author | paulson <lp15@cam.ac.uk> |
Wed, 17 Jul 2019 14:02:42 +0100 | |
changeset 70365 | 4df0628e8545 |
parent 70136 | f03a01a18c6e |
child 74362 | 0135a0c77b64 |
permissions | -rw-r--r-- |
63627 | 1 |
(* Title: HOL/Analysis/Finite_Product_Measure.thy |
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Author: Johannes Hölzl, TU München |
3 |
*) |
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69722
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minor tagging updates in 13 theories
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69687
diff
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section \<open>Finite Product Measure\<close> |
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|
42146
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split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
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theory Finite_Product_Measure |
69918
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paulson <lp15@cam.ac.uk>
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imports Binary_Product_Measure Function_Topology |
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begin |
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||
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lemma PiE_choice: "(\<exists>f\<in>Pi\<^sub>E I F. \<forall>i\<in>I. P i (f i)) \<longleftrightarrow> (\<forall>i\<in>I. \<exists>x\<in>F i. P i x)" |
57447
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import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
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by (auto simp: Bex_def PiE_iff Ball_def dest!: choice_iff'[THEN iffD1]) |
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import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
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parents:
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13 |
(force intro: exI[of _ "restrict f I" for f]) |
87429bdecad5
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parents:
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diff
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|
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parents:
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|
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lemma case_prod_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)" |
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by auto |
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||
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subsection\<^marker>\<open>tag unimportant\<close> \<open>More about Function restricted by \<^const>\<open>extensional\<close>\<close> |
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|
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definition |
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"merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)" |
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|
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lemma merge_apply[simp]: |
|
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"I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i" |
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"I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i" |
|
26 |
"J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i" |
|
27 |
"J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I J (x, y) i = y i" |
|
28 |
"i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I J (x, y) i = undefined" |
|
40859 | 29 |
unfolding merge_def by auto |
30 |
||
31 |
lemma merge_commute: |
|
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"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) = merge J I (y, x)" |
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by (force simp: merge_def) |
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|
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lemma Pi_cancel_merge_range[simp]: |
|
49780 | 36 |
"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A" |
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"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A" |
|
38 |
"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I J (A, B)) \<longleftrightarrow> x \<in> Pi I A" |
|
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"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A" |
|
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by (auto simp: Pi_def) |
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||
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lemma Pi_cancel_merge[simp]: |
|
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"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
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"J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B" |
|
45 |
"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" |
|
46 |
"J \<inter> I = {} \<Longrightarrow> merge I J (x, y) \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B" |
|
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by (auto simp: Pi_def) |
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||
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lemma extensional_merge[simp]: "merge I J (x, y) \<in> extensional (I \<union> J)" |
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by (auto simp: extensional_def) |
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||
52 |
lemma restrict_merge[simp]: |
|
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"I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I" |
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"I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J" |
|
55 |
"J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I" |
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"J \<inter> I = {} \<Longrightarrow> restrict (merge I J (x, y)) J = restrict y J" |
|
47694 | 57 |
by (auto simp: restrict_def) |
40859 | 58 |
|
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moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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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
immler@in.tum.de
parents:
50041
diff
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|
60 |
unfolding merge_def by auto |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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diff
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61 |
|
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parents:
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lemma PiE_cancel_merge[simp]: |
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"I \<inter> J = {} \<Longrightarrow> |
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merge I J (x, y) \<in> Pi\<^sub>E (I \<union> J) B \<longleftrightarrow> x \<in> Pi I B \<and> y \<in> Pi J B" |
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hoelzl
parents:
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diff
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65 |
by (auto simp: PiE_def restrict_Pi_cancel) |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
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diff
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66 |
|
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
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parents:
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lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I {i} (x,y) = restrict (x(i := y i)) (insert i I)" |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
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|
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unfolding merge_def by (auto simp: fun_eq_iff) |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
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69 |
|
50042
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
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lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I J (x, y) \<in> extensional K" |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
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|
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unfolding merge_def extensional_def by auto |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
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72 |
|
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
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lemma merge_restrict[simp]: |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
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parents:
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diff
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"merge I J (restrict x I, y) = merge I J (x, y)" |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
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diff
changeset
|
75 |
"merge I J (x, restrict y J) = merge I J (x, y)" |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
76 |
unfolding merge_def by auto |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
77 |
|
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
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|
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lemma merge_x_x_eq_restrict[simp]: |
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merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
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parents:
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|
79 |
"merge I J (x, x) = restrict x (I \<union> J)" |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
80 |
unfolding merge_def by auto |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
81 |
|
50042
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
82 |
lemma injective_vimage_restrict: |
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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
|
83 |
assumes J: "J \<subseteq> I" |
53015
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standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
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diff
changeset
|
84 |
and sets: "A \<subseteq> (\<Pi>\<^sub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^sub>E i\<in>J. S i)" and ne: "(\<Pi>\<^sub>E i\<in>I. S i) \<noteq> {}" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
85 |
and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^sub>E i\<in>I. S i)" |
50042
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
86 |
shows "A = B" |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
87 |
proof (intro set_eqI) |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
88 |
fix x |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
89 |
from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
90 |
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
|
91 |
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
|
92 |
proof cases |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
93 |
assume x: "x \<in> (\<Pi>\<^sub>E i\<in>J. S i)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
94 |
have "x \<in> A \<longleftrightarrow> merge J (I - J) (x,y) \<in> (\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i)" |
61808 | 95 |
using y x \<open>J \<subseteq> I\<close> PiE_cancel_merge[of "J" "I - J" x y S] |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
96 |
by (auto simp del: PiE_cancel_merge simp add: Un_absorb1) |
50042
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
97 |
then show "x \<in> A \<longleftrightarrow> x \<in> B" |
61808 | 98 |
using y x \<open>J \<subseteq> I\<close> PiE_cancel_merge[of "J" "I - J" x y S] |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
99 |
by (auto simp del: PiE_cancel_merge simp add: Un_absorb1 eq) |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
100 |
qed (insert sets, auto) |
50042
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moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
101 |
qed |
6fe18351e9dd
moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
50041
diff
changeset
|
102 |
|
41095 | 103 |
lemma restrict_vimage: |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
104 |
"I \<inter> J = {} \<Longrightarrow> |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
105 |
(\<lambda>x. (restrict x I, restrict x J)) -` (Pi\<^sub>E I E \<times> Pi\<^sub>E J F) = Pi (I \<union> J) (merge I J (E, F))" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
106 |
by (auto simp: restrict_Pi_cancel PiE_def) |
41095 | 107 |
|
108 |
lemma merge_vimage: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
109 |
"I \<inter> J = {} \<Longrightarrow> merge I J -` Pi\<^sub>E (I \<union> J) E = Pi I E \<times> Pi J E" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
110 |
by (auto simp: restrict_Pi_cancel PiE_def) |
50104 | 111 |
|
69683 | 112 |
subsection \<open>Finite product spaces\<close> |
40859 | 113 |
|
70136 | 114 |
definition\<^marker>\<open>tag important\<close> prod_emb where |
64910 | 115 |
"prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (\<Pi>\<^sub>E i\<in>I. space (M i))" |
47694 | 116 |
|
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redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
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|
117 |
lemma prod_emb_iff: |
47694 | 118 |
"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))" |
70136 | 119 |
unfolding prod_emb_def PiE_def by auto |
40859 | 120 |
|
69681
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redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
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diff
changeset
|
121 |
lemma |
47694 | 122 |
shows prod_emb_empty[simp]: "prod_emb M L K {} = {}" |
123 |
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" |
|
124 |
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" |
|
125 |
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))" |
|
126 |
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))" |
|
127 |
and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B" |
|
69681
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redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
128 |
by (auto simp: prod_emb_def) |
40859 | 129 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
130 |
lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
131 |
prod_emb I M J (\<Pi>\<^sub>E i\<in>J. E i) = (\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i))" |
62390 | 132 |
