author | immler |
Thu, 15 Nov 2012 10:49:58 +0100 | |
changeset 50087 | 635d73673b5e |
parent 50042 | 6fe18351e9dd |
child 50099 | a58bb401af80 |
permissions | -rw-r--r-- |
42146
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split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
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1 |
(* Title: HOL/Probability/Finite_Product_Measure.thy |
42067 | 2 |
Author: Johannes Hölzl, TU München |
3 |
*) |
|
4 |
||
42146
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split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
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parents:
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|
5 |
header {*Finite product measures*} |
42067 | 6 |
|
42146
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split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
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parents:
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7 |
theory Finite_Product_Measure |
5b52c6a9c627
split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
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|
8 |
imports Binary_Product_Measure |
35833 | 9 |
begin |
10 |
||
47694 | 11 |
lemma split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)" |
12 |
by auto |
|
13 |
||
40859 | 14 |
abbreviation |
15 |
"Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A" |
|
39094 | 16 |
|
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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17 |
syntax |
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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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18 |
"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3PIE _:_./ _)" 10) |
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parents:
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diff
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19 |
|
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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:
41661
diff
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20 |
syntax (xsymbols) |
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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:
41661
diff
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|
21 |
"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>\<^isub>E _\<in>_./ _)" 10) |
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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:
41661
diff
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|
22 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
23 |
syntax (HTML output) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
24 |
"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>\<^isub>E _\<in>_./ _)" 10) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
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|
25 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
26 |
translations |
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
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27 |
"PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)" |
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|
28 |
|
40859 | 29 |
abbreviation |
30 |
funcset_extensional :: "['a set, 'b set] => ('a => 'b) set" |
|
31 |
(infixr "->\<^isub>E" 60) where |
|
32 |
"A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)" |
|
33 |
||
34 |
notation (xsymbols) |
|
35 |
funcset_extensional (infixr "\<rightarrow>\<^isub>E" 60) |
|
36 |
||
37 |
lemma extensional_insert[intro, simp]: |
|
38 |
assumes "a \<in> extensional (insert i I)" |
|
39 |
shows "a(i := b) \<in> extensional (insert i I)" |
|
40 |
using assms unfolding extensional_def by auto |
|
41 |
||
42 |
lemma extensional_Int[simp]: |
|
43 |
"extensional I \<inter> extensional I' = extensional (I \<inter> I')" |
|
44 |
unfolding extensional_def by auto |
|
38656 | 45 |
|
50038 | 46 |
lemma extensional_UNIV[simp]: "extensional UNIV = UNIV" |
47 |
by (auto simp: extensional_def) |
|
48 |
||
49 |
lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B" |
|
50 |
unfolding restrict_def extensional_def by auto |
|
51 |
||
52 |
lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)" |
|
53 |
unfolding restrict_def by (simp add: fun_eq_iff) |
|
54 |
||
35833 | 55 |
definition |
49780 | 56 |
"merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)" |
40859 | 57 |
|
58 |
lemma merge_apply[simp]: |
|
49780 | 59 |
"I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i" |
60 |
"I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i" |
|
61 |
"J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i" |
|
62 |
"J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i" |
|
63 |
"i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I J (x, y) i = undefined" |
|
40859 | 64 |
unfolding merge_def by auto |
65 |
||
66 |
lemma merge_commute: |
|
49780 | 67 |
"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) = merge J I (y, x)" |
50003 | 68 |
by (force simp: merge_def) |
40859 | 69 |
|
70 |
lemma Pi_cancel_merge_range[simp]: |
|
49780 | 71 |
"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A" |
72 |
"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A" |
|
73 |
"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A" |
|
74 |
"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A" |
|
40859 | 75 |
by (auto simp: Pi_def) |
76 |
||
77 |
lemma Pi_cancel_merge[simp]: |
|
49780 | 78 |
"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
79 |
"J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
|
80 |
"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" |
|
81 |
"J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" |
|
40859 | 82 |
by (auto simp: Pi_def) |
83 |
||
49780 | 84 |
lemma extensional_merge[simp]: "merge I J (x, y) \<in> extensional (I \<union> J)" |
40859 | 85 |
by (auto simp: extensional_def) |
86 |
||
87 |
lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A" |
|
88 |
by (auto simp: restrict_def Pi_def) |
|
89 |
||
90 |
lemma restrict_merge[simp]: |
|
49780 | 91 |
"I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I" |
92 |
"I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J" |
|
93 |
"J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I" |
|
94 |
"J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J" |
|
47694 | 95 |
by (auto simp: restrict_def) |
40859 | 96 |
|
50042
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moved lemmas into projective_family; added header for theory Projective_Family
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parents:
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97 |
lemma split_merge: "P (merge I J (x,y) i) \<longleftrightarrow> (i \<in> I \<longrightarrow> P (x i)) \<and> (i \<in> J - I \<longrightarrow> P (y i)) \<and> (i \<notin> I \<union> J \<longrightarrow> P undefined)" |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
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parents:
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|
98 |
unfolding merge_def by auto |
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moved lemmas into projective_family; added header for theory Projective_Family
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parents:
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|
99 |
|
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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100 |
lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I J (x, y) \<in> extensional K" |
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parents:
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|
101 |
unfolding merge_def extensional_def by auto |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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|
102 |
|
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
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parents:
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|
103 |
lemma injective_vimage_restrict: |
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parents:
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|
104 |
assumes J: "J \<subseteq> I" |
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moved lemmas into projective_family; added header for theory Projective_Family
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parents:
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|
105 |
and sets: "A \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" and ne: "(\<Pi>\<^isub>E i\<in>I. S i) \<noteq> {}" |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
106 |
and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^isub>E i\<in>I. S i)" |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
107 |
shows "A = B" |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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|
108 |
proof (intro set_eqI) |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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|
109 |
fix x |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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|
110 |
from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto |
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moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
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|
111 |
have "J \<inter> (I - J) = {}" by auto |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
112 |
show "x \<in> A \<longleftrightarrow> x \<in> B" |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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changeset
|
113 |
proof cases |
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moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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diff
changeset
|
114 |
assume x: "x \<in> (\<Pi>\<^isub>E i\<in>J. S i)" |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
115 |
have "x \<in> A \<longleftrightarrow> merge J (I - J) (x,y) \<in> (\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i)" |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
116 |
using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub split: split_merge) |
6fe18351e9dd
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immler@in.tum.de
parents:
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diff
