author | Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk> |
Tue, 22 Jan 2019 22:57:16 +0000 | |
changeset 69722 | b5163b2132c5 |
parent 69683 | 8b3458ca0762 |
child 69739 | 8b47c021666e |
permissions | -rw-r--r-- |
63627 | 1 |
(* Title: HOL/Analysis/Borel_Space.thy |
42067 | 2 |
Author: Johannes Hölzl, TU München |
3 |
Author: Armin Heller, TU München |
|
4 |
*) |
|
38656 | 5 |
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69722
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minor tagging updates in 13 theories
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6 |
section \<open>Borel Space\<close> |
33533
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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7 |
|
40859 | 8 |
theory Borel_Space |
50387 | 9 |
imports |
63626
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move measure theory from HOL-Probability to HOL-Multivariate_Analysis
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parents:
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diff
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10 |
Measurable Derivative Ordered_Euclidean_Space Extended_Real_Limits |
33533
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New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
11 |
begin |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
12 |
|
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parents:
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13 |
lemma sets_Collect_eventually_sequentially[measurable]: |
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generalized some Borel measurable statements to support ennreal
hoelzl
parents:
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14 |
"(\<And>i. {x\<in>space M. P x i} \<in> sets M) \<Longrightarrow> {x\<in>space M. eventually (P x) sequentially} \<in> sets M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
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parents:
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15 |
unfolding eventually_sequentially by simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
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16 |
|
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parents:
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17 |
lemma topological_basis_trivial: "topological_basis {A. open A}" |
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generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
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parents:
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18 |
by (auto simp: topological_basis_def) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
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19 |
|
69722
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minor tagging updates in 13 theories
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69683
diff
changeset
|
20 |
proposition open_prod_generated: "open = generate_topology {A \<times> B | A B. open A \<and> open B}" |
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parents:
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21 |
proof - |
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generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
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parents:
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22 |
have "{A \<times> B :: ('a \<times> 'b) set | A B. open A \<and> open B} = ((\<lambda>(a, b). a \<times> b) ` ({A. open A} \<times> {A. open A}))" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
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23 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
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24 |
then show ?thesis |
62372 | 25 |
by (auto intro: topological_basis_prod topological_basis_trivial topological_basis_imp_subbasis) |
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generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
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26 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
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diff
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27 |
|
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tagged 21 theories in the Analysis library for the manual
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
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28 |
definition%important "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s" |
62083 | 29 |
|
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parents:
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30 |
lemma mono_onI: |
62083 | 31 |
"(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r \<le> s \<Longrightarrow> f r \<le> f s) \<Longrightarrow> mono_on f A" |
32 |
unfolding mono_on_def by simp |
|
33 |
||
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parents:
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lemma mono_onD: |
62083 | 35 |
"\<lbrakk>mono_on f A; r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s" |
36 |
unfolding mono_on_def by simp |
|
37 |
||
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parents:
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38 |
lemma mono_imp_mono_on: "mono f \<Longrightarrow> mono_on f A" |
62083 | 39 |
unfolding mono_def mono_on_def by auto |
40 |
||
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parents:
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41 |
lemma mono_on_subset: "mono_on f A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> mono_on f B" |
62083 | 42 |
unfolding mono_on_def by auto |
43 |
||
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tagged 21 theories in the Analysis library for the manual
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
68635
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44 |
definition%important "strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s" |
62083 | 45 |
|
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parents:
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46 |
lemma strict_mono_onI: |
62083 | 47 |
"(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r < s \<Longrightarrow> f r < f s) \<Longrightarrow> strict_mono_on f A" |
48 |
unfolding strict_mono_on_def by simp |
|
49 |
||
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Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
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|
50 |
lemma strict_mono_onD: |
62083 | 51 |
"\<lbrakk>strict_mono_on f A; r \<in> A; s \<in> A; r < s\<rbrakk> \<Longrightarrow> f r < f s" |
52 |
unfolding strict_mono_on_def by simp |
|
53 |
||
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Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
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|
54 |
lemma mono_on_greaterD: |
62083 | 55 |
assumes "mono_on g A" "x \<in> A" "y \<in> A" "g x > (g (y::_::linorder) :: _ :: linorder)" |
56 |
shows "x > y" |
|
57 |
proof (rule ccontr) |
|
58 |
assume "\<not>x > y" |
|
59 |
hence "x \<le> y" by (simp add: not_less) |
|
60 |
from assms(1-3) and this have "g x \<le> g y" by (rule mono_onD) |
|
61 |
with assms(4) show False by simp |
|
62 |
qed |
|
63 |
||
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parents:
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diff
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|
64 |
lemma strict_mono_inv: |
62083 | 65 |
fixes f :: "('a::linorder) \<Rightarrow> ('b::linorder)" |
66 |
assumes "strict_mono f" and "surj f" and inv: "\<And>x. g (f x) = x" |
|
67 |
shows "strict_mono g" |
|
68 |
proof |
|
69 |
fix x y :: 'b assume "x < y" |
|
70 |
from \<open>surj f\<close> obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast |
|
71 |
with \<open>x < y\<close> and \<open>strict_mono f\<close> have "x' < y'" by (simp add: strict_mono_less) |
|
72 |
with inv show "g x < g y" by simp |
|
73 |
qed |
|
74 |
||
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Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
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|
75 |
lemma strict_mono_on_imp_inj_on: |
62083 | 76 |
assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> (_ :: preorder)) A" |
77 |
shows "inj_on f A" |
|
78 |
proof (rule inj_onI) |
|
79 |
fix x y assume "x \<in> A" "y \<in> A" "f x = f y" |
|
80 |
thus "x = y" |
|
81 |
by (cases x y rule: linorder_cases) |
|
62372 | 82 |
(auto dest: strict_mono_onD[OF assms, of x y] strict_mono_onD[OF assms, of y x]) |
62083 | 83 |
qed |
84 |
||
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parents:
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|
85 |
lemma strict_mono_on_leD: |
62083 | 86 |
assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A" "x \<in> A" "y \<in> A" "x \<le> y" |
87 |
shows "f x \<le> f y" |
|
88 |
proof (insert le_less_linear[of y x], elim disjE) |
|
89 |
assume "x < y" |
|
90 |
with assms have "f x < f y" by (rule_tac strict_mono_onD[OF assms(1)]) simp_all |
|
91 |
thus ?thesis by (rule less_imp_le) |
|
92 |
qed (insert assms, simp) |
|
93 |
||
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parents:
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|
94 |
lemma strict_mono_on_eqD: |
62083 | 95 |
fixes f :: "(_ :: linorder) \<Rightarrow> (_ :: preorder)" |
96 |
assumes "strict_mono_on f A" "f x = f y" "x \<in> A" "y \<in> A" |
|
97 |
shows "y = x" |
|
98 |
using assms by (rule_tac linorder_cases[of x y]) (auto dest: strict_mono_onD) |
|
99 |
||
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parents:
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|
100 |
proposition mono_on_imp_deriv_nonneg: |
62083 | 101 |
assumes mono: "mono_on f A" and deriv: "(f has_real_derivative D) (at x)" |
102 |
assumes "x \<in> interior A" |
|
103 |
shows "D \<ge> 0" |
|
69652
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Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