by (force simp: prod_emb_def PiE_iff if_split_mem2) |
47694 | 133 |
|
69681
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redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
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|
134 |
lemma prod_emb_PiE_same_index[simp]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
135 |
"(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^sub>E I E) = Pi\<^sub>E I E" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
136 |
by (auto simp: prod_emb_def PiE_iff) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
137 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
138 |
lemma prod_emb_trans[simp]: |
50038 | 139 |
"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" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
140 |
by (auto simp add: Int_absorb1 prod_emb_def PiE_def) |
50038 | 141 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
142 |
lemma prod_emb_Pi: |
50038 | 143 |
assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
144 |
shows "prod_emb K M J (Pi\<^sub>E J X) = (\<Pi>\<^sub>E i\<in>K. if i \<in> J then X i else space (M i))" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
145 |
using assms sets.space_closed |
62390 | 146 |
by (auto simp: prod_emb_def PiE_iff split: if_split_asm) blast+ |
50038 | 147 |
|
69681
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redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
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|
148 |
lemma prod_emb_id: |
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diff
changeset
|
149 |
"B \<subseteq> (\<Pi>\<^sub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
150 |
by (auto simp: prod_emb_def subset_eq extensional_restrict) |
50038 | 151 |
|
69681
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redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
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diff
changeset
|
152 |
lemma prod_emb_mono: |
50041
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removed redundant/unnecessary assumptions from projective_family
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parents:
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diff
changeset
|
153 |
"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
|
154 |
by (auto simp: prod_emb_def) |
afe886a04198
removed redundant/unnecessary assumptions from projective_family
immler@in.tum.de
parents:
50038
diff
changeset
|
155 |
|
70136 | 156 |
definition\<^marker>\<open>tag important\<close> PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where |
53015
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wenzelm
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changeset
|
157 |
"PiM I M = extend_measure (\<Pi>\<^sub>E i\<in>I. space (M i)) |
47694 | 158 |
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} |
53015
a1119cf551e8
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parents:
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changeset
|
159 |
(\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j)) |
47694 | 160 |
(\<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)))" |
161 |
||
70136 | 162 |
definition\<^marker>\<open>tag important\<close> prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where |
53015
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parents:
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diff
changeset
|
163 |
"prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j)) ` |
47694 | 164 |
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" |
165 |
||
166 |
abbreviation |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
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changeset
|
167 |
"Pi\<^sub>M I M \<equiv> PiM I M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
168 |
|
40859 | 169 |
syntax |
53015
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changeset
|
170 |
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^sub>M _\<in>_./ _)" 10) |
40859 | 171 |
translations |
61988 | 172 |
"\<Pi>\<^sub>M x\<in>I. M" == "CONST PiM I (%x. M)" |
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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diff
changeset
|
173 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
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diff
changeset
|
174 |
lemma extend_measure_cong: |
59425 | 175 |
assumes "\<Omega> = \<Omega>'" "I = I'" "G = G'" "\<And>i. i \<in> I' \<Longrightarrow> \<mu> i = \<mu>' i" |
176 |
shows "extend_measure \<Omega> I G \<mu> = extend_measure \<Omega>' I' G' \<mu>'" |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
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diff
changeset
|
177 |
unfolding extend_measure_def by (auto simp add: assms) |
59425 | 178 |
|
69681
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redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
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parents:
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changeset
|
179 |
lemma Pi_cong_sets: |
59425 | 180 |
"\<lbrakk>I = J; \<And>x. x \<in> I \<Longrightarrow> M x = N x\<rbrakk> \<Longrightarrow> Pi I M = Pi J N" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
181 |
unfolding Pi_def by auto |
59425 | 182 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
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changeset
|
183 |
lemma PiM_cong: |
59425 | 184 |
assumes "I = J" "\<And>x. x \<in> I \<Longrightarrow> M x = N x" |
185 |
shows "PiM I M = PiM J N" |
|
60580 | 186 |
unfolding PiM_def |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
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diff
changeset
|
187 |
proof (rule extend_measure_cong, goal_cases) |
60580 | 188 |
case 1 |
189 |
show ?case using assms |
|
59425 | 190 |
by (subst assms(1), intro PiE_cong[of J "\<lambda>i. space (M i)" "\<lambda>i. space (N i)"]) simp_all |
191 |
next |
|
60580 | 192 |
case 2 |
59425 | 193 |
have "\<And>K. K \<subseteq> J \<Longrightarrow> (\<Pi> j\<in>K. sets (M j)) = (\<Pi> j\<in>K. sets (N j))" |
194 |
using assms by (intro Pi_cong_sets) auto |
|
195 |
thus ?case by (auto simp: assms) |
|
196 |
next |
|
60580 | 197 |
case 3 |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
198 |
show ?case using assms |
59425 | 199 |
by (intro ext) (auto simp: prod_emb_def dest: PiE_mem) |
200 |
next |
|
60580 | 201 |
case (4 x) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
202 |
thus ?case using assms |
64272 | 203 |
by (auto intro!: prod.cong split: if_split_asm) |
59425 | 204 |
qed |
205 |
||
206 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
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diff
changeset
|
207 |
lemma prod_algebra_sets_into_space: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
208 |
"prod_algebra I M \<subseteq> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))" |
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
209 |
by (auto simp: prod_emb_def prod_algebra_def) |
40859 | 210 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
211 |
lemma prod_algebra_eq_finite: |
47694 | 212 |
assumes I: "finite I" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
213 |
shows "prod_algebra I M = {(\<Pi>\<^sub>E i\<in>I. X i) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R") |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
214 |
proof (intro iffI set_eqI) |
47694 | 215 |
fix A assume "A \<in> ?L" |
216 |
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)" |
|
64910 | 217 |
and A: "A = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)" |
47694 | 218 |
by (auto simp: prod_algebra_def) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
219 |
let ?A = "\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i)" |
47694 | 220 |
have A: "A = ?A" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
221 |
unfolding A using J by (intro prod_emb_PiE sets.sets_into_space) auto |
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
222 |
show "A \<in> ?R" unfolding A using J sets.top |
47694 | 223 |
by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp |
224 |
next |
|
225 |
fix A assume "A \<in> ?R" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
226 |
then obtain X where A: "A = (\<Pi>\<^sub>E i\<in>I. X i)" and X: "X \<in> (\<Pi> j\<in>I. sets (M j))" by auto |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
227 |
then have A: "A = prod_emb I M I (\<Pi>\<^sub>E i\<in>I. X i)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
228 |
by (simp add: prod_emb_PiE_same_index[OF sets.sets_into_space] Pi_iff) |
47694 | 229 |
from X I show "A \<in> ?L" unfolding A |
230 |
by (auto simp: prod_algebra_def) |
|
231 |
qed |
|
41095 | 232 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
233 |
lemma prod_algebraI: |
47694 | 234 |
"finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)) |
64910 | 235 |
\<Longrightarrow> prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) \<in> prod_algebra I M" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
236 |
by (auto simp: prod_algebra_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
|
237 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
238 |
lemma prod_algebraI_finite: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
239 |
"finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^sub>E I E) \<in> prod_algebra I M" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
240 |
using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets.sets_into_space] by simp |
50038 | 241 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
242 |
lemma Int_stable_PiE: "Int_stable {Pi\<^sub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}" |
50038 | 243 |
proof (safe intro!: Int_stableI) |
244 |
fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
245 |
then show "\<exists>G. Pi\<^sub>E J E \<inter> Pi\<^sub>E J F = Pi\<^sub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
246 |
by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int) |
50038 | 247 |
qed |
248 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
249 |
lemma prod_algebraE: |
47694 | 250 |
assumes A: "A \<in> prod_algebra I M" |
64910 | 251 |
obtains J E where "A = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
252 |
"finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" |
47694 | 253 |
using A by (auto simp: prod_algebra_def) |
42988 | 254 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
255 |
lemma prod_algebraE_all: |
47694 | 256 |
assumes A: "A \<in> prod_algebra I M" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
257 |
obtains E where "A = Pi\<^sub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))" |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
258 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
259 |
from A obtain E J where A: "A = prod_emb I M J (Pi\<^sub>E J E)" |
47694 | 260 |
and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))" |
261 |
by (auto simp: prod_algebra_def) |
|
262 |
from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
263 |
using sets.sets_into_space by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
264 |
then have "A = (\<Pi>\<^sub>E i\<in>I. if i\<in>J then E i else space (M i))" |
47694 | 265 |
using A J by (auto simp: prod_emb_PiE) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
266 |
moreover have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
267 |
using sets.top E by auto |
47694 | 268 |
ultimately show ?thesis using that by auto |
269 |
qed |
|
40859 | 270 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
271 |
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)" |
47694 | 272 |
proof (unfold Int_stable_def, safe) |
273 |
fix A assume "A \<in> prod_algebra I M" |
|
274 |
from prod_algebraE[OF this] guess J E . note A = this |
|
275 |
fix B assume "B \<in> prod_algebra I M" |
|
276 |
from prod_algebraE[OF this] guess K F . note B = this |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
277 |
have "A \<inter> B = prod_emb I M (J \<union> K) (\<Pi>\<^sub>E i\<in>J \<union> K. (if i \<in> J then E i else space (M i)) \<inter> |
47694 | 278 |
(if i \<in> K then F i else space (M i)))" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
279 |
unfolding A B using A(2,3,4) A(5)[THEN sets.sets_into_space] B(2,3,4) |
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
280 |
B(5)[THEN sets.sets_into_space] |
47694 | 281 |
apply (subst (1 2 3) prod_emb_PiE) |
282 |
apply (simp_all add: subset_eq PiE_Int) |
|
283 |
apply blast |
|
284 |
apply (intro PiE_cong) |
|
285 |
apply auto |
|
286 |
done |
|
287 |
also have "\<dots> \<in> prod_algebra I M" |
|
288 |
using A B by (auto intro!: prod_algebraI) |
|
289 |
finally show "A \<inter> B \<in> prod_algebra I M" . |
|
290 |
qed |
|
291 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
292 |
proposition prod_algebra_mono: |
47694 | 293 |
assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)" |
294 |
assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)" |
|
295 |
shows "prod_algebra I E \<subseteq> prod_algebra I F" |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
296 |
proof |
47694 | 297 |
fix A assume "A \<in> prod_algebra I E" |
298 |
then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
299 |
and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)" |
47694 | 300 |
and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)" |
301 |
by (auto simp: prod_algebra_def) |
|
302 |
moreover |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
303 |
from space have "(\<Pi>\<^sub>E i\<in>I. space (E i)) = (\<Pi>\<^sub>E i\<in>I. space (F i))" |
47694 | 304 |
by (rule PiE_cong) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
305 |
with A have "A = prod_emb I F J (\<Pi>\<^sub>E i\<in>J. G i)" |
47694 | 306 |
by (simp add: prod_emb_def) |
307 |
moreover |
|
308 |
from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)" |
|
309 |
by auto |
|
310 |
ultimately show "A \<in> prod_algebra I F" |
|
311 |
apply (simp add: prod_algebra_def image_iff) |
|
312 |
apply (intro exI[of _ J] exI[of _ G] conjI) |
|
313 |
apply auto |
|
314 |
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
|
315 |
qed |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
316 |
proposition prod_algebra_cong: |
50104 | 317 |
assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))" |
318 |
shows "prod_algebra I M = prod_algebra J N" |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
319 |
proof - |
50104 | 320 |
have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)" |
321 |