changeset
|
117 |
then show "x \<in> A \<longleftrightarrow> x \<in> B" |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
118 |
using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub eq split: split_merge) |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
119 |
next |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
120 |
assume "x \<notin> (\<Pi>\<^isub>E i\<in>J. S i)" with sets show "x \<in> A \<longleftrightarrow> x \<in> B" by auto |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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diff
changeset
|
121 |
qed |
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moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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diff
changeset
|
122 |
qed |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
123 |
|
40859 | 124 |
lemma extensional_insert_undefined[intro, simp]: |
125 |
assumes "a \<in> extensional (insert i I)" |
|
126 |
shows "a(i := undefined) \<in> extensional I" |
|
127 |
using assms unfolding extensional_def by auto |
|
128 |
||
129 |
lemma extensional_insert_cancel[intro, simp]: |
|
130 |
assumes "a \<in> extensional I" |
|
131 |
shows "a \<in> extensional (insert i I)" |
|
132 |
using assms unfolding extensional_def by auto |
|
133 |
||
49780 | 134 |
lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I {i} (x,y) = restrict (x(i := y i)) (insert i I)" |
41095 | 135 |
unfolding merge_def by (auto simp: fun_eq_iff) |
136 |
||
137 |
lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)" |
|
138 |
by auto |
|
139 |
||
40859 | 140 |
lemma PiE_Int: "(Pi\<^isub>E I A) \<inter> (Pi\<^isub>E I B) = Pi\<^isub>E I (\<lambda>x. A x \<inter> B x)" |
141 |
by auto |
|
142 |
||
143 |
lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B" |
|
144 |
by (auto simp: Pi_def) |
|
145 |
||
146 |
lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i" |
|
147 |
by (auto simp: Pi_def) |
|
39088
ca17017c10e6
Measurable on product space is equiv. to measurable components
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parents:
39082
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148 |
|
40859 | 149 |
lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X" |
150 |
by (auto simp: Pi_def) |
|
151 |
||
152 |
lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
|
153 |
by (auto simp: Pi_def) |
|
154 |
||
41095 | 155 |
lemma restrict_vimage: |
156 |
assumes "I \<inter> J = {}" |
|
49780 | 157 |
shows "(\<lambda>x. (restrict x I, restrict x J)) -` (Pi\<^isub>E I E \<times> Pi\<^isub>E J F) = Pi (I \<union> J) (merge I J (E, F))" |
41095 | 158 |
using assms by (auto simp: restrict_Pi_cancel) |
159 |
||
160 |
lemma merge_vimage: |
|
161 |
assumes "I \<inter> J = {}" |
|
49780 | 162 |
shows "merge I J -` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E" |
41095 | 163 |
using assms by (auto simp: restrict_Pi_cancel) |
164 |
||
165 |
lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I" |
|
47694 | 166 |
by (auto simp: restrict_def) |
41095 | 167 |
|
168 |
lemma merge_restrict[simp]: |
|
49780 | 169 |
"merge I J (restrict x I, y) = merge I J (x, y)" |
170 |
"merge I J (x, restrict y J) = merge I J (x, y)" |
|
47694 | 171 |
unfolding merge_def by auto |
41095 | 172 |
|
173 |
lemma merge_x_x_eq_restrict[simp]: |
|
49780 | 174 |
"merge I J (x, x) = restrict x (I \<union> J)" |
47694 | 175 |
unfolding merge_def by auto |
41095 | 176 |
|
177 |
lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A" |
|
178 |
apply auto |
|
179 |
apply (drule_tac x=x in Pi_mem) |
|
180 |
apply (simp_all split: split_if_asm) |
|
181 |
apply (drule_tac x=i in Pi_mem) |
|
182 |
apply (auto dest!: Pi_mem) |
|
183 |
done |
|
184 |
||
185 |
lemma Pi_UN: |
|
186 |
fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set" |
|
187 |
assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i" |
|
188 |
shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)" |
|
189 |
proof (intro set_eqI iffI) |
|
190 |
fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)" |
|
191 |
then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto |
|
192 |
from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto |
|
193 |
obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k" |
|
194 |
using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto |
|
195 |
have "f \<in> Pi I (A k)" |
|
196 |
proof (intro Pi_I) |
|
197 |
fix i assume "i \<in> I" |
|
198 |
from mono[OF this, of "n i" k] k[OF this] n[OF this] |
|
199 |
show "f i \<in> A k i" by auto |
|
200 |
qed |
|
201 |
then show "f \<in> (\<Union>n. Pi I (A n))" by auto |
|
202 |
qed auto |
|
203 |
||
204 |
lemma PiE_cong: |
|
205 |
assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i" |
|
206 |
shows "Pi\<^isub>E I A = Pi\<^isub>E I B" |
|
207 |
using assms by (auto intro!: Pi_cong) |
|
208 |
||
209 |
lemma restrict_upd[simp]: |
|
210 |
"i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)" |
|
211 |
by (auto simp: fun_eq_iff) |
|
212 |
||
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
|
213 |
lemma Pi_eq_subset: |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
214 |
assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
215 |
assumes eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and "i \<in> I" |
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diff
changeset
|
216 |
shows "F i \<subseteq> F' i" |
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parents:
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changeset
|
217 |
proof |
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parents:
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changeset
|
218 |
fix x assume "x \<in> F i" |
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parents:
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diff
changeset
|
219 |
with ne have "\<forall>j. \<exists>y. ((j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined))" by auto |
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hoelzl
parents:
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diff
changeset
|
220 |
from choice[OF this] guess f .. note f = this |
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parents:
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diff
changeset
|
221 |
then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def) |
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parents:
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diff
changeset
|
222 |
then have "f \<in> Pi\<^isub>E I F'" using assms by simp |
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parents:
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changeset
|
223 |
then show "x \<in> F' i" using f `i \<in> I` by auto |
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parents:
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changeset
|
224 |
qed |
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parents:
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diff
changeset
|
225 |
|
3e39b0e730d6
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parents:
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diff
changeset
|
226 |
lemma Pi_eq_iff_not_empty: |
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parents:
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changeset
|
227 |
assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}" |
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hoelzl
parents:
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diff
changeset
|
228 |
shows "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)" |
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parents:
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changeset
|
229 |
proof (intro iffI ballI) |
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parents:
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diff
changeset
|
230 |
fix i assume eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and i: "i \<in> I" |
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parents:
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diff
changeset
|
231 |
show "F i = F' i" |
3e39b0e730d6
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parents:
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changeset
|
232 |
using Pi_eq_subset[of I F F', OF ne eq i] |
3e39b0e730d6
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parents:
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diff
changeset
|
233 |
using Pi_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i] |
3e39b0e730d6
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parents:
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diff
changeset
|
234 |
by auto |
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parents:
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changeset
|
235 |
qed auto |
3e39b0e730d6
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parents:
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diff
changeset
|
236 |
|
3e39b0e730d6
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hoelzl
parents:
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diff
changeset
|
237 |
lemma Pi_eq_empty_iff: |
3e39b0e730d6
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parents:
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diff
changeset
|
238 |
"Pi\<^isub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})" |
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parents:
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diff
changeset
|
239 |
proof |
3e39b0e730d6
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parents:
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changeset
|
240 |
assume "Pi\<^isub>E I F = {}" |
3e39b0e730d6
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parents:
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diff
changeset
|
241 |
show "\<exists>i\<in>I. F i = {}" |
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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changeset
|
242 |
proof (rule ccontr) |
3e39b0e730d6
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parents:
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changeset
|
243 |
assume "\<not> ?thesis" |
3e39b0e730d6
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parents:
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changeset
|
244 |
then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto |
3e39b0e730d6
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hoelzl
parents:
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diff
changeset
|
245 |
from choice[OF this] guess f .. |
3e39b0e730d6
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hoelzl
parents:
41661
diff
changeset
|
246 |
then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def) |
3e39b0e730d6
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hoelzl
parents:
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diff
changeset
|
247 |
with `Pi\<^isub>E I F = {}` show False by auto |
3e39b0e730d6
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parents:
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diff
changeset
|
248 |
qed |
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parents:
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diff
changeset
|
249 |
qed auto |
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parents:
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diff
changeset
|
250 |
|
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parents:
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changeset
|
251 |
lemma Pi_eq_iff: |
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parents:
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changeset
|
252 |
"Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))" |
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
|
253 |
proof (intro iffI disjCI) |
3e39b0e730d6
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parents:
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diff
changeset
|
254 |
assume eq[simp]: "Pi\<^isub>E I F = Pi\<^isub>E I F'" |
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parents:
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diff
changeset
|
255 |
assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))" |
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hoelzl
parents:
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diff
changeset
|
256 |
then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})" |
3e39b0e730d6
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hoelzl
parents:
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diff
changeset
|
257 |
using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto |
3e39b0e730d6
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hoelzl
parents:
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diff
changeset
|
258 |
with Pi_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto |
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parents:
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diff
changeset
|
259 |
next |
3e39b0e730d6
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parents:
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diff
changeset
|
260 |
assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
261 |
then show "Pi\<^isub>E I F = Pi\<^isub>E I F'" |
3e39b0e730d6
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hoelzl
parents:
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diff
changeset
|
262 |
using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto |
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parents:
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changeset
|
263 |
qed |
3e39b0e730d6
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parents:
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diff
changeset
|
264 |
|
40859 | 265 |
section "Finite product spaces" |
266 |
||
267 |
section "Products" |
|
268 |
||
47694 | 269 |
definition prod_emb where |
270 |
"prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))" |
|
271 |
||
272 |
lemma prod_emb_iff: |
|
273 |
"f \<in> prod_emb I M K X \<longleftrightarrow> f \<in> extensional I \<and> (restrict f K \<in> X) \<and> (\<forall>i\<in>I. f i \<in> space (M i))" |
|
274 |
unfolding prod_emb_def by auto |
|
40859 | 275 |
|
47694 | 276 |
lemma |
277 |
shows prod_emb_empty[simp]: "prod_emb M L K {} = {}" |
|
278 |
and prod_emb_Un[simp]: "prod_emb M L K (A \<union> B) = prod_emb M L K A \<union> prod_emb M L K B" |
|
279 |
and prod_emb_Int: "prod_emb M L K (A \<inter> B) = prod_emb M L K A \<inter> prod_emb M L K B" |
|
280 |
and prod_emb_UN[simp]: "prod_emb M L K (\<Union>i\<in>I. F i) = (\<Union>i\<in>I. prod_emb M L K (F i))" |
|
281 |
and prod_emb_INT[simp]: "I \<noteq> {} \<Longrightarrow> prod_emb M L K (\<Inter>i\<in>I. F i) = (\<Inter>i\<in>I. prod_emb M L K (F i))" |
|
282 |
and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B" |
|
283 |
by (auto simp: prod_emb_def) |
|
40859 | 284 |
|
47694 | 285 |
lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> |
286 |
prod_emb I M J (\<Pi>\<^isub>E i\<in>J. E i) = (\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i))" |
|
287 |
by (force simp: prod_emb_def Pi_iff split_if_mem2) |
|
288 |
||
289 |
lemma prod_emb_PiE_same_index[simp]: "(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^isub>E I E) = Pi\<^isub>E I E" |
|
290 |
by (auto simp: prod_emb_def Pi_iff) |
|
41689
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|
291 |
|
50038 | 292 |
lemma prod_emb_trans[simp]: |
293 |
"J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> prod_emb L M K (prod_emb K M J X) = prod_emb L M J X" |
|
294 |
by (auto simp add: Int_absorb1 prod_emb_def) |
|
295 |
||
296 |
lemma prod_emb_Pi: |
|
297 |
assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K" |
|
298 |
shows "prod_emb K M J (Pi\<^isub>E J X) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then X i else space (M i))" |
|
299 |
using assms space_closed |
|
300 |
by (auto simp: prod_emb_def Pi_iff split: split_if_asm) blast+ |
|
301 |
||
302 |
lemma prod_emb_id: |
|
303 |
"B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B" |
|
304 |
by (auto simp: prod_emb_def Pi_iff subset_eq extensional_restrict) |
|
305 |
||
50041
afe886a04198
removed redundant/unnecessary assumptions from projective_family
immler@in.tum.de
parents:
50038
diff
changeset
|
306 |
lemma prod_emb_mono: |
afe886a04198
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immler@in.tum.de
parents:
50038
diff
changeset
|
307 |
"F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G" |
afe886a04198
removed redundant/unnecessary assumptions from projective_family
immler@in.tum.de
parents:
50038
diff
changeset
|
308 |
by (auto simp: prod_emb_def) |
afe886a04198
removed redundant/unnecessary assumptions from projective_family
immler@in.tum.de
parents:
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diff
changeset
|
309 |
|
47694 | 310 |
definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where |
311 |
"PiM I M = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i)) |
|
312 |
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} |
|
313 |
(\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) |
|
314 |
(\<lambda>(J, X). \<Prod>j\<in>J \<union> {i\<in>I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))" |
|
315 |
||
316 |
definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where |
|
317 |
"prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j)) ` |
|
318 |
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" |
|
319 |
||
320 |
abbreviation |
|
321 |
"Pi\<^isub>M I M \<equiv> PiM I M" |
|
41689
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|
322 |
|
40859 | 323 |
syntax |
47694 | 324 |
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3PIM _:_./ _)" 10) |
40859 | 325 |
|
326 |
syntax (xsymbols) |
|
47694 | 327 |
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10) |
40859 | 328 |
|
329 |
syntax (HTML output) |
|
47694 | 330 |
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10) |
40859 | 331 |
|
332 |
translations |
|
47694 | 333 |
"PIM x:I. M" == "CONST PiM I (%x. M)" |
41689
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changeset
|
334 |
|
47694 | 335 |
lemma prod_algebra_sets_into_space: |
336 |
"prod_algebra I M \<subseteq> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
337 |
using assms by (auto simp: prod_emb_def prod_algebra_def) |
|
40859 | 338 |
|
47694 | 339 |
lemma prod_algebra_eq_finite: |
340 |
assumes I: "finite I" |
|
341 |
shows "prod_algebra I M = {(\<Pi>\<^isub>E i\<in>I. X i) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R") |
|
342 |
proof (intro iffI set_eqI) |
|
343 |
fix A assume "A \<in> ?L" |
|
344 |
then obtain J E where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)" |
|
345 |
and A: "A = prod_emb I M J (PIE j:J. E j)" |
|
346 |
by (auto simp: prod_algebra_def) |
|
347 |
let ?A = "\<Pi>\<^isub>E i\<in>I. if i \<in> J then E i else space (M i)" |
|
348 |
have A: "A = ?A" |
|
349 |
unfolding A using J by (intro prod_emb_PiE sets_into_space) auto |
|
350 |
show "A \<in> ?R" unfolding A using J top |
|
351 |
by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp |
|
352 |
next |
|
353 |
fix A assume "A \<in> ?R" |
|
354 |
then obtain X where "A = (\<Pi>\<^isub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto |
|
355 |
then have A: "A = prod_emb I M I (\<Pi>\<^isub>E i\<in>I. X i)" |
|
356 |
using sets_into_space by (force simp: prod_emb_def Pi_iff) |
|
357 |
from X I show "A \<in> ?L" unfolding A |
|
358 |
by (auto simp: prod_algebra_def) |
|
359 |
qed |
|
41095 | 360 |
|
47694 | 361 |
lemma prod_algebraI: |
362 |
"finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)) |
|
363 |
\<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M" |
|
364 |
by (auto simp: prod_algebra_def Pi_iff) |
|
41689
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parents:
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diff
changeset
|
365 |
|
50038 | 366 |
lemma prod_algebraI_finite: |
367 |
"finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^isub>E I E) \<in> prod_algebra I M" |
|
368 |
using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets_into_space] by simp |
|
369 |
||
370 |
lemma Int_stable_PiE: "Int_stable {Pi\<^isub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}" |
|
371 |
proof (safe intro!: Int_stableI) |
|
372 |
fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)" |
|
373 |
then show "\<exists>G. Pi\<^isub>E J E \<inter> Pi\<^isub>E J F = Pi\<^isub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))" |
|
374 |
by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"]) |
|
375 |
qed |
|
376 |
||
47694 | 377 |
lemma prod_algebraE: |
378 |
assumes A: "A \<in> prod_algebra I M" |
|
379 |