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diff
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|
104 |
proof (rule tendsto_lowerbound) |
62083 | 105 |
let ?A' = "(\<lambda>y. y - x) ` interior A" |
106 |
from deriv show "((\<lambda>h. (f (x + h) - f x) / h) \<longlongrightarrow> D) (at 0)" |
|
107 |
by (simp add: field_has_derivative_at has_field_derivative_def) |
|
108 |
from mono have mono': "mono_on f (interior A)" by (rule mono_on_subset) (rule interior_subset) |
|
109 |
||
110 |
show "eventually (\<lambda>h. (f (x + h) - f x) / h \<ge> 0) (at 0)" |
|
111 |
proof (subst eventually_at_topological, intro exI conjI ballI impI) |
|
112 |
have "open (interior A)" by simp |
|
67399 | 113 |
hence "open ((+) (-x) ` interior A)" by (rule open_translation) |
114 |
also have "((+) (-x) ` interior A) = ?A'" by auto |
|
62083 | 115 |
finally show "open ?A'" . |
116 |
next |
|
117 |
from \<open>x \<in> interior A\<close> show "0 \<in> ?A'" by auto |
|
118 |
next |
|
119 |
fix h assume "h \<in> ?A'" |
|
120 |
hence "x + h \<in> interior A" by auto |
|
121 |
with mono' and \<open>x \<in> interior A\<close> show "(f (x + h) - f x) / h \<ge> 0" |
|
122 |
by (cases h rule: linorder_cases[of _ 0]) |
|
123 |
(simp_all add: divide_nonpos_neg divide_nonneg_pos mono_onD field_simps) |
|
124 |
qed |
|
125 |
qed simp |
|
126 |
||
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parents:
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|
127 |
lemma strict_mono_on_imp_mono_on: |
62083 | 128 |
"strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A \<Longrightarrow> mono_on f A" |
129 |
by (rule mono_onI, rule strict_mono_on_leD) |
|
130 |
||
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parents:
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diff
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|
131 |
proposition mono_on_ctble_discont: |
62083 | 132 |
fixes f :: "real \<Rightarrow> real" |
133 |
fixes A :: "real set" |
|
134 |
assumes "mono_on f A" |
|
135 |
shows "countable {a\<in>A. \<not> continuous (at a within A) f}" |
|
69652
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Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
136 |
proof - |
62083 | 137 |
have mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
63167 | 138 |
using \<open>mono_on f A\<close> by (simp add: mono_on_def) |
62083 | 139 |
have "\<forall>a \<in> {a\<in>A. \<not> continuous (at a within A) f}. \<exists>q :: nat \<times> rat. |
140 |
(fst q = 0 \<and> of_rat (snd q) < f a \<and> (\<forall>x \<in> A. x < a \<longrightarrow> f x < of_rat (snd q))) \<or> |
|
141 |
(fst q = 1 \<and> of_rat (snd q) > f a \<and> (\<forall>x \<in> A. x > a \<longrightarrow> f x > of_rat (snd q)))" |
|
142 |
proof (clarsimp simp del: One_nat_def) |
|
143 |
fix a assume "a \<in> A" assume "\<not> continuous (at a within A) f" |
|
144 |
thus "\<exists>q1 q2. |
|
145 |
q1 = 0 \<and> real_of_rat q2 < f a \<and> (\<forall>x\<in>A. x < a \<longrightarrow> f x < real_of_rat q2) \<or> |
|
146 |
q1 = 1 \<and> f a < real_of_rat q2 \<and> (\<forall>x\<in>A. a < x \<longrightarrow> real_of_rat q2 < f x)" |
|
147 |
proof (auto simp add: continuous_within order_tendsto_iff eventually_at) |
|
148 |
fix l assume "l < f a" |
|
149 |
then obtain q2 where q2: "l < of_rat q2" "of_rat q2 < f a" |
|
150 |
using of_rat_dense by blast |
|
151 |
assume * [rule_format]: "\<forall>d>0. \<exists>x\<in>A. x \<noteq> a \<and> dist x a < d \<and> \<not> l < f x" |
|
152 |
from q2 have "real_of_rat q2 < f a \<and> (\<forall>x\<in>A. x < a \<longrightarrow> f x < real_of_rat q2)" |
|
153 |
proof auto |
|
154 |
fix x assume "x \<in> A" "x < a" |
|
155 |
with q2 *[of "a - x"] show "f x < real_of_rat q2" |
|
156 |
apply (auto simp add: dist_real_def not_less) |
|
157 |
apply (subgoal_tac "f x \<le> f xa") |
|
158 |
by (auto intro: mono) |
|
62372 | 159 |
qed |
62083 | 160 |
thus ?thesis by auto |
161 |
next |
|
162 |
fix u assume "u > f a" |
|
163 |
then obtain q2 where q2: "f a < of_rat q2" "of_rat q2 < u" |
|
164 |
using of_rat_dense by blast |
|
165 |
assume *[rule_format]: "\<forall>d>0. \<exists>x\<in>A. x \<noteq> a \<and> dist x a < d \<and> \<not> u > f x" |
|
166 |
from q2 have "real_of_rat q2 > f a \<and> (\<forall>x\<in>A. x > a \<longrightarrow> f x > real_of_rat q2)" |
|
167 |
proof auto |
|
168 |
fix x assume "x \<in> A" "x > a" |
|
169 |
with q2 *[of "x - a"] show "f x > real_of_rat q2" |
|
170 |
apply (auto simp add: dist_real_def) |
|
171 |
apply (subgoal_tac "f x \<ge> f xa") |
|
172 |
by (auto intro: mono) |
|
62372 | 173 |
qed |
62083 | 174 |
thus ?thesis by auto |
175 |
qed |
|
176 |
qed |
|
62372 | 177 |
hence "\<exists>g :: real \<Rightarrow> nat \<times> rat . \<forall>a \<in> {a\<in>A. \<not> continuous (at a within A) f}. |
62083 | 178 |
(fst (g a) = 0 \<and> of_rat (snd (g a)) < f a \<and> (\<forall>x \<in> A. x < a \<longrightarrow> f x < of_rat (snd (g a)))) | |