using sets_eq_imp_space_eq[OF sets] by auto |
|
61808 | 322 |
with sets show ?thesis unfolding \<open>I = J\<close> |
50104 | 323 |
by (intro antisym prod_algebra_mono) auto |
324 |
qed |
|
325 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
326 |
lemma space_in_prod_algebra: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
327 |
"(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M" |
50104 | 328 |
proof cases |
329 |
assume "I = {}" then show ?thesis |
|
330 |
by (auto simp add: prod_algebra_def image_iff prod_emb_def) |
|
331 |
next |
|
332 |
assume "I \<noteq> {}" |
|
333 |
then obtain i where "i \<in> I" by auto |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
334 |
then have "(\<Pi>\<^sub>E i\<in>I. space (M i)) = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. space (M i))" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
335 |
by (auto simp: prod_emb_def) |
50104 | 336 |
also have "\<dots> \<in> prod_algebra I M" |
61808 | 337 |
using \<open>i \<in> I\<close> by (intro prod_algebraI) auto |
50104 | 338 |
finally show ?thesis . |
339 |
qed |
|
340 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
341 |
lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))" |
47694 | 342 |
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp |
343 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
344 |
lemma prod_emb_subset_PiM[simp]: "prod_emb I M K X \<subseteq> space (PiM I M)" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
345 |
by (auto simp: prod_emb_def space_PiM) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
346 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
347 |
lemma space_PiM_empty_iff[simp]: "space (PiM I M) = {} \<longleftrightarrow> (\<exists>i\<in>I. space (M i) = {})" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
348 |
by (auto simp: space_PiM PiE_eq_empty_iff) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
349 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
350 |
lemma undefined_in_PiM_empty[simp]: "(\<lambda>x. undefined) \<in> space (PiM {} M)" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
351 |
by (auto simp: space_PiM) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
352 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
353 |
lemma sets_PiM: "sets (\<Pi>\<^sub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)" |
47694 | 354 |
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
|
355 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
356 |
proposition sets_PiM_single: "sets (PiM I M) = |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
357 |
sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}" |
47694 | 358 |
(is "_ = sigma_sets ?\<Omega> ?R") |
359 |
unfolding sets_PiM |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
360 |
proof (rule sigma_sets_eqI) |
47694 | 361 |
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto |
362 |
fix A assume "A \<in> prod_algebra I M" |
|
363 |
from prod_algebraE[OF this] guess J X . note X = this |
|
364 |
show "A \<in> sigma_sets ?\<Omega> ?R" |
|
365 |
proof cases |
|
366 |
assume "I = {}" |
|
367 |
with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def) |
|
61808 | 368 |
with \<open>I = {}\<close> show ?thesis by (auto intro!: sigma_sets_top) |
47694 | 369 |
next |
370 |
assume "I \<noteq> {}" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
371 |
with X have "A = (\<Inter>j\<in>J. {f\<in>(\<Pi>\<^sub>E i\<in>I. space (M i)). f j \<in> X j})" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
372 |
by (auto simp: prod_emb_def) |
47694 | 373 |
also have "\<dots> \<in> sigma_sets ?\<Omega> ?R" |
61808 | 374 |
using X \<open>I \<noteq> {}\<close> by (intro R.finite_INT sigma_sets.Basic) auto |
47694 | 375 |
finally show "A \<in> sigma_sets ?\<Omega> ?R" . |
376 |
qed |
|
377 |
next |
|
378 |
fix A assume "A \<in> ?R" |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
379 |
then obtain i B where A: "A = {f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> B}" "i \<in> I" "B \<in> sets (M i)" |
47694 | 380 |
by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
381 |
then have "A = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. B)" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
382 |
by (auto simp: prod_emb_def) |
47694 | 383 |
also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)" |
384 |
using A by (intro sigma_sets.Basic prod_algebraI) auto |
|
385 |
finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" . |
|
386 |
qed |
|
387 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
388 |
lemma sets_PiM_eq_proj: |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69064
diff
changeset
|
389 |
"I \<noteq> {} \<Longrightarrow> sets (PiM I M) = sets (SUP i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>x. x i) (M i))" |
63333
158ab2239496
Probability: show that measures form a complete lattice
hoelzl
parents:
63040
diff
changeset
|
390 |
apply (simp add: sets_PiM_single) |
158ab2239496
Probability: show that measures form a complete lattice
hoelzl
parents:
63040
diff
changeset
|
391 |
apply (subst sets_Sup_eq[where X="\<Pi>\<^sub>E i\<in>I. space (M i)"]) |
158ab2239496
Probability: show that measures form a complete lattice
hoelzl
parents:
63040
diff
changeset
|
392 |
apply auto [] |
158ab2239496
Probability: show that measures form a complete lattice
hoelzl
parents:
63040
diff
changeset
|
393 |
apply auto [] |
158ab2239496
Probability: show that measures form a complete lattice
hoelzl
parents:
63040
diff
changeset
|
394 |
apply simp |
69661 | 395 |
apply (subst arg_cong [of _ _ Sup, OF image_cong, OF refl]) |
58606 | 396 |
apply (rule sets_vimage_algebra2) |
397 |
apply (auto intro!: arg_cong2[where f=sigma_sets]) |
|
398 |
done |
|
399 |
||
69739 | 400 |
lemma |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
401 |
shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
402 |
and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
403 |
by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
404 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
405 |
proposition sets_PiM_sigma: |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
406 |
assumes \<Omega>_cover: "\<And>i. i \<in> I \<Longrightarrow> \<exists>S\<subseteq>E i. countable S \<and> \<Omega> i = \<Union>S" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
407 |
assumes E: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (\<Omega> i)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
408 |
assumes J: "\<And>j. j \<in> J \<Longrightarrow> finite j" "\<Union>J = I" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
409 |
defines "P \<equiv> {{f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A i} | A j. j \<in> J \<and> A \<in> Pi j E}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
410 |
shows "sets (\<Pi>\<^sub>M i\<in>I. sigma (\<Omega> i) (E i)) = sets (sigma (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P)" |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
411 |
proof cases |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
412 |
assume "I = {}" |
61808 | 413 |
with \<open>\<Union>J = I\<close> have "P = {{\<lambda>_. undefined}} \<or> P = {}" |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
414 |
by (auto simp: P_def) |
61808 | 415 |
with \<open>I = {}\<close> show ?thesis |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
416 |
by (auto simp add: sets_PiM_empty sigma_sets_empty_eq) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
417 |
next |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
418 |
let ?F = "\<lambda>i. {(\<lambda>x. x i) -` A \<inter> Pi\<^sub>E I \<Omega> |A. A \<in> E i}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
419 |
assume "I \<noteq> {}" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
420 |
then have "sets (Pi\<^sub>M I (\<lambda>i. sigma (\<Omega> i) (E i))) = |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69064
diff
changeset
|
421 |
sets (SUP i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. \<Omega> i) (\<lambda>x. x i) (sigma (\<Omega> i) (E i)))" |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
422 |
by (subst sets_PiM_eq_proj) (auto simp: space_measure_of_conv) |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69064
diff
changeset
|
423 |
also have "\<dots> = sets (SUP i\<in>I. sigma (Pi\<^sub>E I \<Omega>) (?F i))" |
63333
158ab2239496
Probability: show that measures form a complete lattice
hoelzl
parents:
63040
diff
changeset
|
424 |
using E by (intro sets_SUP_cong arg_cong[where f=sets] vimage_algebra_sigma) auto |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
425 |
also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i))" |
61808 | 426 |
using \<open>I \<noteq> {}\<close> by (intro arg_cong[where f=sets] SUP_sigma_sigma) auto |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
427 |
also have "\<dots> = sets (sigma (Pi\<^sub>E I \<Omega>) P)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
428 |
proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
429 |
show "(\<Union>i\<in>I. ?F i) \<subseteq> Pow (Pi\<^sub>E I \<Omega>)" "P \<subseteq> Pow (Pi\<^sub>E I \<Omega>)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
430 |
by (auto simp: P_def) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
431 |
next |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
432 |
interpret P: sigma_algebra "\<Pi>\<^sub>E i\<in>I. \<Omega> i" "sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
433 |
by (auto intro!: sigma_algebra_sigma_sets simp: P_def) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
434 |
|
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
435 |
fix Z assume "Z \<in> (\<Union>i\<in>I. ?F i)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
436 |
then obtain i A where i: "i \<in> I" "A \<in> E i" and Z_def: "Z = (\<lambda>x. x i) -` A \<inter> Pi\<^sub>E I \<Omega>" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
437 |
by auto |
61808 | 438 |
from \<open>i \<in> I\<close> J obtain j where j: "i \<in> j" "j \<in> J" "j \<subseteq> I" "finite j" |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
439 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
440 |
obtain S where S: "\<And>i. i \<in> j \<Longrightarrow> S i \<subseteq> E i" "\<And>i. i \<in> j \<Longrightarrow> countable (S i)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
441 |
"\<And>i. i \<in> j \<Longrightarrow> \<Omega> i = \<Union>(S i)" |
61808 | 442 |
by (metis subset_eq \<Omega>_cover \<open>j \<subseteq> I\<close>) |
63040 | 443 |
define A' where "A' n = n(i := A)" for n |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
444 |
then have A'_i: "\<And>n. A' n i = A" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
445 |
by simp |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
446 |
{ fix n assume "n \<in> Pi\<^sub>E (j - {i}) S" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
447 |
then have "A' n \<in> Pi j E" |
61808 | 448 |
unfolding PiE_def Pi_def using S(1) by (auto simp: A'_def \<open>A \<in> E i\<close> ) |
449 |
with \<open>j \<in> J\<close> have "{f \<in> Pi\<^sub>E I \<Omega>. \<forall>i\<in>j. f i \<in> A' n i} \<in> P" |
|
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
450 |
by (auto simp: P_def) } |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
451 |
note A'_in_P = this |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
452 |
|
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
453 |
{ fix x assume "x i \<in> A" "x \<in> Pi\<^sub>E I \<Omega>" |
61808 | 454 |
with S(3) \<open>j \<subseteq> I\<close> have "\<forall>i\<in>j. \<exists>s\<in>S i. x i \<in> s" |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
455 |
by (auto simp: PiE_def Pi_def) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
456 |
then obtain s where s: "\<And>i. i \<in> j \<Longrightarrow> s i \<in> S i" "\<And>i. i \<in> j \<Longrightarrow> x i \<in> s i" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
457 |
by metis |
64910 | 458 |
with \<open>x i \<in> A\<close> have "\<exists>n\<in>Pi\<^sub>E (j-{i}) S. \<forall>i\<in>j. x i \<in> A' n i" |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
459 |
by (intro bexI[of _ "restrict (s(i := A)) (j-{i})"]) (auto simp: A'_def split: if_splits) } |
64910 | 460 |
then have "Z = (\<Union>n\<in>Pi\<^sub>E (j-{i}) S. {f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A' n i})" |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
461 |
unfolding Z_def |
61808 | 462 |
by (auto simp add: set_eq_iff ball_conj_distrib \<open>i\<in>j\<close> A'_i dest: bspec[OF _ \<open>i\<in>j\<close>] |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
463 |
cong: conj_cong) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
464 |
also have "\<dots> \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P" |
61808 | 465 |
using \<open>finite j\<close> S(2) |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
466 |
by (intro P.countable_UN' countable_PiE) (simp_all add: image_subset_iff A'_in_P) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
467 |
finally show "Z \<in> sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P" . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
468 |
next |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
469 |
interpret F: sigma_algebra "\<Pi>\<^sub>E i\<in>I. \<Omega> i" "sigma_sets (\<Pi>\<^sub>E i\<in>I. \<Omega> i) (\<Union>i\<in>I. ?F i)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
470 |
by (auto intro!: sigma_algebra_sigma_sets) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
471 |
|
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
472 |
fix b assume "b \<in> P" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
473 |
then obtain A j where b: "b = {f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A i}" "j \<in> J" "A \<in> Pi j E" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
474 |
by (auto simp: P_def) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
475 |
show "b \<in> sigma_sets (Pi\<^sub>E I \<Omega>) (\<Union>i\<in>I. ?F i)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
476 |
proof cases |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
477 |
assume "j = {}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
478 |
with b have "b = (\<Pi>\<^sub>E i\<in>I. \<Omega> i)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
479 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
480 |
then show ?thesis |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
481 |
by blast |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