obtains J E where "A = prod_emb I M J (PIE j:J. E j)" |
|
380 |
"finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" |
|
381 |
using A by (auto simp: prod_algebra_def) |
|
42988 | 382 |
|
47694 | 383 |
lemma prod_algebraE_all: |
384 |
assumes A: "A \<in> prod_algebra I M" |
|
385 |
obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))" |
|
386 |
proof - |
|
387 |
from A obtain E J where A: "A = prod_emb I M J (Pi\<^isub>E J E)" |
|
388 |
and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))" |
|
389 |
by (auto simp: prod_algebra_def) |
|
390 |
from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)" |
|
391 |
using sets_into_space by auto |
|
392 |
then have "A = (\<Pi>\<^isub>E i\<in>I. if i\<in>J then E i else space (M i))" |
|
393 |
using A J by (auto simp: prod_emb_PiE) |
|
394 |
moreover then have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))" |
|
395 |
using top E by auto |
|
396 |
ultimately show ?thesis using that by auto |
|
397 |
qed |
|
40859 | 398 |
|
47694 | 399 |
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)" |
400 |
proof (unfold Int_stable_def, safe) |
|
401 |
fix A assume "A \<in> prod_algebra I M" |
|
402 |
from prod_algebraE[OF this] guess J E . note A = this |
|
403 |
fix B assume "B \<in> prod_algebra I M" |
|
404 |
from prod_algebraE[OF this] guess K F . note B = this |
|
405 |
have "A \<inter> B = prod_emb I M (J \<union> K) (\<Pi>\<^isub>E i\<in>J \<union> K. (if i \<in> J then E i else space (M i)) \<inter> |
|
406 |
(if i \<in> K then F i else space (M i)))" |
|
407 |
unfolding A B using A(2,3,4) A(5)[THEN sets_into_space] B(2,3,4) B(5)[THEN sets_into_space] |
|
408 |
apply (subst (1 2 3) prod_emb_PiE) |
|
409 |
apply (simp_all add: subset_eq PiE_Int) |
|
410 |
apply blast |
|
411 |
apply (intro PiE_cong) |
|
412 |
apply auto |
|
413 |
done |
|
414 |
also have "\<dots> \<in> prod_algebra I M" |
|
415 |
using A B by (auto intro!: prod_algebraI) |
|
416 |
finally show "A \<inter> B \<in> prod_algebra I M" . |
|
417 |
qed |
|
418 |
||
419 |
lemma prod_algebra_mono: |
|
420 |
assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)" |
|
421 |
assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)" |
|
422 |
shows "prod_algebra I E \<subseteq> prod_algebra I F" |
|
423 |
proof |
|
424 |
fix A assume "A \<in> prod_algebra I E" |
|
425 |
then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" |
|
426 |
and A: "A = prod_emb I E J (\<Pi>\<^isub>E i\<in>J. G i)" |
|
427 |
and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)" |
|
428 |
by (auto simp: prod_algebra_def) |
|
429 |
moreover |
|
430 |
from space have "(\<Pi>\<^isub>E i\<in>I. space (E i)) = (\<Pi>\<^isub>E i\<in>I. space (F i))" |
|
431 |
by (rule PiE_cong) |
|
432 |
with A have "A = prod_emb I F J (\<Pi>\<^isub>E i\<in>J. G i)" |
|
433 |
by (simp add: prod_emb_def) |
|
434 |
moreover |
|
435 |
from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)" |
|
436 |
by auto |
|
437 |
ultimately show "A \<in> prod_algebra I F" |
|
438 |
apply (simp add: prod_algebra_def image_iff) |
|
439 |
apply (intro exI[of _ J] exI[of _ G] conjI) |
|
440 |
apply auto |
|
441 |
done |
|
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
|
442 |
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
|
443 |
|
47694 | 444 |
lemma space_PiM: "space (\<Pi>\<^isub>M i\<in>I. M i) = (\<Pi>\<^isub>E i\<in>I. space (M i))" |
445 |
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp |
|
446 |
||
447 |
lemma sets_PiM: "sets (\<Pi>\<^isub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) (prod_algebra I M)" |
|
448 |
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) 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
|
449 |
|
47694 | 450 |
lemma sets_PiM_single: "sets (PiM I M) = |
451 |
sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}" |
|
452 |
(is "_ = sigma_sets ?\<Omega> ?R") |
|
453 |
unfolding sets_PiM |
|
454 |
proof (rule sigma_sets_eqI) |
|
455 |
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto |
|
456 |
fix A assume "A \<in> prod_algebra I M" |
|
457 |
from prod_algebraE[OF this] guess J X . note X = this |
|
458 |
show "A \<in> sigma_sets ?\<Omega> ?R" |
|
459 |
proof cases |
|
460 |
assume "I = {}" |
|
461 |
with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def) |
|
462 |
with `I = {}` show ?thesis by (auto intro!: sigma_sets_top) |
|
463 |
next |
|
464 |
assume "I \<noteq> {}" |
|
465 |
with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^isub>E i\<in>I. space (M i)). f j \<in> X j})" |
|
466 |
using sets_into_space[OF X(5)] |
|
467 |
by (auto simp: prod_emb_PiE[OF _ sets_into_space] Pi_iff split: split_if_asm) blast |
|
468 |
also have "\<dots> \<in> sigma_sets ?\<Omega> ?R" |
|
469 |
using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto |
|
470 |
finally show "A \<in> sigma_sets ?\<Omega> ?R" . |
|
471 |
qed |
|
472 |
next |
|
473 |
fix A assume "A \<in> ?R" |
|
474 |
then obtain i B where A: "A = {f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)" |
|
475 |
by auto |
|
476 |
then have "A = prod_emb I M {i} (\<Pi>\<^isub>E i\<in>{i}. B)" |
|
477 |
using sets_into_space[OF A(3)] |
|
478 |
apply (subst prod_emb_PiE) |
|
479 |
apply (auto simp: Pi_iff split: split_if_asm) |
|
480 |
apply blast |
|
481 |
done |
|
482 |
also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)" |
|
483 |
using A by (intro sigma_sets.Basic prod_algebraI) auto |
|
484 |
finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" . |
|
485 |
qed |
|
486 |
||
487 |
lemma sets_PiM_I: |
|
488 |
assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)" |
|
489 |
shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)" |
|
490 |
proof cases |
|
491 |
assume "J = {}" |
|
492 |
then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))" |
|
493 |
by (auto simp: prod_emb_def) |
|
494 |
then show ?thesis |
|
495 |
by (auto simp add: sets_PiM intro!: sigma_sets_top) |
|
496 |
next |
|
497 |
assume "J \<noteq> {}" with assms show ?thesis |
|
50003 | 498 |
by (force simp add: sets_PiM prod_algebra_def) |
40859 | 499 |
qed |
500 |
||
47694 | 501 |
lemma measurable_PiM: |
502 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
503 |
assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow> |
|
504 |
f -` prod_emb I M J (Pi\<^isub>E J X) \<inter> space N \<in> sets N" |
|
505 |
shows "f \<in> measurable N (PiM I M)" |
|
506 |
using sets_PiM prod_algebra_sets_into_space space |
|
507 |
proof (rule measurable_sigma_sets) |
|
508 |
fix A assume "A \<in> prod_algebra I M" |
|
509 |
from prod_algebraE[OF this] guess J X . |
|
510 |
with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto |
|
511 |
qed |
|
512 |
||
513 |
lemma measurable_PiM_Collect: |
|
514 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
515 |
assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow> |
|
516 |
{\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" |
|
517 |
shows "f \<in> measurable N (PiM I M)" |
|
518 |
using sets_PiM prod_algebra_sets_into_space space |
|
519 |
proof (rule measurable_sigma_sets) |
|
520 |
fix A assume "A \<in> prod_algebra I M" |
|
521 |
from prod_algebraE[OF this] guess J X . note X = this |
|
522 |
have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}" |
|
523 |
using sets_into_space[OF X(5)] X(2-) space unfolding X(1) |
|
524 |
by (subst prod_emb_PiE) (auto simp: Pi_iff split: split_if_asm) |
|
525 |
also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets) |
|
526 |
finally show "f -` A \<inter> space N \<in> sets N" . |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
527 |
qed |
41095 | 528 |
|
47694 | 529 |
lemma measurable_PiM_single: |
530 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
531 |
assumes sets: "\<And>A i. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N" |
|
532 |
shows "f \<in> measurable N (PiM I M)" |
|
533 |
using sets_PiM_single |
|
534 |
proof (rule measurable_sigma_sets) |
|
535 |
fix A assume "A \<in> {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}" |
|
536 |
then obtain B i where "A = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)" |
|
537 |
by auto |
|
538 |
with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto |
|
539 |
also have "\<dots> \<in> sets N" using B by (rule sets) |
|
540 |
finally show "f -` A \<inter> space N \<in> sets N" . |
|
541 |
qed (auto simp: space) |
|
40859 | 542 |
|
50003 | 543 |
lemma sets_PiM_I_finite[measurable]: |
47694 | 544 |
assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))" |
545 |
shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)" |
|
546 |
using sets_PiM_I[of I I E M] sets_into_space[OF sets] `finite I` sets by auto |
|
547 |
||
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
548 |
lemma measurable_component_singleton: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
549 |
assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
550 |
proof (unfold measurable_def, intro CollectI conjI ballI) |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
551 |
fix A assume "A \<in> sets (M i)" |
47694 | 552 |
then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) = prod_emb I M {i} (\<Pi>\<^isub>E j\<in>{i}. A)" |
553 |
using sets_into_space `i \<in> I` |
|
554 |
by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm) |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
555 |
then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)" |
47694 | 556 |
using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I) |
557 |
qed (insert `i \<in> I`, auto simp: space_PiM) |
|
558 |
||
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
559 |
lemma measurable_component_singleton'[measurable_app]: |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
560 |
assumes f: "f \<in> measurable N (Pi\<^isub>M I M)" |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
561 |
assumes i: "i \<in> I" |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
562 |
shows "(\<lambda>x. (f x) i) \<in> measurable N (M i)" |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
563 |
using measurable_compose[OF f measurable_component_singleton, OF i] . |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
564 |
|
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
565 |
lemma measurable_nat_case[measurable (raw)]: |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
566 |
assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N" |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