179 |
(fst (g a) = 1 \<and> of_rat (snd (g a)) > f a \<and> (\<forall>x \<in> A. x > a \<longrightarrow> f x > of_rat (snd (g a))))" |
|
180 |
by (rule bchoice) |
|
181 |
then guess g .. |
|
182 |
hence g: "\<And>a x. a \<in> A \<Longrightarrow> \<not> continuous (at a within A) f \<Longrightarrow> x \<in> A \<Longrightarrow> |
|
183 |
(fst (g a) = 0 \<and> of_rat (snd (g a)) < f a \<and> (x < a \<longrightarrow> f x < of_rat (snd (g a)))) | |
|
184 |
(fst (g a) = 1 \<and> of_rat (snd (g a)) > f a \<and> (x > a \<longrightarrow> f x > of_rat (snd (g a))))" |
|
185 |
by auto |
|
186 |
have "inj_on g {a\<in>A. \<not> continuous (at a within A) f}" |
|
187 |
proof (auto simp add: inj_on_def) |
|
188 |
fix w z |
|
189 |
assume 1: "w \<in> A" and 2: "\<not> continuous (at w within A) f" and |
|
190 |
3: "z \<in> A" and 4: "\<not> continuous (at z within A) f" and |
|
191 |
5: "g w = g z" |
|
62372 | 192 |
from g [OF 1 2 3] g [OF 3 4 1] 5 |
62083 | 193 |
show "w = z" by auto |
194 |
qed |
|
62372 | 195 |
thus ?thesis |
196 |
by (rule countableI') |
|
62083 | 197 |
qed |
198 |
||
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parents:
69597
diff
changeset
|
199 |
lemma mono_on_ctble_discont_open: |
62083 | 200 |
fixes f :: "real \<Rightarrow> real" |
201 |
fixes A :: "real set" |
|
202 |
assumes "open A" "mono_on f A" |
|
203 |
shows "countable {a\<in>A. \<not>isCont f a}" |
|
69652
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parents:
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|
204 |
proof - |
62083 | 205 |
have "{a\<in>A. \<not>isCont f a} = {a\<in>A. \<not>(continuous (at a within A) f)}" |
63167 | 206 |
by (auto simp add: continuous_within_open [OF _ \<open>open A\<close>]) |
62083 | 207 |
thus ?thesis |
208 |
apply (elim ssubst) |
|
209 |
by (rule mono_on_ctble_discont, rule assms) |
|
210 |
qed |
|
211 |
||
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parents:
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|
212 |
lemma mono_ctble_discont: |
62083 | 213 |
fixes f :: "real \<Rightarrow> real" |
214 |
assumes "mono f" |
|
215 |
shows "countable {a. \<not> isCont f a}" |
|
69652
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Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
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|
216 |
using assms mono_on_ctble_discont [of f UNIV] unfolding mono_on_def mono_def by auto |
62083 | 217 |
|
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parents:
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|
218 |
lemma has_real_derivative_imp_continuous_on: |
62083 | 219 |
assumes "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)" |
220 |
shows "continuous_on A f" |
|
221 |
apply (intro differentiable_imp_continuous_on, unfold differentiable_on_def) |
|
69022
e2858770997a
removal of more redundancies, and fixes
paulson <lp15@cam.ac.uk>
parents:
68833
diff
changeset
|
222 |
using assms differentiable_at_withinI real_differentiable_def by blast |
62083 | 223 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
224 |
lemma closure_contains_Sup: |
62083 | 225 |
fixes S :: "real set" |
226 |
assumes "S \<noteq> {}" "bdd_above S" |
|
227 |
shows "Sup S \<in> closure S" |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
228 |
proof - |
62372 | 229 |
have "Inf (uminus ` S) \<in> closure (uminus ` S)" |
62083 | 230 |
using assms by (intro closure_contains_Inf) auto |
231 |
also have "Inf (uminus ` S) = -Sup S" by (simp add: Inf_real_def) |
|
232 |
also have "closure (uminus ` S) = uminus ` closure S" |
|
233 |
by (rule sym, intro closure_injective_linear_image) (auto intro: linearI) |
|
234 |
finally show ?thesis by auto |
|
235 |
qed |
|
236 |
||
69652
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updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
237 |
lemma closed_contains_Sup: |
62083 | 238 |
fixes S :: "real set" |
239 |
shows "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> closed S \<Longrightarrow> Sup S \<in> S" |
|
240 |
by (subst closure_closed[symmetric], assumption, rule closure_contains_Sup) |
|
241 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
242 |
lemma closed_subset_contains_Sup: |
67685
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67399
diff
changeset
|
243 |
fixes A C :: "real set" |
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67399
diff
changeset
|
244 |
shows "closed C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> Sup A \<in> C" |
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67399
diff
changeset
|
245 |
by (metis closure_contains_Sup closure_minimal subset_eq) |
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67399
diff
changeset
|
246 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
247 |
proposition deriv_nonneg_imp_mono: |
62083 | 248 |
assumes deriv: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)" |
249 |
assumes nonneg: "\<And>x. x \<in> {a..b} \<Longrightarrow> g' x \<ge> 0" |
|
250 |