482 |
next |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
483 |
assume "j \<noteq> {}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
484 |
with J b(2,3) have eq: "b = (\<Inter>i\<in>j. ((\<lambda>x. x i) -` A i \<inter> Pi\<^sub>E I \<Omega>))" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
485 |
unfolding b(1) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
486 |
by (auto simp: PiE_def Pi_def) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
487 |
show ?thesis |
61808 | 488 |
unfolding eq using \<open>A \<in> Pi j E\<close> \<open>j \<in> J\<close> J(2) |
489 |
by (intro F.finite_INT J \<open>j \<in> J\<close> \<open>j \<noteq> {}\<close> sigma_sets.Basic) blast |
|
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
490 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
491 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
492 |
finally show "?thesis" . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
493 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59048
diff
changeset
|
494 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
495 |
lemma sets_PiM_in_sets: |
58606 | 496 |
assumes space: "space N = (\<Pi>\<^sub>E i\<in>I. space (M i))" |
497 |
assumes sets: "\<And>i A. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {x\<in>space N. x i \<in> A} \<in> sets N" |
|
498 |
shows "sets (\<Pi>\<^sub>M i \<in> I. M i) \<subseteq> sets N" |
|
499 |
unfolding sets_PiM_single space[symmetric] |
|
500 |
by (intro sets.sigma_sets_subset subsetI) (auto intro: sets) |
|
501 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
502 |
lemma sets_PiM_cong[measurable_cong]: |
59048 | 503 |
assumes "I = J" "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (PiM I M) = sets (PiM J N)" |
58606 | 504 |
using assms sets_eq_imp_space_eq[OF assms(2)] by (simp add: sets_PiM_single cong: PiE_cong conj_cong) |
505 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
506 |
lemma sets_PiM_I: |
47694 | 507 |
assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)" |
64910 | 508 |
shows "prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) \<in> sets (\<Pi>\<^sub>M i\<in>I. M i)" |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
509 |
proof cases |
47694 | 510 |
assume "J = {}" |
64910 | 511 |
then have "prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) = (\<Pi>\<^sub>E j\<in>I. space (M j))" |
47694 | 512 |
by (auto simp: prod_emb_def) |
513 |
then show ?thesis |
|
514 |
by (auto simp add: sets_PiM intro!: sigma_sets_top) |
|
515 |
next |
|
516 |
assume "J \<noteq> {}" with assms show ?thesis |
|
50003 | 517 |
by (force simp add: sets_PiM prod_algebra_def) |
40859 | 518 |
qed |
519 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
520 |
proposition measurable_PiM: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
521 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))" |
47694 | 522 |
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> |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
523 |
f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N" |
47694 | 524 |
shows "f \<in> measurable N (PiM I M)" |
525 |
using sets_PiM prod_algebra_sets_into_space space |
|
526 |
proof (rule measurable_sigma_sets) |
|
527 |
fix A assume "A \<in> prod_algebra I M" |
|
528 |
from prod_algebraE[OF this] guess J X . |
|
529 |
with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto |
|
530 |
qed |
|
531 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
532 |
lemma measurable_PiM_Collect: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
533 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))" |
47694 | 534 |
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> |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
535 |
{\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" |
47694 | 536 |
shows "f \<in> measurable N (PiM I M)" |
537 |
using sets_PiM prod_algebra_sets_into_space space |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
538 |
proof (rule measurable_sigma_sets) |
47694 | 539 |
fix A assume "A \<in> prod_algebra I M" |
540 |
from prod_algebraE[OF this] guess J X . note X = this |
|
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
541 |
then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}" |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
542 |
using space by (auto simp: prod_emb_def del: PiE_I) |
47694 | 543 |
also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets) |
544 |
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
|
545 |
qed |
41095 | 546 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
547 |
lemma measurable_PiM_single: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
548 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
549 |
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" |
47694 | 550 |
shows "f \<in> measurable N (PiM I M)" |
551 |
using sets_PiM_single |
|
552 |
proof (rule measurable_sigma_sets) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
553 |
fix A assume "A \<in> {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
554 |
then obtain B i where "A = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> B}" and B: "i \<in> I" "B \<in> sets (M i)" |
47694 | 555 |
by auto |
556 |
with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto |
|
557 |
also have "\<dots> \<in> sets N" using B by (rule sets) |
|
558 |
finally show "f -` A \<inter> space N \<in> sets N" . |
|
559 |
qed (auto simp: space) |
|
40859 | 560 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
561 |
lemma measurable_PiM_single': |
50099 | 562 |
assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
563 |
and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
564 |
shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)" |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
565 |
proof (rule measurable_PiM_single) |
50099 | 566 |
fix A i assume A: "i \<in> I" "A \<in> sets (M i)" |
567 |
then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N" |
|
568 |
by auto |
|
569 |
then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N" |
|
570 |
using A f by (auto intro!: measurable_sets) |
|
571 |
qed fact |
|
572 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
573 |
lemma sets_PiM_I_finite[measurable]: |
47694 | 574 |
assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))" |
64910 | 575 |
shows "(\<Pi>\<^sub>E j\<in>I. E j) \<in> sets (\<Pi>\<^sub>M i\<in>I. M i)" |
61808 | 576 |
using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] \<open>finite I\<close> sets by auto |
47694 | 577 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
578 |
lemma measurable_component_singleton[measurable (raw)]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
579 |
assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (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
|
580 |
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
|
581 |
fix A assume "A \<in> sets (M i)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
582 |
then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) = prod_emb I M {i} (\<Pi>\<^sub>E j\<in>{i}. A)" |
61808 | 583 |
using sets.sets_into_space \<open>i \<in> I\<close> |
62390 | 584 |
by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: if_split_asm) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
585 |
then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) \<in> sets (Pi\<^sub>M I M)" |
61808 | 586 |
using \<open>A \<in> sets (M i)\<close> \<open>i \<in> I\<close> by (auto intro!: sets_PiM_I) |
587 |
qed (insert \<open>i \<in> I\<close>, auto simp: space_PiM) |
|
47694 | 588 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
589 |
lemma measurable_component_singleton'[measurable_dest]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
590 |
assumes f: "f \<in> measurable N (Pi\<^sub>M I M)" |
59353
f0707dc3d9aa
measurability prover: removed app splitting, replaced by more powerful destruction rules
hoelzl
parents:
59088
diff
changeset
|
591 |
assumes g: "g \<in> measurable L N" |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
592 |
assumes i: "i \<in> I" |
59353
f0707dc3d9aa
measurability prover: removed app splitting, replaced by more powerful destruction rules
hoelzl
parents:
59088
diff
changeset
|
593 |
shows "(\<lambda>x. (f (g x)) i) \<in> measurable L (M i)" |
f0707dc3d9aa
measurability prover: removed app splitting, replaced by more powerful destruction rules
hoelzl
parents:
59088
diff
changeset
|
594 |
using measurable_compose[OF measurable_compose[OF g f] measurable_component_singleton, OF i] . |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
595 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
596 |
lemma measurable_PiM_component_rev: |
50099 | 597 |
"i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N" |
598 |
by simp |
|
599 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
600 |
lemma measurable_case_nat[measurable (raw)]: |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
601 |
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
|
602 |
"\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N" |
55415 | 603 |
shows "(\<lambda>x. case_nat (f x) (g x) i) \<in> measurable M N" |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
604 |
by (cases i) simp_all |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
605 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
606 |
lemma measurable_case_nat'[measurable (raw)]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
607 |
assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)" |
55415 | 608 |
shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)" |
50099 | 609 |
using fg[THEN measurable_space] |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
610 |
by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split) |
50099 | 611 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
612 |
lemma measurable_add_dim[measurable]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
613 |
"(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M)" |
47694 | 614 |
(is "?f \<in> measurable ?P ?I") |
615 |
proof (rule measurable_PiM_single) |
|
616 |
fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)" |
|
617 |
have "{\<omega> \<in> space ?P. (\<lambda>(f, x). fun_upd f i x) \<omega> j \<in> A} = |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
618 |
(if j = i then space (Pi\<^sub>M I M) \<times> A else ((\<lambda>x. x j) \<circ> fst) -` A \<inter> space ?P)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
619 |
using sets.sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM) |
47694 | 620 |
also have "\<dots> \<in> sets ?P" |
621 |
using A j |
|
622 |
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
623 |
finally show "{\<omega> \<in> space ?P. case_prod (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" . |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
624 |
qed (auto simp: space_pair_measure space_PiM PiE_def) |
41661 | 625 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
626 |
proposition measurable_fun_upd: |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
627 |
assumes I: "I = J \<union> {i}" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
628 |
assumes f[measurable]: "f \<in> measurable N (PiM J M)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
629 |
assumes h[measurable]: "h \<in> measurable N (M i)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
630 |
shows "(\<lambda>x. (f x) (i := h x)) \<in> measurable N (PiM I M)" |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
631 |
proof (intro measurable_PiM_single') |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
632 |
fix j assume "j \<in> I" then show "(\<lambda>\<omega>. ((f \<omega>)(i := h \<omega>)) j) \<in> measurable N (M j)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
633 |
unfolding I by (cases "j = i") auto |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
634 |
next |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
635 |
show "(\<lambda>x. (f x)(i := h x)) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
636 |
using I f[THEN measurable_space] h[THEN measurable_space] |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
637 |
by (auto simp: space_PiM PiE_iff extensional_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
638 |
qed |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
639 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
640 |
lemma measurable_component_update: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
641 |
"x \<in> space (Pi\<^sub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^sub>M (insert i I) M)" |
50003 | 642 |
by simp |
643 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
644 |
lemma measurable_merge[measurable]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
645 |
"merge I J \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M)" |
47694 | 646 |
(is "?f \<in> measurable ?P ?U") |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
647 |
proof (rule measurable_PiM_single) |
47694 | 648 |
fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J" |
49780 | 649 |
then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} = |
47694 | 650 |
(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 | 651 |
by (auto simp: merge_def) |
47694 | 652 |
also have "\<dots> \<in> sets ?P" |
653 |
using A |
|
654 |
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) |
|
49780 | 655 |
finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" . |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
656 |
qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def) |
42988 | 657 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
658 |
lemma measurable_restrict[measurable (raw)]: |
47694 | 659 |
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
660 |
shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)" |
47694 | 661 |
proof (rule measurable_PiM_single) |
662 |
fix A i assume A: "i \<in> I" "A \<in> sets (M i)" |
|
663 |
then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N" |
|
664 |
by auto |
|
665 |
then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N" |
|
666 |
using A X by (auto intro!: measurable_sets) |
|
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
667 |
qed (insert X, auto simp add: PiE_def dest: measurable_space) |
47694 | 668 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
669 |
lemma measurable_abs_UNIV: |
57025 | 670 |
"(\<And>n. (\<lambda>\<omega>. f n \<omega>) \<in> measurable M (N n)) \<Longrightarrow> (\<lambda>\<omega> n. f n \<omega>) \<in> measurable M (PiM UNIV N)" |