567 |
"\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N" |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
568 |
shows "(\<lambda>x. nat_case (f x) (g x) i) \<in> measurable M N" |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
569 |
by (cases i) simp_all |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
570 |
|
50003 | 571 |
lemma measurable_add_dim[measurable]: |
49776 | 572 |
"(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)" |
47694 | 573 |
(is "?f \<in> measurable ?P ?I") |
574 |
proof (rule measurable_PiM_single) |
|
575 |
fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)" |
|
576 |
have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} = |
|
577 |
(if j = i then space (Pi\<^isub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)" |
|
578 |
using sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM) |
|
579 |
also have "\<dots> \<in> sets ?P" |
|
580 |
using A j |
|
581 |
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) |
|
582 |
finally show "{\<omega> \<in> space ?P. prod_case (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" . |
|
583 |
qed (auto simp: space_pair_measure space_PiM) |
|
41661 | 584 |
|
50003 | 585 |
lemma measurable_component_update: |
586 |
"x \<in> space (Pi\<^isub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" |
|
587 |
by simp |
|
588 |
||
589 |
lemma measurable_merge[measurable]: |
|
49780 | 590 |
"merge I J \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)" |
47694 | 591 |
(is "?f \<in> measurable ?P ?U") |
592 |
proof (rule measurable_PiM_single) |
|
593 |
fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J" |
|
49780 | 594 |
then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} = |
47694 | 595 |
(if i \<in> I then ((\<lambda>x. x i) \<circ> fst) -` A \<inter> space ?P else ((\<lambda>x. x i) \<circ> snd) -` A \<inter> space ?P)" |
49776 | 596 |
by (auto simp: merge_def) |
47694 | 597 |
also have "\<dots> \<in> sets ?P" |
598 |
using A |
|
599 |
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) |
|
49780 | 600 |
finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" . |
49776 | 601 |
qed (auto simp: space_pair_measure space_PiM Pi_iff merge_def extensional_def) |
42988 | 602 |
|
50003 | 603 |
lemma measurable_restrict[measurable (raw)]: |
47694 | 604 |
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)" |
605 |
shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^isub>M I M)" |
|
606 |
proof (rule measurable_PiM_single) |
|
607 |
fix A i assume A: "i \<in> I" "A \<in> sets (M i)" |
|
608 |
then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N" |
|
609 |
by auto |
|
610 |
then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N" |
|
611 |
using A X by (auto intro!: measurable_sets) |
|
612 |
qed (insert X, auto dest: measurable_space) |
|
613 |
||
50038 | 614 |
lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^isub>M L M) (Pi\<^isub>M J M)" |
615 |
by (intro measurable_restrict measurable_component_singleton) auto |
|
616 |
||
617 |
lemma measurable_prod_emb[intro, simp]: |
|
618 |
"J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^isub>M L M)" |
|
619 |
unfolding prod_emb_def space_PiM[symmetric] |
|
620 |
by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton) |
|
621 |
||
50003 | 622 |
lemma sets_in_Pi_aux: |
623 |
"finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow> |
|
624 |
{x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)" |
|
625 |
by (simp add: subset_eq Pi_iff) |
|
626 |
||
627 |
lemma sets_in_Pi[measurable (raw)]: |
|
628 |
"finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow> |
|
629 |
(\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow> |
|
630 |
Sigma_Algebra.pred N (\<lambda>x. f x \<in> Pi I F)" |
|
631 |
unfolding pred_def |
|
632 |
by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto |
|
633 |
||
634 |
lemma sets_in_extensional_aux: |
|
635 |
"{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)" |
|
636 |
proof - |
|
637 |
have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)" |
|
638 |
by (auto simp add: extensional_def space_PiM) |
|
639 |
then show ?thesis by simp |
|
640 |
qed |
|
641 |
||
642 |
lemma sets_in_extensional[measurable (raw)]: |
|
643 |
"f \<in> measurable N (PiM I M) \<Longrightarrow> Sigma_Algebra.pred N (\<lambda>x. f x \<in> extensional I)" |
|
644 |
unfolding pred_def |
|
645 |
by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto |
|
646 |
||
47694 | 647 |
locale product_sigma_finite = |
648 |
fixes M :: "'i \<Rightarrow> 'a measure" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
649 |
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)" |
40859 | 650 |
|
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
|
651 |
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i |
40859 | 652 |
by (rule sigma_finite_measures) |
653 |
||
47694 | 654 |
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" + |
655 |
fixes I :: "'i set" |
|
656 |
assumes finite_index: "finite I" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
657 |
|
40859 | 658 |
lemma (in finite_product_sigma_finite) sigma_finite_pairs: |
659 |
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set. |
|
660 |
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and> |
|
47694 | 661 |
(\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and> |
662 |
(\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space (PiM I M)" |
|
40859 | 663 |
proof - |
47694 | 664 |
have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. emeasure (M i) (F k) \<noteq> \<infinity>)" |
665 |
using M.sigma_finite_incseq by metis |
|
40859 | 666 |
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. |
47694 | 667 |
then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. emeasure (M i) (F i k) \<noteq> \<infinity>" |
40859 | 668 |
by auto |
669 |
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k" |
|
47694 | 670 |
note space_PiM[simp] |
40859 | 671 |
show ?thesis |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
672 |
proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI) |
40859 | 673 |
fix i show "range (F i) \<subseteq> sets (M i)" by fact |
674 |
next |
|
47694 | 675 |
fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact |
40859 | 676 |
next |
47694 | 677 |
fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space (PiM I M)" |
678 |
using `\<And>i. range (F i) \<subseteq> sets (M i)` sets_into_space |
|
679 |
by auto blast |
|
40859 | 680 |
next |
47694 | 681 |
fix f assume "f \<in> space (PiM I M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
682 |
with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
683 |
show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def) |
40859 | 684 |
next |
685 |
fix i show "?F i \<subseteq> ?F (Suc i)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
686 |
using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto |
40859 | 687 |
qed |
688 |
qed |
|
689 |
||
49780 | 690 |
lemma |
691 |
shows space_PiM_empty: "space (Pi\<^isub>M {} M) = {\<lambda>k. undefined}" |
|
692 |
and sets_PiM_empty: "sets (Pi\<^isub>M {} M) = { {}, {\<lambda>k. undefined} }" |
|
693 |
by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq) |
|
694 |
||
695 |
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1" |
|
696 |
proof - |
|
697 |
let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)" |
|
698 |
have "emeasure (Pi\<^isub>M {} M) (prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = 1" |
|
699 |
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) |
|
700 |
show "positive (PiM {} M) ?\<mu>" |
|
701 |
by (auto simp: positive_def) |
|
702 |
show "countably_additive (PiM {} M) ?\<mu>" |
|
703 |
by (rule countably_additiveI_finite) |
|
704 |
(auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: ) |
|
705 |
qed (auto simp: prod_emb_def) |
|
706 |
also have "(prod_emb {} M {} (\<Pi>\<^isub>E i\<in>{}. {})) = {\<lambda>_. undefined}" |
|
707 |
by (auto simp: prod_emb_def) |
|
708 |
finally show ?thesis |
|
709 |
by simp |
|
710 |
qed |
|
711 |
||
712 |
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}" |
|
713 |
by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def) |
|
714 |
||
49776 | 715 |
lemma (in product_sigma_finite) emeasure_PiM: |
716 |
"finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" |
|
717 |
proof (induct I arbitrary: A rule: finite_induct) |
|
40859 | 718 |
case (insert i I) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
719 |
interpret finite_product_sigma_finite M I by default fact |
40859 | 720 |
have "finite (insert i I)" using `finite I` by auto |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
721 |
interpret I': finite_product_sigma_finite M "insert i I" by default fact |
41661 | 722 |
let ?h = "(\<lambda>(f, y). f(i := y))" |
47694 | 723 |
|
724 |
let ?P = "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M) ?h" |
|
725 |
let ?\<mu> = "emeasure ?P" |
|
726 |
let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}" |
|
727 |
let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))" |
|
728 |
||
49776 | 729 |
have "emeasure (Pi\<^isub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^isub>E (insert i I) A)) = |
730 |
(\<Prod>i\<in>insert i I. emeasure (M i) (A i))" |
|
731 |
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) |
|
732 |
fix J E assume "(J \<noteq> {} \<or> insert i I = {}) \<and> finite J \<and> J \<subseteq> insert i I \<and> E \<in> (\<Pi> j\<in>J. sets (M j))" |
|
733 |
then have J: "J \<noteq> {}" "finite J" "J \<subseteq> insert i I" and E: "\<forall>j\<in>J. E j \<in> sets (M j)" by auto |
|
734 |
let ?p = "prod_emb (insert i I) M J (Pi\<^isub>E J E)" |
|
735 |
let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^isub>E j\<in>J-{i}. E j)" |
|
736 |
have "?\<mu> ?p = |
|
737 |
emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i))" |
|
738 |
by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+ |
|
739 |
also have "?h -` ?p \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))" |
|
740 |
using J E[rule_format, THEN sets_into_space] |
|
741 |
by (force simp: space_pair_measure space_PiM Pi_iff prod_emb_iff split: split_if_asm) |
|
742 |
also have "emeasure (Pi\<^isub>M I M \<Otimes>\<^isub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) = |
|
743 |
emeasure (Pi\<^isub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))" |
|
744 |