assumes ab: "a \<le> b" |
|
251 |
shows "g a \<le> g b" |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
252 |
proof (cases "a < b") |
62083 | 253 |
assume "a < b" |
68635 | 254 |
from deriv have "\<And>x. \<lbrakk>x \<ge> a; x \<le> b\<rbrakk> \<Longrightarrow> (g has_real_derivative g' x) (at x)" by simp |
255 |
with MVT2[OF \<open>a < b\<close>] and deriv |
|
62083 | 256 |
obtain \<xi> where \<xi>_ab: "\<xi> > a" "\<xi> < b" and g_ab: "g b - g a = (b - a) * g' \<xi>" by blast |
257 |
from \<xi>_ab ab nonneg have "(b - a) * g' \<xi> \<ge> 0" by simp |
|
258 |
with g_ab show ?thesis by simp |
|
259 |
qed (insert ab, simp) |
|
260 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
261 |
lemma continuous_interval_vimage_Int: |
62083 | 262 |
assumes "continuous_on {a::real..b} g" and mono: "\<And>x y. a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> b \<Longrightarrow> g x \<le> g y" |
263 |
assumes "a \<le> b" "(c::real) \<le> d" "{c..d} \<subseteq> {g a..g b}" |
|
264 |
obtains c' d' where "{a..b} \<inter> g -` {c..d} = {c'..d'}" "c' \<le> d'" "g c' = c" "g d' = d" |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
265 |
proof- |
63040 | 266 |
let ?A = "{a..b} \<inter> g -` {c..d}" |
267 |
from IVT'[of g a c b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5) |
|
268 |
obtain c'' where c'': "c'' \<in> ?A" "g c'' = c" by auto |
|
269 |
from IVT'[of g a d b, OF _ _ \<open>a \<le> b\<close> assms(1)] assms(4,5) |
|
270 |
obtain d'' where d'': "d'' \<in> ?A" "g d'' = d" by auto |
|
271 |
hence [simp]: "?A \<noteq> {}" by blast |
|
62083 | 272 |
|
63040 | 273 |
define c' where "c' = Inf ?A" |
274 |
define d' where "d' = Sup ?A" |
|
275 |
have "?A \<subseteq> {c'..d'}" unfolding c'_def d'_def |
|
276 |
by (intro subsetI) (auto intro: cInf_lower cSup_upper) |
|
277 |
moreover from assms have "closed ?A" |
|
278 |
using continuous_on_closed_vimage[of "{a..b}" g] by (subst Int_commute) simp |
|
279 |
hence c'd'_in_set: "c' \<in> ?A" "d' \<in> ?A" unfolding c'_def d'_def |
|
280 |
by ((intro closed_contains_Inf closed_contains_Sup, simp_all)[])+ |
|
281 |
hence "{c'..d'} \<subseteq> ?A" using assms |
|
282 |
by (intro subsetI) |
|
283 |
(auto intro!: order_trans[of c "g c'" "g x" for x] order_trans[of "g x" "g d'" d for x] |
|
284 |
intro!: mono) |
|
285 |
moreover have "c' \<le> d'" using c'd'_in_set(2) unfolding c'_def by (intro cInf_lower) auto |
|
286 |
moreover have "g c' \<le> c" "g d' \<ge> d" |
|
287 |
apply (insert c'' d'' c'd'_in_set) |
|
288 |
apply (subst c''(2)[symmetric]) |
|
289 |
apply (auto simp: c'_def intro!: mono cInf_lower c'') [] |
|
290 |
apply (subst d''(2)[symmetric]) |
|
291 |
apply (auto simp: d'_def intro!: mono cSup_upper d'') [] |
|
292 |
done |
|
293 |
with c'd'_in_set have "g c' = c" "g d' = d" by auto |
|
294 |
ultimately show ?thesis using that by blast |
|
62083 | 295 |
qed |
296 |
||
69683 | 297 |
subsection \<open>Generic Borel spaces\<close> |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
298 |
|
68833
fde093888c16
tagged 21 theories in the Analysis library for the manual
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
68635
diff
changeset
|
299 |
definition%important (in topological_space) borel :: "'a measure" where |
47694 | 300 |
"borel = sigma UNIV {S. open S}" |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
301 |
|
47694 | 302 |
abbreviation "borel_measurable M \<equiv> measurable M borel" |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
303 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
304 |
lemma in_borel_measurable: |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
305 |
"f \<in> borel_measurable M \<longleftrightarrow> |
47694 | 306 |
(\<forall>S \<in> sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
307 |
by (auto simp add: measurable_def borel_def) |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
308 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
309 |
lemma in_borel_measurable_borel: |
38656 | 310 |
"f \<in> borel_measurable M \<longleftrightarrow> |
40859 | 311 |
(\<forall>S \<in> sets borel. |
38656 | 312 |
f -` S \<inter> space M \<in> sets M)" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
313 |
by (auto simp add: measurable_def borel_def) |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
314 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
315 |
lemma space_borel[simp]: "space borel = UNIV" |
40859 | 316 |
unfolding borel_def by auto |
38656 | 317 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
318 |
lemma space_in_borel[measurable]: "UNIV \<in> sets borel" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
319 |
unfolding borel_def by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
320 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
321 |
lemma sets_borel: "sets borel = sigma_sets UNIV {S. open S}" |
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
322 |
unfolding borel_def by (rule sets_measure_of) simp |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
