author | hoelzl |
Mon, 24 Nov 2014 12:20:14 +0100 | |
changeset 59048 | 7dc8ac6f0895 |
parent 58876 | 1888e3cb8048 |
child 59088 | ff2bd4a14ddb |
permissions | -rw-r--r-- |
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(* Title: HOL/Probability/Finite_Product_Measure.thy |
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Author: Johannes Hölzl, TU München |
3 |
*) |
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section {*Finite product measures*} |
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theory Finite_Product_Measure |
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imports Binary_Product_Measure |
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begin |
10 |
||
57447
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lemma PiE_choice: "(\<exists>f\<in>PiE I F. \<forall>i\<in>I. P i (f i)) \<longleftrightarrow> (\<forall>i\<in>I. \<exists>x\<in>F i. P i x)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
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12 |
by (auto simp: Bex_def PiE_iff Ball_def dest!: choice_iff'[THEN iffD1]) |
87429bdecad5
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parents:
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13 |
(force intro: exI[of _ "restrict f I" for f]) |
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|
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lemma split_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)" |
16 |
by auto |
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||
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subsubsection {* More about Function restricted by @{const extensional} *} |
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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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|
23 |
lemma merge_apply[simp]: |
|
49780 | 24 |
"I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I J (x, y) i = x i" |
25 |
"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: |
|
49780 | 32 |
"I \<inter> J = {} \<Longrightarrow> merge I J (x, y) = merge J I (y, x)" |
50003 | 33 |
by (force simp: merge_def) |
40859 | 34 |
|
35 |
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" |
37 |
"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" |
|
39 |
"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J I (B, A)) \<longleftrightarrow> x \<in> Pi I A" |
|
40859 | 40 |
by (auto simp: Pi_def) |
41 |
||
42 |
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" |
44 |
"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" |
|
40859 | 47 |
by (auto simp: Pi_def) |
48 |
||
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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) |
51 |
||
52 |
lemma restrict_merge[simp]: |
|
49780 | 53 |
"I \<inter> J = {} \<Longrightarrow> restrict (merge I J (x, y)) I = restrict x I" |
54 |
"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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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
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|
60 |
unfolding merge_def by auto |
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immler@in.tum.de
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61 |
|
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62 |
lemma PiE_cancel_merge[simp]: |
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63 |
"I \<inter> J = {} \<Longrightarrow> |
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64 |
merge I J (x, y) \<in> PiE (I \<union> J) B \<longleftrightarrow> x \<in> Pi I B \<and> y \<in> Pi J B" |
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|
65 |
by (auto simp: PiE_def restrict_Pi_cancel) |
69b35a75caf3
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parents:
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66 |
|
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
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lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I {i} (x,y) = restrict (x(i := y i)) (insert i I)" |
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parents:
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68 |
unfolding merge_def by (auto simp: fun_eq_iff) |
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hoelzl
parents:
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69 |
|
50042
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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 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:
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|
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unfolding merge_def extensional_def by auto |
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moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents:
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72 |
|
50123
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73 |
lemma merge_restrict[simp]: |
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"merge I J (restrict x I, y) = merge I J (x, y)" |
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parents:
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75 |
"merge I J (x, restrict y J) = merge I J (x, y)" |
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hoelzl
parents:
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diff
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|
76 |
unfolding merge_def by auto |
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
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77 |
|
69b35a75caf3
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|
78 |
lemma merge_x_x_eq_restrict[simp]: |
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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:
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diff
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|
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:
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|
82 |
lemma injective_vimage_restrict: |
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parents:
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|
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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|
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:
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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:
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diff
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|
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)" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
95 |
using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I - J" x y S] |
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" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
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|
98 |
using y x `J \<subseteq> I` PiE_cancel_merge[of "J" "I - J" x y S] |
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
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parents:
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diff
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|
100 |
qed (insert sets, auto) |
50042
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immler@in.tum.de
parents:
50041
diff
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|
101 |
qed |
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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|
102 |
|
41095 | 103 |
lemma restrict_vimage: |
50123
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merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
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|
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
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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
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hoelzl
parents:
50104
diff
changeset
|
110 |
by (auto simp: restrict_Pi_cancel PiE_def) |
50104 | 111 |
|
56994 | 112 |
subsection {* Finite product spaces *} |
40859 | 113 |
|
56994 | 114 |
subsubsection {* Products *} |
40859 | 115 |
|
47694 | 116 |
definition prod_emb where |
117 |
"prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (PIE i:I. space (M i))" |
|
118 |
||
119 |
lemma prod_emb_iff: |
|
120 |
"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))" |
|
50123
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|
121 |
unfolding prod_emb_def PiE_def by auto |
40859 | 122 |
|
47694 | 123 |
lemma |
124 |
shows prod_emb_empty[simp]: "prod_emb M L K {} = {}" |
|
125 |
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" |
|
126 |
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" |
|
127 |
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))" |
|
128 |
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))" |