using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto |
|
745 |
also have "?p' = (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))" |
|
746 |
using J E[rule_format, THEN sets_into_space] |
|
747 |
by (auto simp: prod_emb_iff Pi_iff split: split_if_asm) blast+ |
|
748 |
also have "emeasure (Pi\<^isub>M I M) (\<Pi>\<^isub>E j\<in>I. if j \<in> J-{i} then E j else space (M j)) = |
|
749 |
(\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))" |
|
750 |
using E by (subst insert) (auto intro!: setprod_cong) |
|
751 |
also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) * |
|
752 |
emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)" |
|
753 |
using insert by (auto simp: mult_commute intro!: arg_cong2[where f="op *"] setprod_cong) |
|
754 |
also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)" |
|
755 |
using insert(1,2) J E by (intro setprod_mono_one_right) auto |
|
756 |
finally show "?\<mu> ?p = \<dots>" . |
|
47694 | 757 |
|
49776 | 758 |
show "prod_emb (insert i I) M J (Pi\<^isub>E J E) \<in> Pow (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" |
759 |
using J E[rule_format, THEN sets_into_space] by (auto simp: prod_emb_iff) |
|
760 |
next |
|
761 |
show "positive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^isub>M (insert i I) M)) ?\<mu>" |
|
762 |
using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all |
|
763 |
next |
|
764 |
show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and> |
|
765 |
insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))" |
|
766 |
using insert by auto |
|
767 |
qed (auto intro!: setprod_cong) |
|
768 |
with insert show ?case |
|
769 |
by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets_into_space) |
|
50003 | 770 |
qed simp |
47694 | 771 |
|
49776 | 772 |
lemma (in product_sigma_finite) sigma_finite: |
773 |
assumes "finite I" |
|
774 |
shows "sigma_finite_measure (PiM I M)" |
|
775 |
proof - |
|
776 |
interpret finite_product_sigma_finite M I by default fact |
|
777 |
||
778 |
from sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. |
|
779 |
then have F: "\<And>j. j \<in> I \<Longrightarrow> range (F j) \<subseteq> sets (M j)" |
|
780 |
"incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k)" |
|
781 |
"(\<Union>k. \<Pi>\<^isub>E j \<in> I. F j k) = space (Pi\<^isub>M I M)" |
|
782 |
"\<And>k. \<And>j. j \<in> I \<Longrightarrow> emeasure (M j) (F j k) \<noteq> \<infinity>" |
|
47694 | 783 |
by blast+ |
49776 | 784 |
let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> I. F j k" |
47694 | 785 |
|
49776 | 786 |
show ?thesis |
47694 | 787 |
proof (unfold_locales, intro exI[of _ ?F] conjI allI) |
49776 | 788 |
show "range ?F \<subseteq> sets (Pi\<^isub>M I M)" using F(1) `finite I` by auto |
47694 | 789 |
next |
49776 | 790 |
from F(3) show "(\<Union>i. ?F i) = space (Pi\<^isub>M I M)" by simp |
47694 | 791 |
next |
792 |
fix j |
|
49776 | 793 |
from F `finite I` setprod_PInf[of I, OF emeasure_nonneg, of M] |
794 |
show "emeasure (\<Pi>\<^isub>M i\<in>I. M i) (?F j) \<noteq> \<infinity>" |
|
795 |
by (subst emeasure_PiM) auto |
|
40859 | 796 |
qed |
797 |
qed |
|
798 |
||
47694 | 799 |
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^isub>M I M" |
800 |
using sigma_finite[OF finite_index] . |
|
40859 | 801 |
|
802 |
lemma (in finite_product_sigma_finite) measure_times: |
|
47694 | 803 |
"(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^isub>M I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" |
804 |
using emeasure_PiM[OF finite_index] by auto |
|
41096 | 805 |
|
40859 | 806 |
lemma (in product_sigma_finite) positive_integral_empty: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
807 |
assumes pos: "0 \<le> f (\<lambda>k. undefined)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
808 |
shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)" |
40859 | 809 |
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
|
810 |
interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI) |
47694 | 811 |
have "\<And>A. emeasure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1" |
40859 | 812 |
using assms by (subst measure_times) auto |
813 |
then show ?thesis |
|
47694 | 814 |
unfolding positive_integral_def simple_function_def simple_integral_def[abs_def] |
815 |
proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
816 |
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
817 |
by (intro SUP_upper) (auto simp: le_fun_def split: split_max) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
818 |
show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
819 |
by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm) |
40859 | 820 |
qed |
821 |
qed |
|
822 |
||
47694 | 823 |
lemma (in product_sigma_finite) distr_merge: |
40859 | 824 |
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J" |
49780 | 825 |
shows "distr (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M) (merge I J) = Pi\<^isub>M (I \<union> J) M" |
47694 | 826 |
(is "?D = ?P") |
40859 | 827 |
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
|
828 |
interpret I: finite_product_sigma_finite M I by default fact |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
829 |
interpret J: finite_product_sigma_finite M J by default fact |
40859 | 830 |
have "finite (I \<union> J)" using fin by auto |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
831 |
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact |
47694 | 832 |
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default |
49780 | 833 |
let ?g = "merge I J" |
47694 | 834 |
|
41661 | 835 |
from IJ.sigma_finite_pairs obtain F where |
836 |
F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
837 |
"incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)" |
47694 | 838 |
"(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space ?P" |
839 |
"\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>" |
|
41661 | 840 |
by auto |
841 |
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k" |
|
47694 | 842 |
|
843 |
show ?thesis |
|
844 |
proof (rule measure_eqI_generator_eq[symmetric]) |
|
845 |
show "Int_stable (prod_algebra (I \<union> J) M)" |
|
846 |
by (rule Int_stable_prod_algebra) |
|
847 |
show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^isub>E i \<in> I \<union> J. space (M i))" |
|
848 |
by (rule prod_algebra_sets_into_space) |
|
849 |
show "sets ?P = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)" |
|
850 |
by (rule sets_PiM) |
|
851 |
then show "sets ?D = sigma_sets (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)" |
|
852 |
by simp |
|
853 |
||
854 |
show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F |
|
855 |
using fin by (auto simp: prod_algebra_eq_finite) |
|
856 |
show "(\<Union>i. \<Pi>\<^isub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^isub>E i\<in>I \<union> J. space (M i))" |
|
857 |
using F(3) by (simp add: space_PiM) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
858 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
859 |
fix k |
47694 | 860 |
from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M] |
861 |
show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto |
|
41661 | 862 |
next |
47694 | 863 |
fix A assume A: "A \<in> prod_algebra (I \<union> J) M" |
50003 | 864 |
with fin obtain F where A_eq: "A = (Pi\<^isub>E (I \<union> J) F)" and F: "\<forall>i\<in>J. F i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)" |
47694 | 865 |
by (auto simp add: prod_algebra_eq_finite) |
866 |
let ?B = "Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M" |
|
867 |
let ?X = "?g -` A \<inter> space ?B" |
|
868 |
have "Pi\<^isub>E I F \<subseteq> space (Pi\<^isub>M I M)" "Pi\<^isub>E J F \<subseteq> space (Pi\<^isub>M J M)" |
|
50003 | 869 |
using F[rule_format, THEN sets_into_space] by (force simp: space_PiM)+ |
47694 | 870 |
then have X: "?X = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)" |
871 |
unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM) |
|
872 |
have "emeasure ?D A = emeasure ?B ?X" |
|
873 |
using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM) |
|
874 |
also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))" |
|
50003 | 875 |
using `finite J` `finite I` F unfolding X |
49776 | 876 |
by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times Pi_iff) |
47694 | 877 |
also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))" |
41661 | 878 |
using `finite J` `finite I` `I \<inter> J = {}` by (simp add: setprod_Un_one) |
47694 | 879 |
also have "\<dots> = emeasure ?P (Pi\<^isub>E (I \<union> J) F)" |
41661 | 880 |
using `finite J` `finite I` F unfolding A |
881 |
by (intro IJ.measure_times[symmetric]) auto |
|
47694 | 882 |
finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp |
883 |
qed |
|
41661 | 884 |
qed |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
885 |
|
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
886 |
lemma (in product_sigma_finite) product_positive_integral_fold: |
47694 | 887 |
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
888 |
and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
889 |
shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f = |
49780 | 890 |
(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
891 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
892 |
interpret I: finite_product_sigma_finite M I by default fact |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
893 |
interpret J: finite_product_sigma_finite M J by default fact |
41831 | 894 |
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default |
49780 | 895 |
have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" |
49776 | 896 |
using measurable_comp[OF measurable_merge f] by (simp add: comp_def) |
41661 | 897 |
show ?thesis |
47694 | 898 |
apply (subst distr_merge[OF IJ, symmetric]) |
49776 | 899 |
apply (subst positive_integral_distr[OF measurable_merge f]) |
49999
dfb63b9b8908
for the product measure it is enough if only one measure is sigma-finite
hoelzl
parents:
49784
diff
changeset
|
900 |
apply (subst J.positive_integral_fst_measurable(2)[symmetric, OF P_borel]) |
47694 | 901 |
apply simp |
902 |
done |
|
40859 | 903 |
qed |
904 |
||
47694 | 905 |
lemma (in product_sigma_finite) distr_singleton: |
906 |
"distr (Pi\<^isub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _") |
|
907 |
proof (intro measure_eqI[symmetric]) |
|
41831 | 908 |
interpret I: finite_product_sigma_finite M "{i}" by default simp |