323 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
324 |
lemma measurable_sets_borel: |
62083 | 325 |
"\<lbrakk>f \<in> measurable borel M; A \<in> sets M\<rbrakk> \<Longrightarrow> f -` A \<in> sets borel" |
326 |
by (drule (1) measurable_sets) simp |
|
327 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
328 |
lemma pred_Collect_borel[measurable (raw)]: "Measurable.pred borel P \<Longrightarrow> {x. P x} \<in> sets borel" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
329 |
unfolding borel_def pred_def by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
330 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
331 |
lemma borel_open[measurable (raw generic)]: |
40859 | 332 |
assumes "open A" shows "A \<in> sets borel" |
38656 | 333 |
proof - |
44537
c10485a6a7af
make HOL-Probability respect set/pred distinction
huffman
parents:
44282
diff
changeset
|
334 |
have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms . |
47694 | 335 |
thus ?thesis unfolding borel_def by auto |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
336 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
337 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
338 |
lemma borel_closed[measurable (raw generic)]: |
40859 | 339 |
assumes "closed A" shows "A \<in> sets borel" |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
340 |
proof - |
40859 | 341 |
have "space borel - (- A) \<in> sets borel" |
342 |
using assms unfolding closed_def by (blast intro: borel_open) |
|
38656 | 343 |
thus ?thesis by simp |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
344 |
qed |
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
345 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
346 |
lemma borel_singleton[measurable]: |
50003 | 347 |
"A \<in> sets borel \<Longrightarrow> insert x A \<in> sets (borel :: 'a::t1_space measure)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
348 |
unfolding insert_def by (rule sets.Un) auto |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
349 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
350 |
lemma sets_borel_eq_count_space: "sets (borel :: 'a::{countable, t2_space} measure) = count_space UNIV" |
64320
ba194424b895
HOL-Probability: move stopping time from AFP/Markov_Models
hoelzl
parents:
64287
diff
changeset
|
351 |
proof - |
ba194424b895
HOL-Probability: move stopping time from AFP/Markov_Models
hoelzl
parents:
64287
diff
changeset
|
352 |
have "(\<Union>a\<in>A. {a}) \<in> sets borel" for A :: "'a set" |
ba194424b895
HOL-Probability: move stopping time from AFP/Markov_Models
hoelzl
parents:
64287
diff
changeset
|
353 |
by (intro sets.countable_UN') auto |
ba194424b895
HOL-Probability: move stopping time from AFP/Markov_Models
hoelzl
parents:
64287
diff
changeset
|
354 |
then show ?thesis |
ba194424b895
HOL-Probability: move stopping time from AFP/Markov_Models
hoelzl
parents:
64287
diff
changeset
|
355 |
by auto |
ba194424b895
HOL-Probability: move stopping time from AFP/Markov_Models
hoelzl
parents:
64287
diff
changeset
|
356 |
qed |
ba194424b895
HOL-Probability: move stopping time from AFP/Markov_Models
hoelzl
parents:
64287
diff
changeset
|
357 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
358 |
lemma borel_comp[measurable]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
359 |
unfolding Compl_eq_Diff_UNIV by simp |
41830 | 360 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
361 |
lemma borel_measurable_vimage: |
38656 | 362 |
fixes f :: "'a \<Rightarrow> 'x::t2_space" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
363 |
assumes borel[measurable]: "f \<in> borel_measurable M" |
38656 | 364 |
shows "f -` {x} \<inter> space M \<in> sets M" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
365 |
by simp |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
366 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
367 |
lemma borel_measurableI: |
61076 | 368 |
fixes f :: "'a \<Rightarrow> 'x::topological_space" |
38656 | 369 |
assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M" |
370 |
shows "f \<in> borel_measurable M" |
|
40859 | 371 |
unfolding borel_def |
47694 | 372 |
proof (rule measurable_measure_of, simp_all) |
44537
c10485a6a7af
make HOL-Probability respect set/pred distinction
huffman
parents:
44282
diff
changeset
|
373 |
fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M" |
c10485a6a7af
make HOL-Probability respect set/pred distinction
huffman
parents:
44282
diff
changeset
|
374 |
using assms[of S] by simp |
40859 | 375 |
qed |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
376 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
377 |
lemma borel_measurable_const: |
38656 | 378 |
"(\<lambda>x. c) \<in> borel_measurable M" |
47694 | 379 |
by auto |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
380 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
381 |
lemma borel_measurable_indicator: |
38656 | 382 |