|
129 |
and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B" |
|
130 |
by (auto simp: prod_emb_def) |
|
40859 | 131 |
|
47694 | 132 |
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
|
133 |
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))" |
50123
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hoelzl
parents:
50104
diff
changeset
|
134 |
by (force simp: prod_emb_def PiE_iff split_if_mem2) |
47694 | 135 |
|
50123
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parents:
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diff
changeset
|
136 |
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
|
137 |
"(\<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
|
138 |
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
|
139 |
|
50038 | 140 |
lemma prod_emb_trans[simp]: |
141 |
"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
|
142 |
by (auto simp add: Int_absorb1 prod_emb_def PiE_def) |
50038 | 143 |
|
144 |
lemma prod_emb_Pi: |
|
145 |
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
|
146 |
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
|
147 |
using assms sets.space_closed |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
148 |
by (auto simp: prod_emb_def PiE_iff split: split_if_asm) blast+ |
50038 | 149 |
|
150 |
lemma prod_emb_id: |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
151 |
"B \<subseteq> (\<Pi>\<^sub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B" |
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152 |
by (auto simp: prod_emb_def subset_eq extensional_restrict) |
50038 | 153 |
|
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|
154 |
lemma prod_emb_mono: |
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|
155 |
"F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G" |
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|
156 |
by (auto simp: prod_emb_def) |
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|
157 |
|
47694 | 158 |
definition PiM :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where |
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159 |
"PiM I M = extend_measure (\<Pi>\<^sub>E i\<in>I. space (M i)) |
47694 | 160 |
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} |
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161 |
(\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j)) |
47694 | 162 |
(\<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)))" |
163 |
||
164 |
definition prod_algebra :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i \<Rightarrow> 'a) set set" where |
|
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165 |
"prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^sub>E j\<in>J. X j)) ` |
47694 | 166 |
{(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}" |
167 |
||
168 |
abbreviation |
|
53015
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169 |
"Pi\<^sub>M I M \<equiv> PiM I M" |
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|
170 |
|
40859 | 171 |
syntax |
47694 | 172 |
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3PIM _:_./ _)" 10) |
40859 | 173 |
|
174 |
syntax (xsymbols) |
|
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|
175 |
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^sub>M _\<in>_./ _)" 10) |
40859 | 176 |
|
177 |
syntax (HTML output) |
|
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178 |
"_PiM" :: "pttrn \<Rightarrow> 'i set \<Rightarrow> 'a measure \<Rightarrow> ('i => 'a) measure" ("(3\<Pi>\<^sub>M _\<in>_./ _)" 10) |
40859 | 179 |
|
180 |
translations |
|
47694 | 181 |
"PIM x:I. M" == "CONST PiM I (%x. M)" |
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182 |
|
47694 | 183 |
lemma prod_algebra_sets_into_space: |
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|
184 |
"prod_algebra I M \<subseteq> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))" |
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|
185 |
by (auto simp: prod_emb_def prod_algebra_def) |
40859 | 186 |
|
47694 | 187 |
lemma prod_algebra_eq_finite: |
188 |
assumes I: "finite I" |
|
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|
189 |
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") |
47694 | 190 |
proof (intro iffI set_eqI) |
191 |
fix A assume "A \<in> ?L" |
|
192 |
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)" |
|
193 |
and A: "A = prod_emb I M J (PIE j:J. E j)" |
|
194 |
by (auto simp: prod_algebra_def) |
|
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195 |
let ?A = "\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i)" |
47694 | 196 |
have A: "A = ?A" |
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197 |
unfolding A using J by (intro prod_emb_PiE sets.sets_into_space) auto |
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|
198 |
show "A \<in> ?R" unfolding A using J sets.top |
47694 | 199 |
by (intro CollectI exI[of _ "\<lambda>i. if i \<in> J then E i else space (M i)"]) simp |
200 |
next |
|
201 |
fix A assume "A \<in> ?R" |
|
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|
202 |
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 |
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|
203 |
then have A: "A = prod_emb I M I (\<Pi>\<^sub>E i\<in>I. X i)" |
50244
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|
204 |
by (simp add: prod_emb_PiE_same_index[OF sets.sets_into_space] Pi_iff) |
47694 | 205 |
from X I show "A \<in> ?L" unfolding A |
206 |
by (auto simp: prod_algebra_def) |
|
207 |
qed |
|
41095 | 208 |
|
47694 | 209 |
lemma prod_algebraI: |
210 |
"finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)) |
|
211 |
\<Longrightarrow> prod_emb I M J (PIE j:J. E j) \<in> prod_algebra I M" |
|
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|
212 |
by (auto simp: prod_algebra_def) |
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|
213 |
|
50038 | 214 |
lemma prod_algebraI_finite: |
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|
215 |
"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
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|
216 |
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 | 217 |
|
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218 |
lemma Int_stable_PiE: "Int_stable {Pi\<^sub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}" |
50038 | 219 |
proof (safe intro!: Int_stableI) |
220 |
fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)" |
|
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|
221 |
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))" |
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|
222 |
by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int) |
50038 | 223 |
qed |
224 |
||
47694 | 225 |
lemma prod_algebraE: |
226 |
assumes A: "A \<in> prod_algebra I M" |
|
227 |
obtains J E where "A = prod_emb I M J (PIE j:J. E j)" |
|
228 |
"finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" |
|
229 |
using A by (auto simp: prod_algebra_def) |
|
42988 | 230 |
|
47694 | 231 |
lemma prod_algebraE_all: |
232 |
assumes A: "A \<in> prod_algebra I M" |
|
53015
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|
233 |
obtains E where "A = Pi\<^sub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))" |
47694 | 234 |
proof - |
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235 |
from A obtain E J where A: "A = prod_emb I M J (Pi\<^sub>E J E)" |
47694 | 236 |
and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))" |
237 |
by (auto simp: prod_algebra_def) |
|
238 |
from E have "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)" |
|
50244
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|
239 |
using sets.sets_into_space by auto |
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|
240 |
then have "A = (\<Pi>\<^sub>E i\<in>I. if i\<in>J then E i else space (M i))" |
47694 | 241 |
using A J by (auto simp: prod_emb_PiE) |
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|
242 |
moreover have "(\<lambda>i. if i\<in>J then E i else space (M i)) \<in> (\<Pi> i\<in>I. sets (M i))" |
50244
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|
243 |
using sets.top E by auto |
47694 | 244 |
ultimately show ?thesis using that by auto |
245 |
qed |
|
40859 | 246 |
|
47694 | 247 |
lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)" |
248 |
proof (unfold Int_stable_def, safe) |
|
249 |
fix A assume "A \<in> prod_algebra I M" |
|
250 |
from prod_algebraE[OF this] guess J E . note A = this |
|
251 |
fix B assume "B \<in> prod_algebra I M" |
|
252 |
from prod_algebraE[OF this] guess K F . note B = this |
|
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|
253 |
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 | 254 |
(if i \<in> K then F i else space (M i)))" |
50244
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changeset
|
255 |