671 |
by (intro measurable_PiM_single) (auto dest: measurable_space) |
|
672 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
673 |
lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)" |
50038 | 674 |
by (intro measurable_restrict measurable_component_singleton) auto |
675 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
676 |
lemma measurable_restrict_subset': |
59425 | 677 |
assumes "J \<subseteq> L" "\<And>x. x \<in> J \<Longrightarrow> sets (M x) = sets (N x)" |
678 |
shows "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)" |
|
679 |
proof- |
|
680 |
from assms(1) have "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)" |
|
681 |
by (rule measurable_restrict_subset) |
|
682 |
also from assms(2) have "measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M) = measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)" |
|
683 |
by (intro sets_PiM_cong measurable_cong_sets) simp_all |
|
684 |
finally show ?thesis . |
|
685 |
qed |
|
686 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
687 |
lemma measurable_prod_emb[intro, simp]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
688 |
"J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^sub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^sub>M L M)" |
50038 | 689 |
unfolding prod_emb_def space_PiM[symmetric] |
690 |
by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton) |
|
691 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
692 |
lemma merge_in_prod_emb: |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
693 |
assumes "y \<in> space (PiM I M)" "x \<in> X" and X: "X \<in> sets (Pi\<^sub>M J M)" and "J \<subseteq> I" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
694 |
shows "merge J I (x, y) \<in> prod_emb I M J X" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
695 |
using assms sets.sets_into_space[OF X] |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
696 |
by (simp add: merge_def prod_emb_def subset_eq space_PiM PiE_def extensional_restrict Pi_iff |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
697 |
cong: if_cong restrict_cong) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
698 |
(simp add: extensional_def) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
699 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
700 |
lemma prod_emb_eq_emptyD: |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
701 |
assumes J: "J \<subseteq> I" and ne: "space (PiM I M) \<noteq> {}" and X: "X \<in> sets (Pi\<^sub>M J M)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
702 |
and *: "prod_emb I M J X = {}" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
703 |
shows "X = {}" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
704 |
proof safe |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
705 |
fix x assume "x \<in> X" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
706 |
obtain \<omega> where "\<omega> \<in> space (PiM I M)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
707 |
using ne by blast |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
708 |
from merge_in_prod_emb[OF this \<open>x\<in>X\<close> X J] * show "x \<in> {}" by auto |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
709 |
qed |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
710 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
711 |
lemma sets_in_Pi_aux: |
50003 | 712 |
"finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow> |
713 |
{x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)" |
|
714 |
by (simp add: subset_eq Pi_iff) |
|
715 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
716 |
lemma sets_in_Pi[measurable (raw)]: |
50003 | 717 |
"finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow> |
718 |
(\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow> |
|
50387 | 719 |
Measurable.pred N (\<lambda>x. f x \<in> Pi I F)" |
50003 | 720 |
unfolding pred_def |
721 |
by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto |
|
722 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
723 |
lemma sets_in_extensional_aux: |
50003 | 724 |
"{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)" |
725 |
proof - |
|
726 |
have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)" |
|
727 |
by (auto simp add: extensional_def space_PiM) |
|
728 |
then show ?thesis by simp |
|
729 |
qed |
|
730 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
731 |
lemma sets_in_extensional[measurable (raw)]: |
50387 | 732 |
"f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)" |
50003 | 733 |
unfolding pred_def |
734 |
by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto |
|
735 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
736 |
lemma sets_PiM_I_countable: |
61363
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
737 |
assumes I: "countable I" and E: "\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> sets (Pi\<^sub>M I M)" |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
738 |
proof cases |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
739 |
assume "I \<noteq> {}" |
64910 | 740 |
then have "Pi\<^sub>E I E = (\<Inter>i\<in>I. prod_emb I M {i} (Pi\<^sub>E {i} E))" |
61363
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
741 |
using E[THEN sets.sets_into_space] by (auto simp: PiE_iff prod_emb_def fun_eq_iff) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
742 |
also have "\<dots> \<in> sets (PiM I M)" |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
743 |
using I \<open>I \<noteq> {}\<close> by (safe intro!: sets.countable_INT' measurable_prod_emb sets_PiM_I_finite E) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
744 |
finally show ?thesis . |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
745 |
qed (simp add: sets_PiM_empty) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
746 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
747 |
lemma sets_PiM_D_countable: |
61363
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
748 |
assumes A: "A \<in> PiM I M" |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
749 |
shows "\<exists>J\<subseteq>I. \<exists>X\<in>PiM J M. countable J \<and> A = prod_emb I M J X" |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
750 |
using A[unfolded sets_PiM_single] |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
751 |
proof induction |
61363
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
752 |
case (Basic A) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
753 |
then obtain i X where *: "i \<in> I" "X \<in> sets (M i)" and "A = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> X}" |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
754 |
by auto |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
755 |
then have A: "A = prod_emb I M {i} (\<Pi>\<^sub>E _\<in>{i}. X)" |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
756 |
by (auto simp: prod_emb_def) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
757 |
then show ?case |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
758 |
by (intro exI[of _ "{i}"] conjI bexI[of _ "\<Pi>\<^sub>E _\<in>{i}. X"]) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
759 |
(auto intro: countable_finite * sets_PiM_I_finite) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
760 |
next |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
761 |
case Empty then show ?case |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
762 |
by (intro exI[of _ "{}"] conjI bexI[of _ "{}"]) auto |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
763 |
next |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
764 |
case (Compl A) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
765 |
then obtain J X where "J \<subseteq> I" "X \<in> sets (Pi\<^sub>M J M)" "countable J" "A = prod_emb I M J X" |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
766 |
by auto |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
767 |
then show ?case |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
768 |
by (intro exI[of _ J] bexI[of _ "space (PiM J M) - X"] conjI) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
769 |
(auto simp add: space_PiM prod_emb_PiE intro!: sets_PiM_I_countable) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
770 |
next |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
771 |
case (Union K) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
772 |
obtain J X where J: "\<And>i. J i \<subseteq> I" "\<And>i. countable (J i)" and X: "\<And>i. X i \<in> sets (Pi\<^sub>M (J i) M)" |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
773 |
and K: "\<And>i. K i = prod_emb I M (J i) (X i)" |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
774 |
by (metis Union.IH) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
775 |
show ?case |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
776 |
proof (intro exI[of _ "\<Union>i. J i"] bexI[of _ "\<Union>i. prod_emb (\<Union>i. J i) M (J i) (X i)"] conjI) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
777 |
show "(\<Union>i. J i) \<subseteq> I" "countable (\<Union>i. J i)" using J by auto |
69313 | 778 |
with J show "\<Union>(K ` UNIV) = prod_emb I M (\<Union>i. J i) (\<Union>i. prod_emb (\<Union>i. J i) M (J i) (X i))" |
61363
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
779 |
by (simp add: K[abs_def] SUP_upper) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
780 |
qed(auto intro: X) |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
781 |
qed |
76ac925927aa
measurable sets on product spaces are embeddings of countable products
hoelzl
parents:
61362
diff
changeset
|
782 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
783 |
proposition measure_eqI_PiM_finite: |
61362 | 784 |
assumes [simp]: "finite I" "sets P = PiM I M" "sets Q = PiM I M" |
785 |
assumes eq: "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> P (Pi\<^sub>E I A) = Q (Pi\<^sub>E I A)" |
|
786 |
assumes A: "range A \<subseteq> prod_algebra I M" "(\<Union>i. A i) = space (PiM I M)" "\<And>i::nat. P (A i) \<noteq> \<infinity>" |
|
787 |
shows "P = Q" |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
788 |
proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space]) |
61362 | 789 |
show "range A \<subseteq> prod_algebra I M" "(\<Union>i. A i) = (\<Pi>\<^sub>E i\<in>I. space (M i))" "\<And>i. P (A i) \<noteq> \<infinity>" |
790 |
unfolding space_PiM[symmetric] by fact+ |
|
791 |
fix X assume "X \<in> prod_algebra I M" |
|
64910 | 792 |
then obtain J E where X: "X = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)" |
61362 | 793 |
and J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)" |
794 |
by (force elim!: prod_algebraE) |
|
795 |
then show "emeasure P X = emeasure Q X" |
|
796 |
unfolding X by (subst (1 2) prod_emb_Pi) (auto simp: eq) |
|
797 |
qed (simp_all add: sets_PiM) |
|
798 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
799 |
proposition measure_eqI_PiM_infinite: |
61362 | 800 |
assumes [simp]: "sets P = PiM I M" "sets Q = PiM I M" |
801 |
assumes eq: "\<And>A J. finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> |
|
802 |
P (prod_emb I M J (Pi\<^sub>E J A)) = Q (prod_emb I M J (Pi\<^sub>E J A))" |
|
803 |
assumes A: "finite_measure P" |
|
804 |
shows "P = Q" |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
805 |
proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space]) |
61362 | 806 |
interpret finite_measure P by fact |
63040 | 807 |
define i where "i = (SOME i. i \<in> I)" |
61362 | 808 |
have i: "I \<noteq> {} \<Longrightarrow> i \<in> I" |
809 |
unfolding i_def by (rule someI_ex) auto |
|
63040 | 810 |
define A where "A n = |
811 |
(if I = {} then prod_emb I M {} (\<Pi>\<^sub>E i\<in>{}. {}) else prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. space (M i)))" |
|
812 |
for n :: nat |
|
61362 | 813 |
then show "range A \<subseteq> prod_algebra I M" |
814 |
using prod_algebraI[of "{}" I "\<lambda>i. space (M i)" M] by (auto intro!: prod_algebraI i) |
|
815 |
have "\<And>i. A i = space (PiM I M)" |
|
816 |
by (auto simp: prod_emb_def space_PiM PiE_iff A_def i ex_in_conv[symmetric] exI) |
|
817 |
then show "(\<Union>i. A i) = (\<Pi>\<^sub>E i\<in>I. space (M i))" "\<And>i. emeasure P (A i) \<noteq> \<infinity>" |
|
818 |
by (auto simp: space_PiM) |
|
819 |
next |
|
820 |
fix X assume X: "X \<in> prod_algebra I M" |
|
64910 | 821 |
then obtain J E where X: "X = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)" |
61362 | 822 |
and J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)" |
823 |
by (force elim!: prod_algebraE) |
|
824 |
then show "emeasure P X = emeasure Q X" |
|
825 |
by (auto intro!: eq) |
|
826 |
qed (auto simp: sets_PiM) |
|
827 |
||
70136 | 828 |
locale\<^marker>\<open>tag unimportant\<close> product_sigma_finite = |
47694 | 829 |
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
|
830 |
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)" |
40859 | 831 |
|
70136 | 832 |
sublocale\<^marker>\<open>tag unimportant\<close> product_sigma_finite \<subseteq> M?: sigma_finite_measure "M i" for i |
40859 | 833 |
by (rule sigma_finite_measures) |
834 |
||
70136 | 835 |
locale\<^marker>\<open>tag unimportant\<close> finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" + |
47694 | 836 |
fixes I :: "'i set" |
837 |
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
|
838 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
839 |
proposition (in finite_product_sigma_finite) sigma_finite_pairs: |
40859 | 840 |
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set. |
841 |
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and> |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
842 |
(\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k) \<and> |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
843 |
(\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)" |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
844 |
proof - |
47694 | 845 |
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>)" |
846 |
using M.sigma_finite_incseq by metis |
|
40859 | 847 |
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. |
47694 | 848 |
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 | 849 |
by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
850 |
let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k" |
47694 | 851 |
note space_PiM[simp] |
40859 | 852 |
show ?thesis |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
853 |
proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI) |
40859 | 854 |
fix i show "range (F i) \<subseteq> sets (M i)" by fact |