assumes A: "A \<in> sets M" |
383 |
shows "indicator A \<in> borel_measurable M" |
|
46905 | 384 |
unfolding indicator_def [abs_def] using A |
47694 | 385 |
by (auto intro!: measurable_If_set) |
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff
changeset
|
386 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
387 |
lemma borel_measurable_count_space[measurable (raw)]: |
50096 | 388 |
"f \<in> borel_measurable (count_space S)" |
389 |
unfolding measurable_def by auto |
|
390 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
391 |
lemma borel_measurable_indicator'[measurable (raw)]: |
50096 | 392 |
assumes [measurable]: "{x\<in>space M. f x \<in> A x} \<in> sets M" |
393 |
shows "(\<lambda>x. indicator (A x) (f x)) \<in> borel_measurable M" |
|
50001
382bd3173584
add syntax and a.e.-rules for (conditional) probability on predicates
hoelzl
parents:
49774
diff
changeset
|
394 |
unfolding indicator_def[abs_def] |
382bd3173584
add syntax and a.e.-rules for (conditional) probability on predicates
hoelzl
parents:
49774
diff
changeset
|
395 |
by (auto intro!: measurable_If) |
382bd3173584
add syntax and a.e.-rules for (conditional) probability on predicates
hoelzl
parents:
49774
diff
changeset
|
396 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
397 |
lemma borel_measurable_indicator_iff: |
40859 | 398 |
"(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M" |
399 |
(is "?I \<in> borel_measurable M \<longleftrightarrow> _") |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
400 |
proof |
40859 | 401 |
assume "?I \<in> borel_measurable M" |
402 |
then have "?I -` {1} \<inter> space M \<in> sets M" |
|
403 |
unfolding measurable_def by auto |
|
404 |
also have "?I -` {1} \<inter> space M = A \<inter> space M" |
|
46905 | 405 |
unfolding indicator_def [abs_def] by auto |
40859 | 406 |
finally show "A \<inter> space M \<in> sets M" . |
407 |
next |
|
408 |
assume "A \<inter> space M \<in> sets M" |
|
409 |
moreover have "?I \<in> borel_measurable M \<longleftrightarrow> |
|
410 |
(indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M" |
|
411 |
by (intro measurable_cong) (auto simp: indicator_def) |
|
412 |
ultimately show "?I \<in> borel_measurable M" by auto |
|
413 |
qed |
|
414 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
415 |
lemma borel_measurable_subalgebra: |
41545 | 416 |
assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N" |
39092 | 417 |
shows "f \<in> borel_measurable M" |
418 |
using assms unfolding measurable_def by auto |
|
419 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
420 |
lemma borel_measurable_restrict_space_iff_ereal: |
57137 | 421 |
fixes f :: "'a \<Rightarrow> ereal" |
422 |
assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M" |
|
423 |
shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow> |
|
424 |
(\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M" |
|
57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
425 |
by (subst measurable_restrict_space_iff) |
63566 | 426 |
(auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong) |
57137 | 427 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
428 |
lemma borel_measurable_restrict_space_iff_ennreal: |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
429 |
fixes f :: "'a \<Rightarrow> ennreal" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
430 |
assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
431 |
shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow> |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
432 |
(\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
433 |
by (subst measurable_restrict_space_iff) |
63566 | 434 |
(auto simp: indicator_def if_distrib[where f="\<lambda>x. a * x" for a] cong del: if_weak_cong) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
435 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
436 |
lemma borel_measurable_restrict_space_iff: |
57137 | 437 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
438 |
assumes \<Omega>[measurable, simp]: "\<Omega> \<inter> space M \<in> sets M" |
|
439 |
shows "f \<in> borel_measurable (restrict_space M \<Omega>) \<longleftrightarrow> |
|
440 |
(\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> borel_measurable M" |
|
57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
441 |
by (subst measurable_restrict_space_iff) |
63566 | 442 |
(auto simp: indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a] ac_simps |
443 |
cong del: if_weak_cong) |
|
57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
444 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
445 |
lemma cbox_borel[measurable]: "cbox a b \<in> sets borel" |
57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
446 |
by (auto intro: borel_closed) |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
447 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
448 |
lemma box_borel[measurable]: "box a b \<in> sets borel" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
449 |