unfolding A B using A(2,3,4) A(5)[THEN sets.sets_into_space] B(2,3,4) |
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|
256 |
B(5)[THEN sets.sets_into_space] |
47694 | 257 |
apply (subst (1 2 3) prod_emb_PiE) |
258 |
apply (simp_all add: subset_eq PiE_Int) |
|
259 |
apply blast |
|
260 |
apply (intro PiE_cong) |
|
261 |
apply auto |
|
262 |
done |
|
263 |
also have "\<dots> \<in> prod_algebra I M" |
|
264 |
using A B by (auto intro!: prod_algebraI) |
|
265 |
finally show "A \<inter> B \<in> prod_algebra I M" . |
|
266 |
qed |
|
267 |
||
268 |
lemma prod_algebra_mono: |
|
269 |
assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)" |
|
270 |
assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)" |
|
271 |
shows "prod_algebra I E \<subseteq> prod_algebra I F" |
|
272 |
proof |
|
273 |
fix A assume "A \<in> prod_algebra I E" |
|
274 |
then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" |
|
53015
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|
275 |
and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)" |
47694 | 276 |
and G: "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (E i)" |
277 |
by (auto simp: prod_algebra_def) |
|
278 |
moreover |
|
53015
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|
279 |
from space have "(\<Pi>\<^sub>E i\<in>I. space (E i)) = (\<Pi>\<^sub>E i\<in>I. space (F i))" |
47694 | 280 |
by (rule PiE_cong) |
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|
281 |
with A have "A = prod_emb I F J (\<Pi>\<^sub>E i\<in>J. G i)" |
47694 | 282 |
by (simp add: prod_emb_def) |
283 |
moreover |
|
284 |
from sets G J have "\<And>i. i \<in> J \<Longrightarrow> G i \<in> sets (F i)" |
|
285 |
by auto |
|
286 |
ultimately show "A \<in> prod_algebra I F" |
|
287 |
apply (simp add: prod_algebra_def image_iff) |
|
288 |
apply (intro exI[of _ J] exI[of _ G] conjI) |
|
289 |
apply auto |
|
290 |
done |
|
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|
291 |
qed |
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|
292 |
|
50104 | 293 |
lemma prod_algebra_cong: |
294 |
assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))" |
|
295 |
shows "prod_algebra I M = prod_algebra J N" |
|
296 |
proof - |
|
297 |
have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)" |
|
298 |
using sets_eq_imp_space_eq[OF sets] by auto |
|
299 |
with sets show ?thesis unfolding `I = J` |
|
300 |
by (intro antisym prod_algebra_mono) auto |
|
301 |
qed |
|
302 |
||
303 |
lemma space_in_prod_algebra: |
|
53015
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|
304 |
"(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M" |
50104 | 305 |
proof cases |
306 |
assume "I = {}" then show ?thesis |
|
307 |
by (auto simp add: prod_algebra_def image_iff prod_emb_def) |
|
308 |
next |
|
309 |
assume "I \<noteq> {}" |
|
310 |
then obtain i where "i \<in> I" by auto |
|
53015
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|
311 |
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
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changeset
|
312 |
by (auto simp: prod_emb_def) |
50104 | 313 |
also have "\<dots> \<in> prod_algebra I M" |
314 |
using `i \<in> I` by (intro prod_algebraI) auto |
|
315 |
finally show ?thesis . |
|
316 |
qed |
|
317 |
||
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|
318 |
lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))" |
47694 | 319 |
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp |
320 |
||
53015
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changeset
|
321 |
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 | 322 |
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp |
41689
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changeset
|
323 |
|
47694 | 324 |
lemma sets_PiM_single: "sets (PiM I M) = |
53015
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changeset
|
325 |
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 | 326 |
(is "_ = sigma_sets ?\<Omega> ?R") |
327 |
unfolding sets_PiM |
|
328 |
proof (rule sigma_sets_eqI) |
|
329 |
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto |
|
330 |
fix A assume "A \<in> prod_algebra I M" |
|
331 |
from prod_algebraE[OF this] guess J X . note X = this |
|
332 |
show "A \<in> sigma_sets ?\<Omega> ?R" |
|
333 |
proof cases |
|
334 |
assume "I = {}" |
|
335 |
with X have "A = {\<lambda>x. undefined}" by (auto simp: prod_emb_def) |
|
336 |
with `I = {}` show ?thesis by (auto intro!: sigma_sets_top) |
|
337 |
next |
|
338 |
assume "I \<noteq> {}" |
|
53015
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standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
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parents:
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diff
changeset
|
339 |
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
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changeset
|
340 |
by (auto simp: prod_emb_def) |
47694 | 341 |
also have "\<dots> \<in> sigma_sets ?\<Omega> ?R" |
342 |
using X `I \<noteq> {}` by (intro R.finite_INT sigma_sets.Basic) auto |
|
343 |
finally show "A \<in> sigma_sets ?\<Omega> ?R" . |
|
344 |
qed |
|
345 |
next |
|
346 |
fix A assume "A \<in> ?R" |
|
53015
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diff
changeset
|
347 |
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 | 348 |
by auto |
53015
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parents:
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changeset
|
349 |
then have "A = prod_emb I M {i} (\<Pi>\<^sub>E i\<in>{i}. B)" |
50123
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|
350 |
by (auto simp: prod_emb_def) |
47694 | 351 |
also have "\<dots> \<in> sigma_sets ?\<Omega> (prod_algebra I M)" |
352 |
using A by (intro sigma_sets.Basic prod_algebraI) auto |
|
353 |
finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" . |
|
354 |
qed |
|
355 |
||
58606 | 356 |
lemma sets_PiM_eq_proj: |
357 |
"I \<noteq> {} \<Longrightarrow> sets (PiM I M) = sets (\<Squnion>\<^sub>\<sigma> i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>x. x i) (M i))" |
|
358 |
apply (simp add: sets_PiM_single sets_Sup_sigma) |
|
359 |
apply (subst SUP_cong[OF refl]) |
|
360 |
apply (rule sets_vimage_algebra2) |
|
361 |
apply auto [] |
|
362 |
apply (auto intro!: arg_cong2[where f=sigma_sets]) |
|
363 |
done |
|
364 |
||
365 |
lemma sets_PiM_in_sets: |
|
366 |
assumes space: "space N = (\<Pi>\<^sub>E i\<in>I. space (M i))" |
|
367 |
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" |
|
368 |
shows "sets (\<Pi>\<^sub>M i \<in> I. M i) \<subseteq> sets N" |
|
369 |
unfolding sets_PiM_single space[symmetric] |
|
370 |
by (intro sets.sigma_sets_subset subsetI) (auto intro: sets) |
|
371 |
||
59048 | 372 |
lemma sets_PiM_cong[measurable_cong]: |
373 |
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 | 374 |
using assms sets_eq_imp_space_eq[OF assms(2)] by (simp add: sets_PiM_single cong: PiE_cong conj_cong) |
375 |
||
47694 | 376 |
lemma sets_PiM_I: |
377 |
assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)" |
|
378 |
shows "prod_emb I M J (PIE j:J. E j) \<in> sets (PIM i:I. M i)" |
|
379 |
proof cases |
|
380 |
assume "J = {}" |
|
381 |
then have "prod_emb I M J (PIE j:J. E j) = (PIE j:I. space (M j))" |
|
382 |
by (auto simp: prod_emb_def) |
|
383 |
then show ?thesis |
|
384 |
by (auto simp add: sets_PiM intro!: sigma_sets_top) |
|
385 |
next |
|
386 |
assume "J \<noteq> {}" with assms show ?thesis |
|
50003 | 387 |
by (force simp add: sets_PiM prod_algebra_def) |
40859 | 388 |
qed |
389 |
||
47694 | 390 |
lemma measurable_PiM: |
53015
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wenzelm
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changeset
|
391 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))" |
47694 | 392 |
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> |
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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changeset
|
393 |
f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N" |
47694 | 394 |
shows "f \<in> measurable N (PiM I M)" |
395 |
using sets_PiM prod_algebra_sets_into_space space |
|
396 |
proof (rule measurable_sigma_sets) |
|
397 |
fix A assume "A \<in> prod_algebra I M" |
|
398 |
from prod_algebraE[OF this] guess J X . |
|
399 |
with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto |
|
400 |
qed |
|
401 |
||
402 |
lemma measurable_PiM_Collect: |
|
53015
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parents:
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diff
changeset
|
403 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))" |
47694 | 404 |
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> |
405 |
{\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N" |
|
406 |
shows "f \<in> measurable N (PiM I M)" |
|
407 |
using sets_PiM prod_algebra_sets_into_space space |
