d469103c0737
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diff
changeset
|
262 |
qed (insert assms, simp_all add: sigma_finite_embed_measure) |
d469103c0737
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hoelzl
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diff
changeset
|
263 |
|
d469103c0737
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hoelzl
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diff
changeset
|
264 |
lemma density_embed_measure: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
265 |
assumes inj: "inj f" and Mg[measurable]: "g \<in> borel_measurable (embed_measure M f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
266 |
shows "density (embed_measure M f) g = embed_measure (density M (g \<circ> f)) f" (is "?M1 = ?M2") |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
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diff
changeset
|
267 |
proof (rule measure_eqI) |
d469103c0737
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hoelzl
parents:
diff
changeset
|
268 |
fix X assume X: "X \<in> sets ?M1" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
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diff
changeset
|
269 |
from inj have Mf[measurable]: "f \<in> measurable M (embed_measure M f)" |
d469103c0737
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hoelzl
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diff
changeset
|
270 |
by (rule measurable_embed_measure2) |
d469103c0737
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diff
changeset
|
271 |
from Mg and X have "emeasure ?M1 X = \<integral>\<^sup>+ x. g x * indicator X x \<partial>embed_measure M f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
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diff
changeset
|
272 |
by (subst emeasure_density) simp_all |
d469103c0737
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hoelzl
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diff
changeset
|
273 |
also from X have "... = \<integral>\<^sup>+ x. g (f x) * indicator X (f x) \<partial>M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
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diff
changeset
|
274 |
by (subst embed_measure_eq_distr[OF inj], subst nn_integral_distr) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
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diff
changeset
|
275 |
(auto intro!: borel_measurable_ereal_times borel_measurable_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
276 |
also have "... = \<integral>\<^sup>+ x. g (f x) * indicator (f -` X \<inter> space M) x \<partial>M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
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diff
changeset
|
277 |
by (intro nn_integral_cong) (auto split: split_indicator) |
d469103c0737
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hoelzl
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diff
changeset
|
278 |
also from X have "... = emeasure (density M (g \<circ> f)) (f -` X \<inter> space M)" |
d469103c0737
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hoelzl
parents:
diff
changeset
|
279 |
by (subst emeasure_density) (simp_all add: measurable_comp[OF Mf Mg] measurable_sets[OF Mf]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
280 |
also from X and inj have "... = emeasure ?M2 X" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
281 |
by (subst emeasure_embed_measure) (simp_all add: sets_embed_measure) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
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diff
changeset
|
282 |
finally show "emeasure ?M1 X = emeasure ?M2 X" . |
d469103c0737
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hoelzl
parents:
diff
changeset
|
283 |
qed (simp_all add: sets_embed_measure inj) |
d469103c0737
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hoelzl
parents:
diff
changeset
|
284 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
285 |
lemma density_embed_measure': |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
286 |
assumes inj: "inj f" and inv: "\<And>x. f' (f x) = x" and Mg[measurable]: "g \<in> borel_measurable M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
287 |
shows "density (embed_measure M f) (g \<circ> f') = embed_measure (density M g) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
288 |
proof- |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
289 |
have "density (embed_measure M f) (g \<circ> f') = embed_measure (density M (g \<circ> f' \<circ> f)) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
290 |
by (rule density_embed_measure[OF inj]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
291 |
(rule measurable_comp, rule measurable_embed_measure1, subst measurable_cong, |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
292 |
rule inv, rule measurable_ident_sets, simp, rule Mg) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
293 |
also have "density M (g \<circ> f' \<circ> f) = density M g" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
294 |
by (intro density_cong) (subst measurable_cong, simp add: o_def inv, simp_all add: Mg inv) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
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diff
changeset
|
295 |
finally show ?thesis . |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
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diff
changeset
|
296 |
qed |
d469103c0737
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hoelzl
parents:
diff
changeset
|
297 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
298 |
lemma inj_on_image_subset_iff: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
299 |
assumes "inj_on f C" "A \<subseteq> C" "B \<subseteq> C" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
300 |
shows "f ` A \<subseteq> f ` B \<longleftrightarrow> A \<subseteq> B" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
301 |
proof (intro iffI subsetI) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
302 |
fix x assume A: "f ` A \<subseteq> f ` B" and B: "x \<in> A" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
303 |
from B have "f x \<in> f ` A" by blast |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
304 |
with A have "f x \<in> f ` B" by blast |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
305 |
then obtain y where "f x = f y" and "y \<in> B" by blast |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
306 |
with assms and B have "x = y" by (auto dest: inj_onD) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
307 |
with `y \<in> B` show "x \<in> B" by simp |
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qed auto |
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309 |
|
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310 |
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311 |
lemma AE_embed_measure': |
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312 |
assumes inj: "inj_on f (space M)" |
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313 |
shows "(AE x in embed_measure M f. P x) \<longleftrightarrow> (AE x in M. P (f x))" |
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proof |
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315 |
let ?M = "embed_measure M f" |
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316 |
assume "AE x in ?M. P x" |
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317 |
then obtain A where A_props: "A \<in> sets ?M" "emeasure ?M A = 0" "{x\<in>space ?M. \<not>P x} \<subseteq> A" |
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318 |
by (force elim: AE_E) |
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319 |
then obtain A' where A'_props: "A = f ` A'" "A' \<in> sets M" by (auto simp: sets_embed_measure' inj) |
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320 |
moreover have B: "{x\<in>space ?M. \<not>P x} = f ` {x\<in>space M. \<not>P (f x)}" |
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321 |
by (auto simp: inj space_embed_measure) |
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322 |
from A_props(3) have "{x\<in>space M. \<not>P (f x)} \<subseteq> A'" |
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323 |
by (subst (asm) B, subst (asm) A'_props, subst (asm) inj_on_image_subset_iff[OF inj]) |
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324 |
(insert A'_props, auto dest: sets.sets_into_space) |
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moreover from A_props A'_props have "emeasure M A' = 0" |
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326 |
by (simp add: emeasure_embed_measure_image' inj) |
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327 |
ultimately show "AE x in M. P (f x)" by (intro AE_I) |
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328 |
next |
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329 |
let ?M = "embed_measure M f" |
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330 |
assume "AE x in M. P (f x)" |
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331 |
then obtain A where A_props: "A \<in> sets M" "emeasure M A = 0" "{x\<in>space M. \<not>P (f x)} \<subseteq> A" |
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332 |
by (force elim: AE_E) |
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333 |
hence "f`A \<in> sets ?M" "emeasure ?M (f`A) = 0" "{x\<in>space ?M. \<not>P x} \<subseteq> f`A" |
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334 |
by (auto simp: space_embed_measure emeasure_embed_measure_image' sets_embed_measure' inj) |
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thus "AE x in ?M. P x" by (intro AE_I) |
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336 |
qed |
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337 |
|
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338 |
lemma AE_embed_measure: |
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339 |
assumes inj: "inj f" |
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340 |
shows "(AE x in embed_measure M f. P x) \<longleftrightarrow> (AE x in M. P (f x))" |
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341 |
using assms by (intro AE_embed_measure') (auto intro!: inj_onI dest: injD) |
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342 |
|
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343 |
end |