47694 | 909 |
fix A assume A: "A \<in> sets (M i)" |
910 |
moreover then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M {i} M) = (\<Pi>\<^isub>E i\<in>{i}. A)" |
|
911 |
using sets_into_space by (auto simp: space_PiM) |
|
912 |
ultimately show "emeasure (M i) A = emeasure ?D A" |
|
913 |
using A I.measure_times[of "\<lambda>_. A"] |
|
914 |
by (simp add: emeasure_distr measurable_component_singleton) |
|
915 |
qed simp |
|
41831 | 916 |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
917 |
lemma (in product_sigma_finite) product_positive_integral_singleton: |
40859 | 918 |
assumes f: "f \<in> borel_measurable (M i)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
919 |
shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f" |
40859 | 920 |
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
|
921 |
interpret I: finite_product_sigma_finite M "{i}" by default simp |
47694 | 922 |
from f show ?thesis |
923 |
apply (subst distr_singleton[symmetric]) |
|
924 |
apply (subst positive_integral_distr[OF measurable_component_singleton]) |
|
925 |
apply simp_all |
|
926 |
done |
|
40859 | 927 |
qed |
928 |
||
41096 | 929 |
lemma (in product_sigma_finite) product_positive_integral_insert: |
49780 | 930 |
assumes I[simp]: "finite I" "i \<notin> I" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
931 |
and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
932 |
shows "integral\<^isup>P (Pi\<^isub>M (insert i I) M) f = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^isub>M I M))" |
41096 | 933 |
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
|
934 |
interpret I: finite_product_sigma_finite M I by default auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
935 |
interpret i: finite_product_sigma_finite M "{i}" by default auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
936 |
have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
937 |
using f by auto |
41096 | 938 |
show ?thesis |
49780 | 939 |
unfolding product_positive_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f] |
940 |
proof (rule positive_integral_cong, subst product_positive_integral_singleton[symmetric]) |
|
47694 | 941 |
fix x assume x: "x \<in> space (Pi\<^isub>M I M)" |
49780 | 942 |
let ?f = "\<lambda>y. f (x(i := y))" |
943 |
show "?f \<in> borel_measurable (M i)" |
|
47694 | 944 |
using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`] |
945 |
unfolding comp_def . |
|
49780 | 946 |
show "(\<integral>\<^isup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^isub>M {i} M) = (\<integral>\<^isup>+ y. f (x(i := y i)) \<partial>Pi\<^isub>M {i} M)" |
947 |
using x |
|
948 |
by (auto intro!: positive_integral_cong arg_cong[where f=f] |
|
949 |
simp add: space_PiM extensional_def) |
|
41096 | 950 |
qed |
951 |
qed |
|
952 |
||
953 |
lemma (in product_sigma_finite) product_positive_integral_setprod: |
|
43920 | 954 |
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal" |
41096 | 955 |
assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
956 |
and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
957 |
shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))" |
41096 | 958 |
using assms proof induct |
959 |
case (insert i I) |
|
960 |
note `finite I`[intro, simp] |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
961 |
interpret I: finite_product_sigma_finite M I by default auto |
41096 | 962 |
have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))" |
963 |
using insert by (auto intro!: setprod_cong) |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
964 |
have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^isub>M J M)" |
41096 | 965 |
using sets_into_space insert |
47694 | 966 |
by (intro borel_measurable_ereal_setprod |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
967 |
measurable_comp[OF measurable_component_singleton, unfolded comp_def]) |
41096 | 968 |
auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
969 |
then show ?case |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
970 |
apply (simp add: product_positive_integral_insert[OF insert(1,2) prod]) |
47694 | 971 |
apply (simp add: insert(2-) * pos borel setprod_ereal_pos positive_integral_multc) |
972 |
apply (subst positive_integral_cmult) |
|
973 |
apply (auto simp add: pos borel insert(2-) setprod_ereal_pos positive_integral_positive) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
974 |
done |
47694 | 975 |
qed (simp add: space_PiM) |
41096 | 976 |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
977 |
lemma (in product_sigma_finite) product_integral_singleton: |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
978 |
assumes f: "f \<in> borel_measurable (M i)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
979 |
shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
980 |
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
|
981 |
interpret I: finite_product_sigma_finite M "{i}" by default simp |
43920 | 982 |
have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)" |
983 |
"(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)" |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
984 |
using assms by auto |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
985 |
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
|
986 |
unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] .. |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
987 |
qed |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
988 |
|
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
989 |
lemma (in product_sigma_finite) product_integral_fold: |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
990 |
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
991 |
and f: "integrable (Pi\<^isub>M (I \<union> J) M) f" |
49780 | 992 |
shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I J (x, y)) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
993 |
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
|
994 |
interpret I: finite_product_sigma_finite M I by default fact |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
995 |
interpret J: finite_product_sigma_finite M J by default fact |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
996 |
have "finite (I \<union> J)" using fin by auto |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
997 |
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact |
47694 | 998 |
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default |
49780 | 999 |
let ?M = "merge I J" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1000 |
let ?f = "\<lambda>x. f (?M x)" |
47694 | 1001 |
from f have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)" |
1002 |
by auto |
|
49780 | 1003 |
have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" |
49776 | 1004 |
using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def) |
47694 | 1005 |
have f_int: "integrable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ?f" |
49776 | 1006 |
by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f) |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
1007 |
show ?thesis |
47694 | 1008 |
apply (subst distr_merge[symmetric, OF IJ fin]) |
49776 | 1009 |
apply (subst integral_distr[OF measurable_merge f_borel]) |
47694 | 1010 |
apply (subst P.integrable_fst_measurable(2)[symmetric, OF f_int]) |
1011 |
apply simp |
|
1012 |
done |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
1013 |
qed |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
1014 |
|
49776 | 1015 |
lemma (in product_sigma_finite) |
1016 |
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)" |
|
1017 |
shows emeasure_fold_integral: |
|
49780 | 1018 |
"emeasure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I) |
49776 | 1019 |
and emeasure_fold_measurable: |
49780 | 1020 |
"(\<lambda>x. emeasure (Pi\<^isub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B) |
49776 | 1021 |
proof - |
1022 |
interpret I: finite_product_sigma_finite M I by default fact |
|
1023 |
interpret J: finite_product_sigma_finite M J by default fact |
|
1024 |
interpret IJ: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" .. |
|
49780 | 1025 |
have merge: "merge I J -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)" |
49776 | 1026 |
by (intro measurable_sets[OF _ A] measurable_merge assms) |
1027 |
||
1028 |
show ?I |
|
1029 |
apply (subst distr_merge[symmetric, OF IJ]) |
|
1030 |
apply (subst emeasure_distr[OF measurable_merge A]) |
|
1031 |
apply (subst J.emeasure_pair_measure_alt[OF merge]) |
|
1032 |
apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure) |
|
1033 |
done |
|
1034 |
||
1035 |
show ?B |
|
1036 |
using IJ.measurable_emeasure_Pair1[OF merge] |
|
1037 |
by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong) |
|
1038 |
qed |
|
1039 |
||
41096 | 1040 |
lemma (in product_sigma_finite) product_integral_insert: |
47694 | 1041 |
assumes I: "finite I" "i \<notin> I" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1042 |
and f: "integrable (Pi\<^isub>M (insert i I) M) f" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1043 |
shows "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^isub>M I M)" |
41096 | 1044 |
proof - |
47694 | 1045 |
have "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = integral\<^isup>L (Pi\<^isub>M (I \<union> {i}) M) f" |
1046 |
by simp |
|
49780 | 1047 |
also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) \<partial>Pi\<^isub>M I M)" |
47694 | 1048 |
using f I by (intro product_integral_fold) auto |
1049 |
also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^isub>M I M)" |
|
1050 |
proof (rule integral_cong, subst product_integral_singleton[symmetric]) |
|
1051 |
fix x assume x: "x \<in> space (Pi\<^isub>M I M)" |
|
1052 |
have f_borel: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)" |
|
1053 |
using f by auto |
|
1054 |
show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)" |
|
1055 |
using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`] |
|
1056 |
unfolding comp_def . |
|
49780 | 1057 |
from x I show "(\<integral> y. f (merge I {i} (x,y)) \<partial>Pi\<^isub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^isub>M {i} M)" |
47694 | 1058 |
by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def) |
41096 | 1059 |
qed |
47694 | 1060 |
finally show ?thesis . |
41096 | 1061 |
qed |
1062 |
||
1063 |
lemma (in product_sigma_finite) product_integrable_setprod: |
|
1064 |