|
408 |
proof (rule measurable_sigma_sets) |
|
409 |
fix A assume "A \<in> prod_algebra I M" |
|
410 |
from prod_algebraE[OF this] guess J X . note X = this |
|
50123
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hoelzl
parents:
50104
diff
changeset
|
411 |
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
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diff
changeset
|
412 |
using space by (auto simp: prod_emb_def del: PiE_I) |
47694 | 413 |
also have "\<dots> \<in> sets N" using X(3,2,4,5) by (rule sets) |
414 |
finally show "f -` A \<inter> space N \<in> sets N" . |
|
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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changeset
|
415 |
qed |
41095 | 416 |
|
47694 | 417 |
lemma measurable_PiM_single: |
53015
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wenzelm
parents:
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changeset
|
418 |
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))" |
47694 | 419 |
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" |
420 |
shows "f \<in> measurable N (PiM I M)" |
|
421 |
using sets_PiM_single |
|
422 |
proof (rule measurable_sigma_sets) |
|
53015
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standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
423 |
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
|
424 |
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 | 425 |
by auto |
426 |
with space have "f -` A \<inter> space N = {\<omega> \<in> space N. f \<omega> i \<in> B}" by auto |
|
427 |
also have "\<dots> \<in> sets N" using B by (rule sets) |
|
428 |
finally show "f -` A \<inter> space N \<in> sets N" . |
|
429 |
qed (auto simp: space) |
|
40859 | 430 |
|
50099 | 431 |
lemma measurable_PiM_single': |
432 |
assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)" |
|
53015
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wenzelm
parents:
50387
diff
changeset
|
433 |
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
|
434 |
shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)" |
50099 | 435 |
proof (rule measurable_PiM_single) |
436 |
fix A i assume A: "i \<in> I" "A \<in> sets (M i)" |
|
437 |
then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N" |
|
438 |
by auto |
|
439 |
then show "{\<omega> \<in> space N. f i \<omega> \<in> A} \<in> sets N" |
|
440 |
using A f by (auto intro!: measurable_sets) |
|
441 |
qed fact |
|
442 |
||
50003 | 443 |
lemma sets_PiM_I_finite[measurable]: |
47694 | 444 |
assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))" |
445 |
shows "(PIE j:I. E j) \<in> sets (PIM i:I. M i)" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
446 |
using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] `finite I` sets by auto |
47694 | 447 |
|
50021
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add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
448 |
lemma measurable_component_singleton: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
449 |
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
|
450 |
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
|
451 |
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
|
452 |
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)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
453 |
using sets.sets_into_space `i \<in> I` |
47694 | 454 |
by (fastforce dest: Pi_mem simp: prod_emb_def space_PiM split: split_if_asm) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
455 |
then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^sub>M I M) \<in> sets (Pi\<^sub>M I M)" |
47694 | 456 |
using `A \<in> sets (M i)` `i \<in> I` by (auto intro!: sets_PiM_I) |
457 |
qed (insert `i \<in> I`, auto simp: space_PiM) |
|
458 |
||
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
459 |
lemma measurable_component_singleton'[measurable_app]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
460 |
assumes f: "f \<in> measurable N (Pi\<^sub>M I M)" |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
461 |
assumes i: "i \<in> I" |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
462 |
shows "(\<lambda>x. (f x) i) \<in> measurable N (M i)" |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
463 |
using measurable_compose[OF f measurable_component_singleton, OF i] . |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
464 |
|
50099 | 465 |
lemma measurable_PiM_component_rev[measurable (raw)]: |
466 |
"i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N" |
|
467 |
by simp |
|
468 |
||
55415 | 469 |
lemma measurable_case_nat[measurable (raw)]: |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
470 |
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
|
471 |
"\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N" |
55415 | 472 |
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
|
473 |
by (cases i) simp_all |
59048 | 474 |
|
55415 | 475 |
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
|
476 |
assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)" |
55415 | 477 |
shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)" |
50099 | 478 |
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
|
479 |
by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split) |
50099 | 480 |
|
50003 | 481 |
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
|
482 |
"(\<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 | 483 |
(is "?f \<in> measurable ?P ?I") |
484 |
proof (rule measurable_PiM_single) |
|
485 |
fix j A assume j: "j \<in> insert i I" and A: "A \<in> sets (M j)" |
|
486 |
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
|
487 |
(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
|
488 |
using sets.sets_into_space[OF A] by (auto simp add: space_pair_measure space_PiM) |
47694 | 489 |
also have "\<dots> \<in> sets ?P" |
490 |
using A j |
|
491 |
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
|
492 |
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
|
493 |
qed (auto simp: space_pair_measure space_PiM PiE_def) |
41661 | 494 |
|
50003 | 495 |
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
|
496 |
"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 | 497 |
by simp |
498 |
||
499 |
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
|
500 |
"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 | 501 |
(is "?f \<in> measurable ?P ?U") |
502 |
proof (rule measurable_PiM_single) |
|
503 |
fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J" |
|
49780 | 504 |
then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} = |
47694 | 505 |
(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 | 506 |
by (auto simp: merge_def) |
47694 | 507 |
also have "\<dots> \<in> sets ?P" |
508 |
using A |
|
509 |
by (auto intro!: measurable_sets[OF measurable_comp, OF _ measurable_component_singleton]) |
|
49780 | 510 |
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
|
511 |
qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def) |
42988 | 512 |
|
50003 | 513 |
lemma measurable_restrict[measurable (raw)]: |
47694 | 514 |
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
|
515 |
shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)" |
47694 | 516 |
proof (rule measurable_PiM_single) |
517 |
fix A i assume A: "i \<in> I" "A \<in> sets (M i)" |
|
518 |
then have "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} = X i -` A \<inter> space N" |
|
519 |
by auto |
|
520 |
then show "{\<omega> \<in> space N. (\<lambda>i\<in>I. X i \<omega>) i \<in> A} \<in> sets N" |
|
521 |
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
|
522 |
qed (insert X, auto simp add: PiE_def dest: measurable_space) |
47694 | 523 |
|
57025 | 524 |
lemma measurable_abs_UNIV: |
525 |
"(\<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)" |
|
526 |
by (intro measurable_PiM_single) (auto dest: measurable_space) |
|
527 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
528 |
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 | 529 |
by (intro measurable_restrict measurable_component_singleton) auto |
530 |
||
531 |
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
|
532 |
"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 | 533 |
unfolding prod_emb_def space_PiM[symmetric] |
534 |
by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton) |
|
535 |
||
50003 | 536 |
lemma sets_in_Pi_aux: |
537 |
"finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow> |
|
538 |
{x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)" |
|
539 |
by (simp add: subset_eq Pi_iff) |
|
540 |
||
541 |
lemma sets_in_Pi[measurable (raw)]: |
|
542 |
"finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow> |
|
543 |
(\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow> |
|
50387 | 544 |
Measurable.pred N (\<lambda>x. f x \<in> Pi I F)" |
50003 | 545 |
unfolding pred_def |
546 |