855 |
next |
|
47694 | 856 |
fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact |
40859 | 857 |
next |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
858 |
fix x assume "x \<in> (\<Union>i. ?F i)" with F(1) show "x \<in> space (PiM I M)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
859 |
by (auto simp: PiE_def dest!: sets.sets_into_space) |
40859 | 860 |
next |
47694 | 861 |
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
|
862 |
with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
863 |
show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def) |
40859 | 864 |
next |
865 |
fix i show "?F i \<subseteq> ?F (Suc i)" |
|
61808 | 866 |
using \<open>\<And>i. incseq (F i)\<close>[THEN incseq_SucD] by auto |
40859 | 867 |
qed |
868 |
qed |
|
869 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
870 |
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1" |
49780 | 871 |
proof - |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
872 |
let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ennreal)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
873 |
have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1" |
49780 | 874 |
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) |
875 |
show "positive (PiM {} M) ?\<mu>" |
|
876 |
by (auto simp: positive_def) |
|
877 |
show "countably_additive (PiM {} M) ?\<mu>" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
878 |
by (rule sets.countably_additiveI_finite) |
49780 | 879 |
(auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: ) |
880 |
qed (auto simp: prod_emb_def) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
881 |
also have "(prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = {\<lambda>_. undefined}" |
49780 | 882 |
by (auto simp: prod_emb_def) |
883 |
finally show ?thesis |
|
884 |
by simp |
|
885 |
qed |
|
886 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
887 |
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
888 |
by (rule measure_eqI) (auto simp add: sets_PiM_empty) |
49780 | 889 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
890 |
lemma (in product_sigma_finite) emeasure_PiM: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
891 |
"finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
892 |
proof (induct I arbitrary: A rule: finite_induct) |
40859 | 893 |
case (insert i I) |
61169 | 894 |
interpret finite_product_sigma_finite M I by standard fact |
61808 | 895 |
have "finite (insert i I)" using \<open>finite I\<close> by auto |
61169 | 896 |
interpret I': finite_product_sigma_finite M "insert i I" by standard fact |
41661 | 897 |
let ?h = "(\<lambda>(f, y). f(i := y))" |
47694 | 898 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
899 |
let ?P = "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M) ?h" |
47694 | 900 |
let ?\<mu> = "emeasure ?P" |
901 |
let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}" |
|
902 |
let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))" |
|
903 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
904 |
have "emeasure (Pi\<^sub>M (insert i I) M) (prod_emb (insert i I) M (insert i I) (Pi\<^sub>E (insert i I) A)) = |
49776 | 905 |
(\<Prod>i\<in>insert i I. emeasure (M i) (A i))" |
906 |
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) |
|
907 |
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))" |
|
908 |
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 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
909 |
let ?p = "prod_emb (insert i I) M J (Pi\<^sub>E J E)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
910 |
let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^sub>E j\<in>J-{i}. E j)" |
49776 | 911 |
have "?\<mu> ?p = |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
912 |
emeasure (Pi\<^sub>M I M \<Otimes>\<^sub>M (M i)) (?h -` ?p \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M M i))" |
49776 | 913 |
by (intro emeasure_distr measurable_add_dim sets_PiM_I) fact+ |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
914 |
also have "?h -` ?p \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) = ?p' \<times> (if i \<in> J then E i else space (M i))" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
915 |
using J E[rule_format, THEN sets.sets_into_space] |
62390 | 916 |
by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: if_split_asm) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
917 |
also have "emeasure (Pi\<^sub>M I M \<Otimes>\<^sub>M (M i)) (?p' \<times> (if i \<in> J then E i else space (M i))) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
918 |
emeasure (Pi\<^sub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))" |
49776 | 919 |
using J E by (intro M.emeasure_pair_measure_Times sets_PiM_I) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
920 |
also have "?p' = (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j))" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
921 |
using J E[rule_format, THEN sets.sets_into_space] |
62390 | 922 |
by (auto simp: prod_emb_iff PiE_def Pi_iff split: if_split_asm) blast+ |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
923 |
also have "emeasure (Pi\<^sub>M I M) (\<Pi>\<^sub>E j\<in>I. if j \<in> J-{i} then E j else space (M j)) = |
49776 | 924 |
(\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))" |
64272 | 925 |
using E by (subst insert) (auto intro!: prod.cong) |
49776 | 926 |
also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) * |
927 |
emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)" |
|
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68833
diff
changeset
|
928 |
using insert by (auto simp: mult.commute intro!: arg_cong2[where f="(*)"] prod.cong) |
49776 | 929 |
also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)" |
64272 | 930 |
using insert(1,2) J E by (intro prod.mono_neutral_right) auto |
49776 | 931 |
finally show "?\<mu> ?p = \<dots>" . |
47694 | 932 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
933 |
show "prod_emb (insert i I) M J (Pi\<^sub>E J E) \<in> Pow (\<Pi>\<^sub>E i\<in>insert i I. space (M i))" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
934 |
using J E[rule_format, THEN sets.sets_into_space] by (auto simp: prod_emb_iff PiE_def) |
49776 | 935 |
next |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
936 |
show "positive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" |
49776 | 937 |
using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all |
938 |
next |
|
939 |
show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and> |
|
940 |
insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))" |
|
941 |
using insert by auto |
|
64272 | 942 |
qed (auto intro!: prod.cong) |
49776 | 943 |
with insert show ?case |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
944 |
by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space) |
50003 | 945 |
qed simp |
47694 | 946 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
947 |
lemma (in product_sigma_finite) PiM_eqI: |
61362 | 948 |
assumes I[simp]: "finite I" and P: "sets P = PiM I M" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
949 |
assumes eq: "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> P (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
950 |
shows "P = PiM I M" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
951 |
proof - |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
952 |
interpret finite_product_sigma_finite M I |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
953 |
proof qed fact |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
954 |
from sigma_finite_pairs guess C .. note C = this |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
955 |
show ?thesis |
61362 | 956 |
proof (rule measure_eqI_PiM_finite[OF I refl P, symmetric]) |
957 |
show "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> (Pi\<^sub>M I M) (Pi\<^sub>E I A) = P (Pi\<^sub>E I A)" for A |
|
958 |
by (simp add: eq emeasure_PiM) |
|
63040 | 959 |
define A where "A n = (\<Pi>\<^sub>E i\<in>I. C i n)" for n |
61362 | 960 |
with C show "range A \<subseteq> prod_algebra I M" "\<And>i. emeasure (Pi\<^sub>M I M) (A i) \<noteq> \<infinity>" "(\<Union>i. A i) = space (PiM I M)" |
64272 | 961 |
by (auto intro!: prod_algebraI_finite simp: emeasure_PiM subset_eq ennreal_prod_eq_top) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
962 |
qed |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
963 |
qed |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
964 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
965 |
lemma (in product_sigma_finite) sigma_finite: |
49776 | 966 |
assumes "finite I" |
967 |
shows "sigma_finite_measure (PiM I M)" |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
968 |
proof |
61169 | 969 |
interpret finite_product_sigma_finite M I by standard fact |
49776 | 970 |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
971 |
obtain F where F: "\<And>j. countable (F j)" "\<And>j f. f \<in> F j \<Longrightarrow> f \<in> sets (M j)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
972 |
"\<And>j f. f \<in> F j \<Longrightarrow> emeasure (M j) f \<noteq> \<infinity>" and |
69745 | 973 |
in_space: "\<And>j. space (M j) = \<Union>(F j)" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
974 |
using sigma_finite_countable by (metis subset_eq) |
64910 | 975 |
moreover have "(\<Union>(Pi\<^sub>E I ` Pi\<^sub>E I F)) = space (Pi\<^sub>M I M)" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
976 |
using in_space by (auto simp: space_PiM PiE_iff intro!: PiE_choice[THEN iffD2]) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
977 |
ultimately show "\<exists>A. countable A \<and> A \<subseteq> sets (Pi\<^sub>M I M) \<and> \<Union>A = space (Pi\<^sub>M I M) \<and> (\<forall>a\<in>A. emeasure (Pi\<^sub>M I M) a \<noteq> \<infinity>)" |
64910 | 978 |
by (intro exI[of _ "Pi\<^sub>E I ` Pi\<^sub>E I F"]) |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
979 |
(auto intro!: countable_PiE sets_PiM_I_finite |
64272 | 980 |
simp: PiE_iff emeasure_PiM finite_index ennreal_prod_eq_top) |
40859 | 981 |
qed |
982 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
983 |
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>M I M" |
47694 | 984 |
using sigma_finite[OF finite_index] . |
40859 | 985 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
986 |
lemma (in finite_product_sigma_finite) measure_times: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
987 |
"(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" |
47694 | 988 |
using emeasure_PiM[OF finite_index] by auto |
41096 | 989 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
990 |
lemma (in product_sigma_finite) nn_integral_empty: |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
991 |
"0 \<le> f (\<lambda>k. undefined) \<Longrightarrow> integral\<^sup>N (Pi\<^sub>M {} M) f = f (\<lambda>k. undefined)" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
992 |
by (simp add: PiM_empty nn_integral_count_space_finite max.absorb2) |
40859 | 993 |
|
70136 | 994 |
lemma\<^marker>\<open>tag important\<close> (in product_sigma_finite) distr_merge: |
40859 | 995 |
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
996 |
shows "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J) = Pi\<^sub>M (I \<union> J) M" |
47694 | 997 |
(is "?D = ?P") |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
998 |
proof (rule PiM_eqI) |
61169 | 999 |
interpret I: finite_product_sigma_finite M I by standard fact |
1000 |
interpret J: finite_product_sigma_finite M J by standard fact |
|
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
1001 |
fix A assume A: "\<And>i. i \<in> I \<union> J \<Longrightarrow> A i \<in> sets (M i)" |
64910 | 1002 |
have *: "(merge I J -` Pi\<^sub>E (I \<union> J) A \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)) = Pi\<^sub>E I A \<times> Pi\<^sub>E J A" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
1003 |
using A[THEN sets.sets_into_space] by (auto simp: space_PiM space_pair_measure) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
1004 |
from A fin show "emeasure (distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J)) (Pi\<^sub>E (I \<union> J) A) = |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
1005 |
(\<Prod>i\<in>I \<union> J. emeasure (M i) (A i))" |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
1006 |
by (subst emeasure_distr) |
64272 | 1007 |
(auto simp: * J.emeasure_pair_measure_Times I.measure_times J.measure_times prod.union_disjoint) |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
1008 |
qed (insert fin, simp_all) |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
1009 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1010 |
proposition (in product_sigma_finite) product_nn_integral_fold: |
47694 | 1011 |
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
1012 |
and f[measurable]: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)" |
56996 | 1013 |
shows "integral\<^sup>N (Pi\<^sub>M (I \<union> J) M) f = |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1014 |
(\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^sub>M J M)) \<partial>(Pi\<^sub>M I M))" |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1015 |
proof - |
61169 | 1016 |
interpret I: finite_product_sigma_finite M I by standard fact |
1017 |
interpret J: finite_product_sigma_finite M J by standard fact |
|
1018 |
interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by standard |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1019 |
have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)" |
49776 | 1020 |
using measurable_comp[OF measurable_merge f] by (simp add: comp_def) |
41661 | 1021 |
show ?thesis |
47694 | 1022 |
apply (subst distr_merge[OF IJ, symmetric]) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
1023 |
apply (subst nn_integral_distr[OF measurable_merge]) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
1024 |
apply measurable [] |
56996 | 1025 |
apply (subst J.nn_integral_fst[symmetric, OF P_borel]) |
47694 | 1026 |
apply simp |
1027 |
done |
|
40859 | 1028 |
qed |
1029 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1030 |
lemma (in product_sigma_finite) distr_singleton: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1031 |
"distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _") |
47694 | 1032 |
proof (intro measure_eqI[symmetric]) |
61169 | 1033 |
interpret I: finite_product_sigma_finite M "{i}" by standard simp |
47694 | 1034 |
fix A assume A: "A \<in> sets (M i)" |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