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1065 |
assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1066 |
shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f") |
41096 | 1067 |
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
|
1068 |
interpret finite_product_sigma_finite M I by default fact |
41096 | 1069 |
have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1070 |
using integrable unfolding integrable_def by auto |
47694 | 1071 |
have borel: "?f \<in> borel_measurable (Pi\<^isub>M I M)" |
1072 |
using measurable_comp[OF measurable_component_singleton[of _ I M] f] by (auto simp: comp_def) |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1073 |
moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)" |
41096 | 1074 |
proof (unfold integrable_def, intro conjI) |
47694 | 1075 |
show "(\<lambda>x. abs (?f x)) \<in> borel_measurable (Pi\<^isub>M I M)" |
41096 | 1076 |
using borel by auto |
47694 | 1077 |
have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>Pi\<^isub>M I M) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>Pi\<^isub>M I M)" |
43920 | 1078 |
by (simp add: setprod_ereal abs_setprod) |
1079 |
also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))" |
|
41096 | 1080 |
using f by (subst product_positive_integral_setprod) auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1081 |
also have "\<dots> < \<infinity>" |
47694 | 1082 |
using integrable[THEN integrable_abs] |
1083 |
by (simp add: setprod_PInf integrable_def positive_integral_positive) |
|
1084 |
finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by auto |
|
1085 |
have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) = (\<integral>\<^isup>+x. 0 \<partial>(Pi\<^isub>M I M))" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1086 |
by (intro positive_integral_cong_pos) auto |
47694 | 1087 |
then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>(Pi\<^isub>M I M)) \<noteq> \<infinity>" by simp |
41096 | 1088 |
qed |
1089 |
ultimately show ?thesis |
|
1090 |
by (rule integrable_abs_iff[THEN iffD1]) |
|
1091 |
qed |
|
1092 |
||
1093 |
lemma (in product_sigma_finite) product_integral_setprod: |
|
1094 |
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real" |
|
49780 | 1095 |
assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1096 |
shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>L (M i) (f i))" |
49780 | 1097 |
using assms proof induct |
1098 |
case empty |
|
1099 |
interpret finite_measure "Pi\<^isub>M {} M" |
|
1100 |
by rule (simp add: space_PiM) |
|
1101 |
show ?case by (simp add: space_PiM measure_def) |
|
41096 | 1102 |
next |
1103 |
case (insert i I) |
|
1104 |
then have iI: "finite (insert i I)" by auto |
|
1105 |
then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1106 |
integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))" |
49780 | 1107 |
by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1108 |
interpret I: finite_product_sigma_finite M I by default fact |
41096 | 1109 |
have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))" |
1110 |
using `i \<notin> I` by (auto intro!: setprod_cong) |
|
1111 |
show ?case |
|
49780 | 1112 |
unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]] |
47694 | 1113 |
by (simp add: * insert integral_multc integral_cmult[OF prod] subset_insertI) |
41096 | 1114 |
qed |
1115 |
||
49776 | 1116 |
lemma sets_Collect_single: |
1117 |
"i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^isub>M I M). x i \<in> A } \<in> sets (Pi\<^isub>M I M)" |
|
50003 | 1118 |
by simp |
49776 | 1119 |
|
1120 |
lemma sigma_prod_algebra_sigma_eq_infinite: |
|
1121 |
fixes E :: "'i \<Rightarrow> 'a set set" |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1122 |
assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" |
49776 | 1123 |
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" |
1124 |
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" |
|
1125 |
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" |
|
1126 |
defines "P == {{f\<in>\<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> E i}" |
|
1127 |
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P" |
|
1128 |
proof |
|
1129 |
let ?P = "sigma (space (Pi\<^isub>M I M)) P" |
|
1130 |
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))" |
|
1131 |
using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq) |
|
1132 |
then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
1133 |
by (simp add: space_PiM) |
|
1134 |
have "sets (PiM I M) = |
|
1135 |
sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}" |
|
1136 |
using sets_PiM_single[of I M] by (simp add: space_P) |
|
1137 |
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)" |
|
1138 |
proof (safe intro!: sigma_sets_subset) |
|
1139 |
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" |
|
1140 |
then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))" |
|
1141 |
apply (subst measurable_iff_measure_of) |
|
1142 |
apply (simp_all add: P_closed) |
|
1143 |
using E_closed |
|
1144 |
apply (force simp: subset_eq space_PiM) |
|
1145 |
apply (force simp: subset_eq space_PiM) |
|
1146 |
apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i]) |
|
1147 |
apply (rule_tac x=Aa in exI) |
|
1148 |
apply (auto simp: space_PiM) |
|
1149 |
done |
|
1150 |
from measurable_sets[OF this, of A] A `i \<in> I` E_closed |
|
1151 |
have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
1152 |
by (simp add: E_generates) |
|
1153 |
also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}" |
|
1154 |
using P_closed by (auto simp: space_PiM) |
|
1155 |
finally show "\<dots> \<in> sets ?P" . |
|
1156 |
qed |
|
1157 |
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P" |
|
1158 |
by (simp add: P_closed) |
|
1159 |
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)" |
|
1160 |
unfolding P_def space_PiM[symmetric] |
|
1161 |
by (intro sigma_sets_subset) (auto simp: E_generates sets_Collect_single) |
|
1162 |
qed |
|
1163 |
||
47694 | 1164 |
lemma sigma_prod_algebra_sigma_eq: |
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1165 |
fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" |
47694 | 1166 |
assumes "finite I" |
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1167 |
assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" |
47694 | 1168 |
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" |
1169 |
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" |
|
1170 |
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" |
|
1171 |
defines "P == { Pi\<^isub>E I F | F. \<forall>i\<in>I. F i \<in> E i }" |
|
1172 |
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P" |
|
1173 |
proof |
|
1174 |
let ?P = "sigma (space (Pi\<^isub>M I M)) P" |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1175 |
from `finite I`[THEN ex_bij_betw_finite_nat] guess T .. |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1176 |
then have T: "\<And>i. i \<in> I \<Longrightarrow> T i < card I" "\<And>i. i\<in>I \<Longrightarrow> the_inv_into I T (T i) = i" |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1177 |
by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f) |
47694 | 1178 |
have P_closed: "P \<subseteq> Pow (space (Pi\<^isub>M I M))" |
1179 |
using E_closed by (auto simp: space_PiM P_def Pi_iff subset_eq) |
|
1180 |
then have space_P: "space ?P = (\<Pi>\<^isub>E i\<in>I. space (M i))" |
|
1181 |
by (simp add: space_PiM) |
|
1182 |
have "sets (PiM I M) = |
|
1183 |
sigma_sets (space ?P) {{f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}" |
|
1184 |
using sets_PiM_single[of I M] by (simp add: space_P) |
|
1185 |
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)" |
|
1186 |
proof (safe intro!: sigma_sets_subset) |
|
1187 |
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" |
|
1188 |
have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))" |
|
1189 |
proof (subst measurable_iff_measure_of) |
|
1190 |
show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact |
|
1191 |
from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" |
|
1192 |
by (auto simp: Pi_iff) |
|
1193 |
show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
1194 |
proof |
|
1195 |
fix A assume A: "A \<in> E i" |
|
1196 |
then have "(\<lambda>x. x i) -` A \<inter> space ?P = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))" |
|
1197 |
using E_closed `i \<in> I` by (auto simp: space_P Pi_iff subset_eq split: split_if_asm) |
|
1198 |
also have "\<dots> = (\<Pi>\<^isub>E j\<in>I. \<Union>n. if i = j then A else S j n)" |
|
1199 |
by (intro PiE_cong) (simp add: S_union) |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1200 |
also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j))" |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1201 |
using T |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1202 |
apply (auto simp: Pi_iff bchoice_iff) |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1203 |
apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI) |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1204 |
apply (auto simp: bij_betw_def) |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1205 |
done |
47694 | 1206 |
also have "\<dots> \<in> sets ?P" |
1207 |
proof (safe intro!: countable_UN) |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1208 |
fix xs show "(\<Pi>\<^isub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P" |
47694 | 1209 |
using A S_in_E |
1210 |
by (simp add: P_closed) |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1211 |
(auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"]) |
47694 | 1212 |
qed |
1213 |
finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
1214 |
using P_closed by simp |
|
1215 |
qed |
|
1216 |
qed |
|
1217 |
from measurable_sets[OF this, of A] A `i \<in> I` E_closed |
|
1218 |
have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
1219 |
by (simp add: E_generates) |
|
1220 |
also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^isub>E i\<in>I. space (M i). f i \<in> A}" |
|
1221 |
using P_closed by (auto simp: space_PiM) |
|
1222 |
finally show "\<dots> \<in> sets ?P" . |
|
1223 |
qed |
|
1224 |
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P" |
|
1225 |
by (simp add: P_closed) |
|
1226 |
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)" |
|
1227 |
using `finite I` |
|
50003 | 1228 |
by (auto intro!: sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def) |
47694 | 1229 |
qed |
1230 |
||
1231 |
end |