by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto |
|
547 |
||
548 |
lemma sets_in_extensional_aux: |
|
549 |
"{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)" |
|
550 |
proof - |
|
551 |
have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)" |
|
552 |
by (auto simp add: extensional_def space_PiM) |
|
553 |
then show ?thesis by simp |
|
554 |
qed |
|
555 |
||
556 |
lemma sets_in_extensional[measurable (raw)]: |
|
50387 | 557 |
"f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)" |
50003 | 558 |
unfolding pred_def |
559 |
by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto |
|
560 |
||
47694 | 561 |
locale product_sigma_finite = |
562 |
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
|
563 |
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)" |
40859 | 564 |
|
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
|
565 |
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i |
40859 | 566 |
by (rule sigma_finite_measures) |
567 |
||
47694 | 568 |
locale finite_product_sigma_finite = product_sigma_finite M for M :: "'i \<Rightarrow> 'a measure" + |
569 |
fixes I :: "'i set" |
|
570 |
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
|
571 |
|
40859 | 572 |
lemma (in finite_product_sigma_finite) sigma_finite_pairs: |
573 |
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set. |
|
574 |
(\<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
|
575 |
(\<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
|
576 |
(\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)" |
40859 | 577 |
proof - |
47694 | 578 |
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>)" |
579 |
using M.sigma_finite_incseq by metis |
|
40859 | 580 |
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" .. |
47694 | 581 |
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 | 582 |
by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
583 |
let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k" |
47694 | 584 |
note space_PiM[simp] |
40859 | 585 |
show ?thesis |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
586 |
proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI) |
40859 | 587 |
fix i show "range (F i) \<subseteq> sets (M i)" by fact |
588 |
next |
|
47694 | 589 |
fix i k show "emeasure (M i) (F i k) \<noteq> \<infinity>" by fact |
40859 | 590 |
next |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
591 |
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
|
592 |
by (auto simp: PiE_def dest!: sets.sets_into_space) |
40859 | 593 |
next |
47694 | 594 |
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
|
595 |
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
|
596 |
show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def PiE_def) |
40859 | 597 |
next |
598 |
fix i show "?F i \<subseteq> ?F (Suc i)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
599 |
using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto |
40859 | 600 |
qed |
601 |
qed |
|
602 |
||
49780 | 603 |
lemma |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
604 |
shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
605 |
and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }" |
49780 | 606 |
by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq) |
607 |
||
608 |
lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1" |
|
609 |
proof - |
|
610 |
let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ereal)" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
611 |
have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1" |
49780 | 612 |
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) |
613 |
show "positive (PiM {} M) ?\<mu>" |
|
614 |
by (auto simp: positive_def) |
|
615 |
show "countably_additive (PiM {} M) ?\<mu>" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
616 |
by (rule sets.countably_additiveI_finite) |
49780 | 617 |
(auto simp: additive_def positive_def sets_PiM_empty space_PiM_empty intro!: ) |
618 |
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
|
619 |
also have "(prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = {\<lambda>_. undefined}" |
49780 | 620 |
by (auto simp: prod_emb_def) |
621 |
finally show ?thesis |
|
622 |
by simp |
|
623 |
qed |
|
624 |
||
625 |
lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}" |
|
626 |
by (rule measure_eqI) (auto simp add: sets_PiM_empty one_ereal_def) |
|
627 |
||
49776 | 628 |
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
|
629 |
"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))" |
49776 | 630 |
proof (induct I arbitrary: A rule: finite_induct) |
40859 | 631 |
case (insert i I) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
632 |
interpret finite_product_sigma_finite M I by default fact |
40859 | 633 |
have "finite (insert i I)" using `finite I` by auto |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
634 |
interpret I': finite_product_sigma_finite M "insert i I" by default fact |
41661 | 635 |
let ?h = "(\<lambda>(f, y). f(i := y))" |
47694 | 636 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
637 |
let ?P = "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M) ?h" |
47694 | 638 |
let ?\<mu> = "emeasure ?P" |
639 |
let ?I = "{j \<in> insert i I. emeasure (M j) (space (M j)) \<noteq> 1}" |
|
640 |
let ?f = "\<lambda>J E j. if j \<in> J then emeasure (M j) (E j) else emeasure (M j) (space (M j))" |
|
641 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
642 |
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 | 643 |
(\<Prod>i\<in>insert i I. emeasure (M i) (A i))" |
644 |
proof (subst emeasure_extend_measure_Pair[OF PiM_def]) |
|
645 |
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))" |
|
646 |
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
|
647 |
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
|
648 |
let ?p' = "prod_emb I M (J - {i}) (\<Pi>\<^sub>E j\<in>J-{i}. E j)" |
49776 | 649 |
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
|
650 |
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 | 651 |
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
|
652 |
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
|
653 |
using J E[rule_format, 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
|
654 |
by (force simp: space_pair_measure space_PiM prod_emb_iff PiE_def Pi_iff split: split_if_asm) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
655 |
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
|
656 |
emeasure (Pi\<^sub>M I M) ?p' * emeasure (M i) (if i \<in> J then (E i) else space (M i))" |
49776 | 657 |
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
|
658 |
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
|
659 |
using J E[rule_format, 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
|
660 |
by (auto simp: prod_emb_iff PiE_def Pi_iff split: split_if_asm) blast+ |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
661 |
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 | 662 |
(\<Prod> j\<in>I. if j \<in> J-{i} then emeasure (M j) (E j) else emeasure (M j) (space (M j)))" |
57418 | 663 |
using E by (subst insert) (auto intro!: setprod.cong) |
49776 | 664 |
also have "(\<Prod>j\<in>I. if j \<in> J - {i} then emeasure (M j) (E j) else emeasure (M j) (space (M j))) * |
665 |
emeasure (M i) (if i \<in> J then E i else space (M i)) = (\<Prod>j\<in>insert i I. ?f J E j)" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57447
diff
changeset
|
666 |
using insert by (auto simp: mult.commute intro!: arg_cong2[where f="op *"] setprod.cong) |
49776 | 667 |
also have "\<dots> = (\<Prod>j\<in>J \<union> ?I. ?f J E j)" |
57418 | 668 |
using insert(1,2) J E by (intro setprod.mono_neutral_right) auto |
49776 | 669 |
finally show "?\<mu> ?p = \<dots>" . |
47694 | 670 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
671 |
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
|
672 |
using J E[rule_format, THEN sets.sets_into_space] by (auto simp: prod_emb_iff PiE_def) |
49776 | 673 |
next |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
674 |
show "positive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" "countably_additive (sets (Pi\<^sub>M (insert i I) M)) ?\<mu>" |
49776 | 675 |
using emeasure_positive[of ?P] emeasure_countably_additive[of ?P] by simp_all |
676 |
next |
|
677 |
show "(insert i I \<noteq> {} \<or> insert i I = {}) \<and> finite (insert i I) \<and> |
|
678 |
insert i I \<subseteq> insert i I \<and> A \<in> (\<Pi> j\<in>insert i I. sets (M j))" |
|
679 |
using insert by auto |
|
57418 | 680 |
qed (auto intro!: setprod.cong) |
49776 | 681 |
with insert show ?case |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
682 |
by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space) |
50003 | 683 |
qed simp |
47694 | 684 |
|
49776 | 685 |
lemma (in product_sigma_finite) sigma_finite: |
686 |
assumes "finite I" |
|
687 |