1035 |
then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M {i} M) = (\<Pi>\<^sub>E i\<in>{i}. A)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
1036 |
using sets.sets_into_space by (auto simp: space_PiM) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
1037 |
then show "emeasure (M i) A = emeasure ?D A" |
47694 | 1038 |
using A I.measure_times[of "\<lambda>_. A"] |
1039 |
by (simp add: emeasure_distr measurable_component_singleton) |
|
1040 |
qed simp |
|
41831 | 1041 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1042 |
lemma (in product_sigma_finite) product_nn_integral_singleton: |
40859 | 1043 |
assumes f: "f \<in> borel_measurable (M i)" |
56996 | 1044 |
shows "integral\<^sup>N (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>N (M i) f" |
40859 | 1045 |
proof - |
61169 | 1046 |
interpret I: finite_product_sigma_finite M "{i}" by standard simp |
47694 | 1047 |
from f show ?thesis |
1048 |
apply (subst distr_singleton[symmetric]) |
|
56996 | 1049 |
apply (subst nn_integral_distr[OF measurable_component_singleton]) |
47694 | 1050 |
apply simp_all |
1051 |
done |
|
40859 | 1052 |
qed |
1053 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1054 |
proposition (in product_sigma_finite) product_nn_integral_insert: |
49780 | 1055 |
assumes I[simp]: "finite I" "i \<notin> I" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1056 |
and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)" |
56996 | 1057 |
shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^sub>M I M))" |
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1058 |
proof - |
61169 | 1059 |
interpret I: finite_product_sigma_finite M I by standard auto |
1060 |
interpret i: finite_product_sigma_finite M "{i}" by standard 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
|
1061 |
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
|
1062 |
using f by auto |
41096 | 1063 |
show ?thesis |
56996 | 1064 |
unfolding product_nn_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f] |
1065 |
proof (rule nn_integral_cong, subst product_nn_integral_singleton[symmetric]) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1066 |
fix x assume x: "x \<in> space (Pi\<^sub>M I M)" |
49780 | 1067 |
let ?f = "\<lambda>y. f (x(i := y))" |
1068 |
show "?f \<in> borel_measurable (M i)" |
|
61808 | 1069 |
using measurable_comp[OF measurable_component_update f, OF x \<open>i \<notin> I\<close>] |
47694 | 1070 |
unfolding comp_def . |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1071 |
show "(\<integral>\<^sup>+ y. f (merge I {i} (x, y)) \<partial>Pi\<^sub>M {i} M) = (\<integral>\<^sup>+ y. f (x(i := y i)) \<partial>Pi\<^sub>M {i} M)" |
49780 | 1072 |
using x |
56996 | 1073 |
by (auto intro!: nn_integral_cong arg_cong[where f=f] |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
1074 |
simp add: space_PiM extensional_def PiE_def) |
41096 | 1075 |
qed |
1076 |
qed |
|
1077 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1078 |
lemma (in product_sigma_finite) product_nn_integral_insert_rev: |
59425 | 1079 |
assumes I[simp]: "finite I" "i \<notin> I" |
1080 |
and [measurable]: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)" |
|
1081 |
shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x(i := y)) \<partial>(Pi\<^sub>M I M)) \<partial>(M i))" |
|
1082 |
apply (subst product_nn_integral_insert[OF assms]) |
|
1083 |
apply (rule pair_sigma_finite.Fubini') |
|
1084 |
apply intro_locales [] |
|
1085 |
apply (rule sigma_finite[OF I(1)]) |
|
1086 |
apply measurable |
|
1087 |
done |
|
1088 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1089 |
lemma (in product_sigma_finite) product_nn_integral_prod: |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
1090 |
assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)" |
56996 | 1091 |
shows "(\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>N (M i) (f i))" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
1092 |
using assms proof (induction I) |
41096 | 1093 |
case (insert i I) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
1094 |
note insert.prems[measurable] |
61808 | 1095 |
note \<open>finite I\<close>[intro, simp] |
61169 | 1096 |
interpret I: finite_product_sigma_finite M I by standard auto |
41096 | 1097 |
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))" |
64272 | 1098 |
using insert by (auto intro!: prod.cong) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1099 |
have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^sub>M J M)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
1100 |
using sets.sets_into_space insert |
64272 | 1101 |
by (intro borel_measurable_prod_ennreal |
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
|
1102 |
measurable_comp[OF measurable_component_singleton, unfolded comp_def]) |
41096 | 1103 |
auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1104 |
then show ?case |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
1105 |
apply (simp add: product_nn_integral_insert[OF insert(1,2)]) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
1106 |
apply (simp add: insert(2-) * nn_integral_multc) |
56996 | 1107 |
apply (subst nn_integral_cmult) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62390
diff
changeset
|
1108 |
apply (auto simp add: insert(2-)) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1109 |
done |
47694 | 1110 |
qed (simp add: space_PiM) |
41096 | 1111 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1112 |
proposition (in product_sigma_finite) product_nn_integral_pair: |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61378
diff
changeset
|
1113 |
assumes [measurable]: "case_prod f \<in> borel_measurable (M x \<Otimes>\<^sub>M M y)" |
59425 | 1114 |
assumes xy: "x \<noteq> y" |
1115 |
shows "(\<integral>\<^sup>+\<sigma>. f (\<sigma> x) (\<sigma> y) \<partial>PiM {x, y} M) = (\<integral>\<^sup>+z. f (fst z) (snd z) \<partial>(M x \<Otimes>\<^sub>M M y))" |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1116 |
proof - |
59425 | 1117 |
interpret psm: pair_sigma_finite "M x" "M y" |
1118 |
unfolding pair_sigma_finite_def using sigma_finite_measures by simp_all |
|
1119 |
have "{x, y} = {y, x}" by auto |
|
1120 |
also have "(\<integral>\<^sup>+\<sigma>. f (\<sigma> x) (\<sigma> y) \<partial>PiM {y, x} M) = (\<integral>\<^sup>+y. \<integral>\<^sup>+\<sigma>. f (\<sigma> x) y \<partial>PiM {x} M \<partial>M y)" |
|
1121 |
using xy by (subst product_nn_integral_insert_rev) simp_all |
|
1122 |
also have "... = (\<integral>\<^sup>+y. \<integral>\<^sup>+x. f x y \<partial>M x \<partial>M y)" |
|
1123 |
by (intro nn_integral_cong, subst product_nn_integral_singleton) simp_all |
|
1124 |
also have "... = (\<integral>\<^sup>+z. f (fst z) (snd z) \<partial>(M x \<Otimes>\<^sub>M M y))" |
|
1125 |
by (subst psm.nn_integral_snd[symmetric]) simp_all |
|
1126 |
finally show ?thesis . |
|
1127 |
qed |
|
1128 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1129 |
lemma (in product_sigma_finite) distr_component: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1130 |
"distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P") |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
1131 |
proof (intro PiM_eqI) |
63540 | 1132 |
fix A assume A: "\<And>ia. ia \<in> {i} \<Longrightarrow> A ia \<in> sets (M ia)" |
1133 |
then have "(\<lambda>x. \<lambda>i\<in>{i}. x) -` Pi\<^sub>E {i} A \<inter> space (M i) = A i" |
|
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64910
diff
changeset
|
1134 |
by (fastforce dest: sets.sets_into_space) |
63540 | 1135 |
with A show "emeasure (distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x)) (Pi\<^sub>E {i} A) = (\<Prod>i\<in>{i}. emeasure (M i) (A i))" |
61359
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
1136 |
by (subst emeasure_distr) (auto intro!: sets_PiM_I_finite measurable_restrict) |
e985b52c3eb3
cleanup projective limit of probability distributions; proved Ionescu-Tulcea; used it to prove infinite prob. distribution
hoelzl
parents:
61169
diff
changeset
|
1137 |
qed simp_all |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
1138 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1139 |
lemma (in product_sigma_finite) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1140 |
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)" |
49776 | 1141 |
shows emeasure_fold_integral: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1142 |
"emeasure (Pi\<^sub>M (I \<union> J) M) A = (\<integral>\<^sup>+x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M)) \<partial>Pi\<^sub>M I M)" (is ?I) |
49776 | 1143 |
and emeasure_fold_measurable: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1144 |
"(\<lambda>x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M))) \<in> borel_measurable (Pi\<^sub>M I M)" (is ?B) |
49776 | 1145 |
proof - |
61169 | 1146 |
interpret I: finite_product_sigma_finite M I by standard fact |
1147 |
interpret J: finite_product_sigma_finite M J by standard fact |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1148 |
interpret IJ: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" .. |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1149 |
have merge: "merge I J -` A \<inter> space (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) \<in> sets (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)" |
49776 | 1150 |
by (intro measurable_sets[OF _ A] measurable_merge assms) |
1151 |
||
1152 |
show ?I |
|
1153 |
apply (subst distr_merge[symmetric, OF IJ]) |
|
1154 |
apply (subst emeasure_distr[OF measurable_merge A]) |
|
1155 |
apply (subst J.emeasure_pair_measure_alt[OF merge]) |
|
56996 | 1156 |
apply (auto intro!: nn_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure) |
49776 | 1157 |
done |
1158 |
||
1159 |
show ?B |
|
1160 |
using IJ.measurable_emeasure_Pair1[OF merge] |
|
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
55415
diff
changeset
|
1161 |
by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong) |
49776 | 1162 |
qed |
1163 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1164 |
lemma sets_Collect_single: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1165 |
"i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^sub>M I M). x i \<in> A } \<in> sets (Pi\<^sub>M I M)" |
50003 | 1166 |
by simp |
49776 | 1167 |
|
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1168 |
lemma pair_measure_eq_distr_PiM: |
50104 | 1169 |
fixes M1 :: "'a measure" and M2 :: "'a measure" |
1170 |
assumes "sigma_finite_measure M1" "sigma_finite_measure M2" |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1171 |
shows "(M1 \<Otimes>\<^sub>M M2) = distr (Pi\<^sub>M UNIV (case_bool M1 M2)) (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. (x True, x False))" |
50104 | 1172 |
(is "?P = ?D") |
1173 |
proof (rule pair_measure_eqI[OF assms]) |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1174 |
interpret B: product_sigma_finite "case_bool M1 M2" |
50104 | 1175 |
unfolding product_sigma_finite_def using assms by (auto split: bool.split) |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1176 |
let ?B = "Pi\<^sub>M UNIV (case_bool M1 M2)" |
50104 | 1177 |
|
1178 |
have [simp]: "fst \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x True)" "snd \<circ> (\<lambda>x. (x True, x False)) = (\<lambda>x. x False)" |
|
1179 |
by auto |
|
1180 |
fix A B assume A: "A \<in> sets M1" and B: "B \<in> sets M2" |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1181 |
have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (case_bool M1 M2 i) (case_bool A B i))" |
50104 | 1182 |
by (simp add: UNIV_bool ac_simps) |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1183 |
also have "\<dots> = emeasure ?B (Pi\<^sub>E UNIV (case_bool A B))" |
50104 | 1184 |
using A B by (subst B.emeasure_PiM) (auto split: bool.split) |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1185 |
also have "Pi\<^sub>E UNIV (case_bool A B) = (\<lambda>x. (x True, x False)) -` (A \<times> B) \<inter> space ?B" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
1186 |
using A[THEN sets.sets_into_space] B[THEN sets.sets_into_space] |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
1187 |
by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split) |
50104 | 1188 |
finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)" |
1189 |
using A B |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1190 |
measurable_component_singleton[of True UNIV "case_bool M1 M2"] |
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1191 |
measurable_component_singleton[of False UNIV "case_bool M1 M2"] |
50104 | 1192 |
by (subst emeasure_distr) (auto simp: measurable_pair_iff) |
1193 |
qed simp |
|
1194 |
||
69681
689997a8a582
redo tagging-related changes from a06b204527e6, 0f4d4a13dc16, and a8faf6f15da7
immler
parents:
69680
diff
changeset
|
1195 |
lemma infprod_in_sets[intro]: |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63627
diff
changeset
|
1196 |
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63627
diff
changeset
|
1197 |
shows "Pi UNIV E \<in> sets (\<Pi>\<^sub>M i\<in>UNIV::nat set. M i)" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63627
diff
changeset
|
1198 |
proof - |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63627
diff
changeset
|
1199 |
have "Pi UNIV E = (\<Inter>i. prod_emb UNIV M {..i} (\<Pi>\<^sub>E j\<in>{..i}. E j))" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63627
diff
changeset
|
1200 |
using E E[THEN sets.sets_into_space] |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63627
diff
changeset
|
1201 |
by (auto simp: prod_emb_def Pi_iff extensional_def) |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63627
diff
changeset
|
1202 |
with E show ?thesis by auto |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63627
diff
changeset
|
1203 |
qed |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63627
diff
changeset
|
1204 |
|
69918
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1205 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1206 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1207 |
subsection \<open>Measurability\<close> |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1208 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1209 |
text \<open>There are two natural sigma-algebras on a product space: the borel sigma algebra, |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1210 |
generated by open sets in the product, and the product sigma algebra, countably generated by |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1211 |