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
|
688 |
proof |
49776 | 689 |
interpret finite_product_sigma_finite M I by default fact |
690 |
||
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
691 |
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
|
692 |
"\<And>j f. f \<in> F j \<Longrightarrow> emeasure (M j) f \<noteq> \<infinity>" and |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
693 |
in_space: "\<And>j. space (M j) = (\<Union>F j)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
694 |
using sigma_finite_countable by (metis subset_eq) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
695 |
moreover have "(\<Union>(PiE I ` PiE I F)) = space (Pi\<^sub>M I M)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
696 |
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
|
697 |
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>)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
698 |
by (intro exI[of _ "PiE I ` PiE I F"]) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
699 |
(auto intro!: countable_PiE sets_PiM_I_finite |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
700 |
simp: PiE_iff emeasure_PiM finite_index setprod_PInf emeasure_nonneg) |
40859 | 701 |
qed |
702 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
703 |
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>M I M" |
47694 | 704 |
using sigma_finite[OF finite_index] . |
40859 | 705 |
|
706 |
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
|
707 |
"(\<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 | 708 |
using emeasure_PiM[OF finite_index] by auto |
41096 | 709 |
|
56996 | 710 |
lemma (in product_sigma_finite) nn_integral_empty: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
711 |
assumes pos: "0 \<le> f (\<lambda>k. undefined)" |
56996 | 712 |
shows "integral\<^sup>N (Pi\<^sub>M {} M) f = f (\<lambda>k. undefined)" |
40859 | 713 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
714 |
interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
715 |
have "\<And>A. emeasure (Pi\<^sub>M {} M) (Pi\<^sub>E {} A) = 1" |
40859 | 716 |
using assms by (subst measure_times) auto |
717 |
then show ?thesis |
|
56996 | 718 |
unfolding nn_integral_def simple_function_def simple_integral_def[abs_def] |
47694 | 719 |
proof (simp add: space_PiM_empty sets_PiM_empty, intro antisym) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
720 |
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
721 |
by (intro SUP_upper) (auto simp: le_fun_def split: split_max) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
722 |
show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
723 |
by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm) |
40859 | 724 |
qed |
725 |
qed |
|
726 |
||
47694 | 727 |
lemma (in product_sigma_finite) distr_merge: |
40859 | 728 |
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
|
729 |
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 | 730 |
(is "?D = ?P") |
40859 | 731 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
732 |
interpret I: finite_product_sigma_finite M I by default fact |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
733 |
interpret J: finite_product_sigma_finite M J by default fact |
40859 | 734 |
have "finite (I \<union> J)" using fin by auto |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
735 |
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
736 |
interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default |
49780 | 737 |
let ?g = "merge I J" |
47694 | 738 |
|
41661 | 739 |
from IJ.sigma_finite_pairs obtain F where |
740 |
F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> 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
|
741 |
"incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k)" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
742 |
"(\<Union>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k) = space ?P" |
47694 | 743 |
"\<And>k. \<forall>i\<in>I\<union>J. emeasure (M i) (F i k) \<noteq> \<infinity>" |
41661 | 744 |
by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
745 |
let ?F = "\<lambda>k. \<Pi>\<^sub>E i\<in>I \<union> J. F i k" |
47694 | 746 |
|
747 |
show ?thesis |
|
748 |
proof (rule measure_eqI_generator_eq[symmetric]) |
|
749 |
show "Int_stable (prod_algebra (I \<union> J) M)" |
|
750 |
by (rule Int_stable_prod_algebra) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
751 |
show "prod_algebra (I \<union> J) M \<subseteq> Pow (\<Pi>\<^sub>E i \<in> I \<union> J. space (M i))" |
47694 | 752 |
by (rule 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
|
753 |
show "sets ?P = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)" |
47694 | 754 |
by (rule sets_PiM) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
755 |
then show "sets ?D = sigma_sets (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i)) (prod_algebra (I \<union> J) M)" |
47694 | 756 |
by simp |
757 |
||
758 |
show "range ?F \<subseteq> prod_algebra (I \<union> J) M" using F |
|
759 |
using fin by (auto simp: prod_algebra_eq_finite) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
760 |
show "(\<Union>i. \<Pi>\<^sub>E ia\<in>I \<union> J. F ia i) = (\<Pi>\<^sub>E i\<in>I \<union> J. space (M i))" |
47694 | 761 |
using F(3) by (simp add: space_PiM) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
762 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
763 |
fix k |
47694 | 764 |
from F `finite I` setprod_PInf[of "I \<union> J", OF emeasure_nonneg, of M] |
765 |
show "emeasure ?P (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto |
|
41661 | 766 |
next |
47694 | 767 |
fix A assume A: "A \<in> prod_algebra (I \<union> J) M" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
768 |
with fin obtain F where A_eq: "A = (Pi\<^sub>E (I \<union> J) F)" and F: "\<forall>i\<in>J. F i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)" |
47694 | 769 |
by (auto simp add: prod_algebra_eq_finite) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
770 |
let ?B = "Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M" |
47694 | 771 |
let ?X = "?g -` A \<inter> space ?B" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
772 |
have "Pi\<^sub>E I F \<subseteq> space (Pi\<^sub>M I M)" "Pi\<^sub>E J F \<subseteq> space (Pi\<^sub>M J M)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
773 |
using F[rule_format, THEN sets.sets_into_space] by (force simp: space_PiM)+ |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
774 |
then have X: "?X = (Pi\<^sub>E I F \<times> Pi\<^sub>E J F)" |
47694 | 775 |
unfolding A_eq by (subst merge_vimage) (auto simp: space_pair_measure space_PiM) |
776 |
have "emeasure ?D A = emeasure ?B ?X" |
|
777 |
using A by (intro emeasure_distr measurable_merge) (auto simp: sets_PiM) |
|
778 |
also have "emeasure ?B ?X = (\<Prod>i\<in>I. emeasure (M i) (F i)) * (\<Prod>i\<in>J. emeasure (M i) (F i))" |
|
50003 | 779 |
using `finite J` `finite I` F unfolding X |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
780 |
by (simp add: J.emeasure_pair_measure_Times I.measure_times J.measure_times) |
47694 | 781 |
also have "\<dots> = (\<Prod>i\<in>I \<union> J. emeasure (M i) (F i))" |
57418 | 782 |
using `finite J` `finite I` `I \<inter> J = {}` by (simp add: setprod.union_inter_neutral) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
783 |
also have "\<dots> = emeasure ?P (Pi\<^sub>E (I \<union> J) F)" |
41661 | 784 |
using `finite J` `finite I` F unfolding A |
785 |
by (intro IJ.measure_times[symmetric]) auto |
|
47694 | 786 |
finally show "emeasure ?P A = emeasure ?D A" using A_eq by simp |
787 |
qed |
|
41661 | 788 |
qed |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
789 |
|
56996 | 790 |
lemma (in product_sigma_finite) product_nn_integral_fold: |
47694 | 791 |
assumes IJ: "I \<inter> J = {}" "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
|
792 |
and f: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)" |
56996 | 793 |
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
|
794 |
(\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^sub>M J M)) \<partial>(Pi\<^sub>M I M))" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
795 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
796 |
interpret I: finite_product_sigma_finite M I by default fact |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
797 |
interpret J: finite_product_sigma_finite M J by default fact |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
798 |
interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
799 |
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 | 800 |
using measurable_comp[OF measurable_merge f] by (simp add: comp_def) |
41661 | 801 |
show ?thesis |
47694 | 802 |
apply (subst distr_merge[OF IJ, symmetric]) |
56996 | 803 |
apply (subst nn_integral_distr[OF measurable_merge f]) |
804 |