products of measurable sets along finitely many coordinates. The second one is defined and studied |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1212 |
in \<^file>\<open>Finite_Product_Measure.thy\<close>. |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1213 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1214 |
These sigma-algebra share a lot of natural properties (measurability of coordinates, for instance), |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1215 |
but there is a fundamental difference: open sets are generated by arbitrary unions, not only |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1216 |
countable ones, so typically many open sets will not be measurable with respect to the product sigma |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1217 |
algebra (while all sets in the product sigma algebra are borel). The two sigma algebras coincide |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1218 |
only when everything is countable (i.e., the product is countable, and the borel sigma algebra in |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1219 |
the factor is countably generated). |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1220 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1221 |
In this paragraph, we develop basic measurability properties for the borel sigma algebra, and |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1222 |
compare it with the product sigma algebra as explained above. |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1223 |
\<close> |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1224 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1225 |
lemma measurable_product_coordinates [measurable (raw)]: |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1226 |
"(\<lambda>x. x i) \<in> measurable borel borel" |
70365
4df0628e8545
a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
1227 |
by (rule borel_measurable_continuous_onI[OF continuous_on_product_coordinates]) |
69918
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1228 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1229 |
lemma measurable_product_then_coordinatewise: |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1230 |
fixes f::"'a \<Rightarrow> 'b \<Rightarrow> ('c::topological_space)" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1231 |
assumes [measurable]: "f \<in> borel_measurable M" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1232 |
shows "(\<lambda>x. f x i) \<in> borel_measurable M" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1233 |
proof - |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1234 |
have "(\<lambda>x. f x i) = (\<lambda>y. y i) o f" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1235 |
unfolding comp_def by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1236 |
then show ?thesis by simp |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1237 |
qed |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1238 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1239 |
text \<open>To compare the Borel sigma algebra with the product sigma algebra, we give a presentation |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1240 |
of the product sigma algebra that is more similar to the one we used above for the product |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1241 |
topology.\<close> |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1242 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1243 |
lemma sets_PiM_finite: |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1244 |
"sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1245 |
{(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1246 |
proof |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1247 |
have "{(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}} \<subseteq> sets (Pi\<^sub>M I M)" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1248 |
proof (auto) |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1249 |
fix X assume H: "\<forall>i. X i \<in> sets (M i)" "finite {i. X i \<noteq> space (M i)}" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1250 |
then have *: "X i \<in> sets (M i)" for i by simp |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1251 |
define J where "J = {i \<in> I. X i \<noteq> space (M i)}" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1252 |
have "finite J" "J \<subseteq> I" unfolding J_def using H by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1253 |
define Y where "Y = (\<Pi>\<^sub>E j\<in>J. X j)" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1254 |
have "prod_emb I M J Y \<in> sets (Pi\<^sub>M I M)" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1255 |
unfolding Y_def apply (rule sets_PiM_I) using \<open>finite J\<close> \<open>J \<subseteq> I\<close> * by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1256 |
moreover have "prod_emb I M J Y = (\<Pi>\<^sub>E i\<in>I. X i)" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1257 |
unfolding prod_emb_def Y_def J_def using H sets.sets_into_space[OF *] |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1258 |
by (auto simp add: PiE_iff, blast) |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1259 |
ultimately show "Pi\<^sub>E I X \<in> sets (Pi\<^sub>M I M)" by simp |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1260 |
qed |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1261 |
then show "sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}} |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1262 |
\<subseteq> sets (Pi\<^sub>M I M)" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1263 |
by (metis (mono_tags, lifting) sets.sigma_sets_subset' sets.top space_PiM) |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1264 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1265 |
have *: "\<exists>X. {f. (\<forall>i\<in>I. f i \<in> space (M i)) \<and> f \<in> extensional I \<and> f i \<in> A} = Pi\<^sub>E I X \<and> |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1266 |
(\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1267 |
if "i \<in> I" "A \<in> sets (M i)" for i A |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1268 |
proof - |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1269 |
define X where "X = (\<lambda>j. if j = i then A else space (M j))" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1270 |
have "{f. (\<forall>i\<in>I. f i \<in> space (M i)) \<and> f \<in> extensional I \<and> f i \<in> A} = Pi\<^sub>E I X" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1271 |
unfolding X_def using sets.sets_into_space[OF \<open>A \<in> sets (M i)\<close>] \<open>i \<in> I\<close> |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1272 |
by (auto simp add: PiE_iff extensional_def, metis subsetCE, metis) |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1273 |
moreover have "X j \<in> sets (M j)" for j |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1274 |
unfolding X_def using \<open>A \<in> sets (M i)\<close> by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1275 |
moreover have "finite {j. X j \<noteq> space (M j)}" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1276 |
unfolding X_def by simp |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1277 |
ultimately show ?thesis by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1278 |
qed |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1279 |
show "sets (Pi\<^sub>M I M) \<subseteq> sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1280 |
unfolding sets_PiM_single |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1281 |
apply (rule sigma_sets_mono') |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1282 |
apply (auto simp add: PiE_iff *) |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1283 |
done |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1284 |
qed |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1285 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1286 |
lemma sets_PiM_subset_borel: |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1287 |
"sets (Pi\<^sub>M UNIV (\<lambda>_. borel)) \<subseteq> sets borel" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1288 |
proof - |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1289 |
have *: "Pi\<^sub>E UNIV X \<in> sets borel" if [measurable]: "\<And>i. X i \<in> sets borel" "finite {i. X i \<noteq> UNIV}" for X::"'a \<Rightarrow> 'b set" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1290 |
proof - |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1291 |
define I where "I = {i. X i \<noteq> UNIV}" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1292 |
have "finite I" unfolding I_def using that by simp |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1293 |
have "Pi\<^sub>E UNIV X = (\<Inter>i\<in>I. (\<lambda>x. x i)-`(X i) \<inter> space borel) \<inter> space borel" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1294 |
unfolding I_def by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1295 |
also have "... \<in> sets borel" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1296 |
using that \<open>finite I\<close> by measurable |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1297 |
finally show ?thesis by simp |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1298 |
qed |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1299 |
then have "{(\<Pi>\<^sub>E i\<in>UNIV. X i) |X::('a \<Rightarrow> 'b set). (\<forall>i. X i \<in> sets borel) \<and> finite {i. X i \<noteq> space borel}} \<subseteq> sets borel" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1300 |
by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1301 |
then show ?thesis unfolding sets_PiM_finite space_borel |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1302 |
by (simp add: * sets.sigma_sets_subset') |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1303 |
qed |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1304 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1305 |
proposition sets_PiM_equal_borel: |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1306 |
"sets (Pi\<^sub>M UNIV (\<lambda>i::('a::countable). borel::('b::second_countable_topology measure))) = sets borel" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1307 |
proof |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1308 |
obtain K::"('a \<Rightarrow> 'b) set set" where K: "topological_basis K" "countable K" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1309 |
"\<And>k. k \<in> K \<Longrightarrow> \<exists>X. (k = Pi\<^sub>E UNIV X) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV}" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1310 |
using product_topology_countable_basis by fast |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1311 |
have *: "k \<in> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))" if "k \<in> K" for k |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1312 |
proof - |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1313 |
obtain X where H: "k = Pi\<^sub>E UNIV X" "\<And>i. open (X i)" "finite {i. X i \<noteq> UNIV}" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1314 |
using K(3)[OF \<open>k \<in> K\<close>] by blast |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1315 |
show ?thesis unfolding H(1) sets_PiM_finite space_borel using borel_open[OF H(2)] H(3) by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1316 |
qed |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1317 |
have **: "U \<in> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))" if "open U" for U::"('a \<Rightarrow> 'b) set" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1318 |
proof - |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1319 |
obtain B where "B \<subseteq> K" "U = (\<Union>B)" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1320 |
using \<open>open U\<close> \<open>topological_basis K\<close> by (metis topological_basis_def) |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1321 |
have "countable B" using \<open>B \<subseteq> K\<close> \<open>countable K\<close> countable_subset by blast |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1322 |
moreover have "k \<in> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))" if "k \<in> B" for k |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1323 |
using \<open>B \<subseteq> K\<close> * that by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1324 |
ultimately show ?thesis unfolding \<open>U = (\<Union>B)\<close> by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1325 |
qed |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1326 |
have "sigma_sets UNIV (Collect open) \<subseteq> sets (Pi\<^sub>M UNIV (\<lambda>i::'a. (borel::('b measure))))" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1327 |
apply (rule sets.sigma_sets_subset') using ** by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1328 |
then show "sets (borel::('a \<Rightarrow> 'b) measure) \<subseteq> sets (Pi\<^sub>M UNIV (\<lambda>_. borel))" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1329 |
unfolding borel_def by auto |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1330 |
qed (simp add: sets_PiM_subset_borel) |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1331 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1332 |
lemma measurable_coordinatewise_then_product: |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1333 |
fixes f::"'a \<Rightarrow> ('b::countable) \<Rightarrow> ('c::second_countable_topology)" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1334 |
assumes [measurable]: "\<And>i. (\<lambda>x. f x i) \<in> borel_measurable M" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1335 |
shows "f \<in> borel_measurable M" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1336 |
proof - |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1337 |
have "f \<in> measurable M (Pi\<^sub>M UNIV (\<lambda>_. borel))" |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1338 |
by (rule measurable_PiM_single', auto simp add: assms) |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1339 |
then show ?thesis using sets_PiM_equal_borel measurable_cong_sets by blast |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1340 |
qed |
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1341 |
|
eddcc7c726f3
new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents:
69745
diff
changeset
|
1342 |
|
47694 | 1343 |
end |