apply (subst J.nn_integral_fst[symmetric, OF P_borel]) |
|
47694 | 805 |
apply simp |
806 |
done |
|
40859 | 807 |
qed |
808 |
||
47694 | 809 |
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
|
810 |
"distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _") |
47694 | 811 |
proof (intro measure_eqI[symmetric]) |
41831 | 812 |
interpret I: finite_product_sigma_finite M "{i}" by default simp |
47694 | 813 |
fix A assume A: "A \<in> sets (M i)" |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
814 |
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
|
815 |
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
|
816 |
then show "emeasure (M i) A = emeasure ?D A" |
47694 | 817 |
using A I.measure_times[of "\<lambda>_. A"] |
818 |
by (simp add: emeasure_distr measurable_component_singleton) |
|
819 |
qed simp |
|
41831 | 820 |
|
56996 | 821 |
lemma (in product_sigma_finite) product_nn_integral_singleton: |
40859 | 822 |
assumes f: "f \<in> borel_measurable (M i)" |
56996 | 823 |
shows "integral\<^sup>N (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>N (M i) f" |
40859 | 824 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
825 |
interpret I: finite_product_sigma_finite M "{i}" by default simp |
47694 | 826 |
from f show ?thesis |
827 |
apply (subst distr_singleton[symmetric]) |
|
56996 | 828 |
apply (subst nn_integral_distr[OF measurable_component_singleton]) |
47694 | 829 |
apply simp_all |
830 |
done |
|
40859 | 831 |
qed |
832 |
||
56996 | 833 |
lemma (in product_sigma_finite) product_nn_integral_insert: |
49780 | 834 |
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
|
835 |
and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)" |
56996 | 836 |
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))" |
41096 | 837 |
proof - |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
838 |
interpret I: finite_product_sigma_finite M I by default auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
839 |
interpret i: finite_product_sigma_finite M "{i}" by default auto |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
840 |
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
|
841 |
using f by auto |
41096 | 842 |
show ?thesis |
56996 | 843 |
unfolding product_nn_integral_fold[OF IJ, unfolded insert, OF I(1) i.finite_index f] |
844 |
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
|
845 |
fix x assume x: "x \<in> space (Pi\<^sub>M I M)" |
49780 | 846 |
let ?f = "\<lambda>y. f (x(i := y))" |
847 |
show "?f \<in> borel_measurable (M i)" |
|
47694 | 848 |
using measurable_comp[OF measurable_component_update f, OF x `i \<notin> I`] |
849 |
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
|
850 |
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 | 851 |
using x |
56996 | 852 |
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
|
853 |
simp add: space_PiM extensional_def PiE_def) |
41096 | 854 |
qed |
855 |
qed |
|
856 |
||
56996 | 857 |
lemma (in product_sigma_finite) product_nn_integral_setprod: |
43920 | 858 |
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal" |
41096 | 859 |
assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
860 |
and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x" |
56996 | 861 |
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))" |
41096 | 862 |
using assms proof induct |
863 |
case (insert i I) |
|
864 |
note `finite I`[intro, simp] |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
865 |
interpret I: finite_product_sigma_finite M I by default auto |
41096 | 866 |
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))" |
57418 | 867 |
using insert by (auto intro!: setprod.cong) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
868 |
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
|
869 |
using sets.sets_into_space insert |
47694 | 870 |
by (intro borel_measurable_ereal_setprod |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
871 |
measurable_comp[OF measurable_component_singleton, unfolded comp_def]) |
41096 | 872 |
auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
873 |
then show ?case |
56996 | 874 |
apply (simp add: product_nn_integral_insert[OF insert(1,2) prod]) |
875 |
apply (simp add: insert(2-) * pos borel setprod_ereal_pos nn_integral_multc) |
|
876 |
apply (subst nn_integral_cmult) |
|
877 |
apply (auto simp add: pos borel insert(2-) setprod_ereal_pos nn_integral_nonneg) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
878 |
done |
47694 | 879 |
qed (simp add: space_PiM) |
41096 | 880 |
|
50104 | 881 |
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
|
882 |
"distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P") |
50104 | 883 |
proof (intro measure_eqI[symmetric]) |
884 |
interpret I: finite_product_sigma_finite M "{i}" by default simp |
|
885 |
||
886 |
have eq: "\<And>x. x \<in> extensional {i} \<Longrightarrow> (\<lambda>j\<in>{i}. x i) = x" |
|
887 |
by (auto simp: extensional_def restrict_def) |
|
888 |
||
59048 | 889 |
have [measurable]: "\<And>j. j \<in> {i} \<Longrightarrow> (\<lambda>x. x) \<in> measurable (M i) (M j)" by simp |
890 |
||
50104 | 891 |
fix A assume A: "A \<in> sets ?P" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
892 |
then have "emeasure ?P A = (\<integral>\<^sup>+x. indicator A x \<partial>?P)" |
50104 | 893 |
by simp |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
894 |
also have "\<dots> = (\<integral>\<^sup>+x. indicator ((\<lambda>x. \<lambda>i\<in>{i}. x) -` A \<inter> space (M i)) (x i) \<partial>PiM {i} M)" |
56996 | 895 |
by (intro nn_integral_cong) (auto simp: space_PiM indicator_def PiE_def eq) |
50104 | 896 |
also have "\<dots> = emeasure ?D A" |
56996 | 897 |
using A by (simp add: product_nn_integral_singleton emeasure_distr) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
898 |
finally show "emeasure (Pi\<^sub>M {i} M) A = emeasure ?D A" . |
50104 | 899 |
qed simp |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
900 |
|
49776 | 901 |
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
|
902 |
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)" |
49776 | 903 |
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
|
904 |
"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 | 905 |
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
|
906 |
"(\<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 | 907 |
proof - |
908 |
interpret I: finite_product_sigma_finite M I by default fact |
|
909 |
interpret J: finite_product_sigma_finite M J by default fact |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
910 |
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
|
911 |
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 | 912 |
by (intro measurable_sets[OF _ A] measurable_merge assms) |
913 |
||
914 |
show ?I |
|
915 |
apply (subst distr_merge[symmetric, OF IJ]) |
|
916 |
apply (subst emeasure_distr[OF measurable_merge A]) |
|
917 |
apply (subst J.emeasure_pair_measure_alt[OF merge]) |
|
56996 | 918 |
apply (auto intro!: nn_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure) |
49776 | 919 |
done |
920 |
||
921 |
show ?B |
|
922 |
using IJ.measurable_emeasure_Pair1[OF merge] |
|
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
55415
diff
changeset
|
923 |
by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong) |
49776 | 924 |
qed |
925 |
||
926 |
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
|
927 |
"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 | 928 |
by simp |
49776 | 929 |
|
930 |
lemma sigma_prod_algebra_sigma_eq_infinite: |
|
931 |
fixes E :: "'i \<Rightarrow> 'a set set" |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
932 |
assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" |
49776 | 933 |
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" |
934 |
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" |
|
935 |
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
936 |
defines "P == {{f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> E i}" |
49776 | 937 |
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P" |
938 |
proof |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
939 |
let ?P = "sigma (space (Pi\<^sub>M I M)) P" |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
940 |
have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>M I M))" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
941 |
using E_closed by (auto simp: space_PiM P_def subset_eq) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
942 |
then have space_P: "space ?P = (\<Pi>\<^sub>E i\<in>I. space (M i))" |
49776 | 943 |
by (simp add: space_PiM) |
944 |
have "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
|
945 |
sigma_sets (space ?P) {{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)}" |
49776 | 946 |
using sets_PiM_single[of I M] by (simp add: space_P) |
947 |
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
948 |
proof (safe intro!: sets.sigma_sets_subset) |
49776 | 949 |
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" |
950 |
then have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))" |
|
951 |
apply (subst measurable_iff_measure_of) |
|
952 |
apply (simp_all add: P_closed) |
|
953 |
using E_closed |
|
954 |
apply (force simp: subset_eq space_PiM) |
|
955 |
apply (force simp: subset_eq space_PiM) |
|
956 |
apply (auto simp: P_def intro!: sigma_sets.Basic exI[of _ i]) |
|
957 |
apply (rule_tac x=Aa in exI) |
|
958 |
apply (auto simp: space_PiM) |
|
959 |
done |
|
960 |
from measurable_sets[OF this, of A] A `i \<in> I` E_closed |
|
961 |
have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
962 |
by (simp add: E_generates) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
963 |
also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A}" |
49776 | 964 |
using P_closed by (auto simp: space_PiM) |
965 |
finally show "\<dots> \<in> sets ?P" . |
|
966 |
qed |
|
967 |
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P" |
|
968 |
by (simp add: P_closed) |
|
969 |
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)" |
|
970 |
unfolding P_def space_PiM[symmetric] |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
971 |
by (intro sets.sigma_sets_subset) (auto simp: E_generates sets_Collect_single) |
49776 | 972 |
qed |
973 |
||
47694 | 974 |
lemma sigma_prod_algebra_sigma_eq: |
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
975 |
fixes E :: "'i \<Rightarrow> 'a set set" and S :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" |
47694 | 976 |
assumes "finite I" |
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
977 |
assumes S_union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (M i)" |
47694 | 978 |
and S_in_E: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i" |
979 |
assumes E_closed: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))" |
|
980 |
and E_generates: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
981 |
defines "P == { Pi\<^sub>E I F | F. \<forall>i\<in>I. F i \<in> E i }" |
47694 | 982 |
shows "sets (PiM I M) = sigma_sets (space (PiM I M)) P" |
983 |
proof |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
984 |
let ?P = "sigma (space (Pi\<^sub>M I M)) P" |
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
985 |
from `finite I`[THEN ex_bij_betw_finite_nat] guess T .. |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
986 |
then have T: "\<And>i. i \<in> I \<Longrightarrow> T i < card I" "\<And>i. i\<in>I \<Longrightarrow> the_inv_into I T (T i) = i" |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
987 |
by (auto simp add: bij_betw_def set_eq_iff image_iff the_inv_into_f_f) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
988 |
have P_closed: "P \<subseteq> Pow (space (Pi\<^sub>M I M))" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
989 |
using E_closed by (auto simp: space_PiM P_def subset_eq) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
990 |
then have space_P: "space ?P = (\<Pi>\<^sub>E i\<in>I. space (M i))" |
47694 | 991 |
by (simp add: space_PiM) |
992 |
have "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
|
993 |
sigma_sets (space ?P) {{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 | 994 |
using sets_PiM_single[of I M] by (simp add: space_P) |
995 |
also have "\<dots> \<subseteq> sets (sigma (space (PiM I M)) P)" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
996 |
proof (safe intro!: sets.sigma_sets_subset) |
47694 | 997 |
fix i A assume "i \<in> I" and A: "A \<in> sets (M i)" |
998 |
have "(\<lambda>x. x i) \<in> measurable ?P (sigma (space (M i)) (E i))" |
|
999 |
proof (subst measurable_iff_measure_of) |
|
1000 |
show "E i \<subseteq> Pow (space (M i))" using `i \<in> I` by fact |
|
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
1001 |
from space_P `i \<in> I` show "(\<lambda>x. x i) \<in> space ?P \<rightarrow> space (M i)" by auto |
47694 | 1002 |
show "\<forall>A\<in>E i. (\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
1003 |
proof |
|
1004 |
fix A assume A: "A \<in> E i" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1005 |
then have "(\<lambda>x. x i) -` A \<inter> space ?P = (\<Pi>\<^sub>E j\<in>I. if i = j then A else space (M j))" |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
1006 |
using E_closed `i \<in> I` by (auto simp: space_P subset_eq split: split_if_asm) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1007 |
also have "\<dots> = (\<Pi>\<^sub>E j\<in>I. \<Union>n. if i = j then A else S j n)" |
47694 | 1008 |
by (intro PiE_cong) (simp add: S_union) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1009 |
also have "\<dots> = (\<Union>xs\<in>{xs. length xs = card I}. \<Pi>\<^sub>E j\<in>I. if i = j then A else S j (xs ! T j))" |
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1010 |
using T |
50123
69b35a75caf3
merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents:
50104
diff
changeset
|
1011 |
apply (auto simp: PiE_iff bchoice_iff) |
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1012 |
apply (rule_tac x="map (\<lambda>n. f (the_inv_into I T n)) [0..<card I]" in exI) |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1013 |
apply (auto simp: bij_betw_def) |
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1014 |
done |
47694 | 1015 |
also have "\<dots> \<in> sets ?P" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
1016 |
proof (safe intro!: sets.countable_UN) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1017 |
fix xs show "(\<Pi>\<^sub>E j\<in>I. if i = j then A else S j (xs ! T j)) \<in> sets ?P" |
47694 | 1018 |
using A S_in_E |
1019 |
by (simp add: P_closed) |
|
49779
1484b4b82855
remove incseq assumption from sigma_prod_algebra_sigma_eq
hoelzl
parents:
49776
diff
changeset
|
1020 |
(auto simp: P_def subset_eq intro!: exI[of _ "\<lambda>j. if i = j then A else S j (xs ! T j)"]) |
47694 | 1021 |
qed |
1022 |
finally show "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
1023 |
using P_closed by simp |
|
1024 |
qed |
|
1025 |
qed |
|
1026 |
from measurable_sets[OF this, of A] A `i \<in> I` E_closed |
|
1027 |
have "(\<lambda>x. x i) -` A \<inter> space ?P \<in> sets ?P" |
|
1028 |
by (simp add: E_generates) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
50387
diff
changeset
|
1029 |
also have "(\<lambda>x. x i) -` A \<inter> space ?P = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A}" |
47694 | 1030 |
using P_closed by (auto simp: space_PiM) |
1031 |
finally show "\<dots> \<in> sets ?P" . |
|
1032 |
qed |
|
1033 |
finally show "sets (PiM I M) \<subseteq> sigma_sets (space (PiM I M)) P" |
|
1034 |
by (simp add: P_closed) |
|
1035 |
show "sigma_sets (space (PiM I M)) P \<subseteq> sets (PiM I M)" |
|
1036 |
using `finite I` |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50123
diff
changeset
|
1037 |
by (auto intro!: sets.sigma_sets_subset sets_PiM_I_finite simp: E_generates P_def) |
47694 | 1038 |
qed |
1039 |
||
50104 | 1040 |
lemma pair_measure_eq_distr_PiM: |
1041 |
fixes M1 :: "'a measure" and M2 :: "'a measure" |
|
1042 |
assumes "sigma_finite_measure M1" "sigma_finite_measure M2" |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1043 |
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 | 1044 |
(is "?P = ?D") |
1045 |
proof (rule pair_measure_eqI[OF assms]) |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1046 |
interpret B: product_sigma_finite "case_bool M1 M2" |
50104 | 1047 |
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
|
1048 |
let ?B = "Pi\<^sub>M UNIV (case_bool M1 M2)" |
50104 | 1049 |
|
1050 |
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)" |
|
1051 |
by auto |
|
1052 |
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
|
1053 |
have "emeasure M1 A * emeasure M2 B = (\<Prod> i\<in>UNIV. emeasure (case_bool M1 M2 i) (case_bool A B i))" |
50104 | 1054 |
by (simp add: UNIV_bool ac_simps) |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1055 |
also have "\<dots> = emeasure ?B (Pi\<^sub>E UNIV (case_bool A B))" |
50104 | 1056 |
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
|
1057 |
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
|
1058 |
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
|
1059 |
by (auto simp: PiE_iff all_bool_eq space_PiM split: bool.split) |
50104 | 1060 |
finally show "emeasure M1 A * emeasure M2 B = emeasure ?D (A \<times> B)" |
1061 |
using A B |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1062 |
measurable_component_singleton[of True UNIV "case_bool M1 M2"] |
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
53374
diff
changeset
|
1063 |
measurable_component_singleton[of False UNIV "case_bool M1 M2"] |
50104 | 1064 |
by (subst emeasure_distr) (auto simp: measurable_pair_iff) |
1065 |
qed simp |
|
1066 |
||
47694 | 1067 |
end |