by (auto intro: borel_open) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
450 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
451 |
lemma borel_compact: "compact (A::'a::t2_space set) \<Longrightarrow> A \<in> sets borel" |
57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
452 |
by (auto intro: borel_closed dest!: compact_imp_closed) |
57137 | 453 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
454 |
lemma borel_sigma_sets_subset: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
455 |
"A \<subseteq> sets borel \<Longrightarrow> sigma_sets UNIV A \<subseteq> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
456 |
using sets.sigma_sets_subset[of A borel] by simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
457 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
458 |
lemma borel_eq_sigmaI1: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
459 |
fixes F :: "'i \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
460 |
assumes borel_eq: "borel = sigma UNIV X" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
461 |
assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV (F ` A))" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
462 |
assumes F: "\<And>i. i \<in> A \<Longrightarrow> F i \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
463 |
shows "borel = sigma UNIV (F ` A)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
464 |
unfolding borel_def |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
465 |
proof (intro sigma_eqI antisym) |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
466 |
have borel_rev_eq: "sigma_sets UNIV {S::'a set. open S} = sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
467 |
unfolding borel_def by simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
468 |
also have "\<dots> = sigma_sets UNIV X" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
469 |
unfolding borel_eq by simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
470 |
also have "\<dots> \<subseteq> sigma_sets UNIV (F`A)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
471 |
using X by (intro sigma_algebra.sigma_sets_subset[OF sigma_algebra_sigma_sets]) auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
472 |
finally show "sigma_sets UNIV {S. open S} \<subseteq> sigma_sets UNIV (F`A)" . |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
473 |
show "sigma_sets UNIV (F`A) \<subseteq> sigma_sets UNIV {S. open S}" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
474 |
unfolding borel_rev_eq using F by (intro borel_sigma_sets_subset) auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
475 |
qed auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
476 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
477 |
lemma borel_eq_sigmaI2: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
478 |
fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
479 |
and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
480 |
assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`B)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
481 |
assumes X: "\<And>i j. (i, j) \<in> B \<Longrightarrow> G i j \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
482 |
assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
483 |
shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
484 |
using assms |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
485 |
by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` B" and F="(\<lambda>(i, j). F i j)"]) auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
486 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
487 |
lemma borel_eq_sigmaI3: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
488 |
fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and X :: "'a::topological_space set set" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
489 |
assumes borel_eq: "borel = sigma UNIV X" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
490 |
assumes X: "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
491 |
assumes F: "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
492 |
shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
493 |
using assms by (intro borel_eq_sigmaI1[where X=X and F="(\<lambda>(i, j). F i j)"]) auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
494 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
495 |
lemma borel_eq_sigmaI4: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
496 |
fixes F :: "'i \<Rightarrow> 'a::topological_space set" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
497 |
and G :: "'l \<Rightarrow> 'k \<Rightarrow> 'a::topological_space set" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
498 |
assumes borel_eq: "borel = sigma UNIV ((\<lambda>(i, j). G i j)`A)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
499 |
assumes X: "\<And>i j. (i, j) \<in> A \<Longrightarrow> G i j \<in> sets (sigma UNIV (range F))" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
500 |
assumes F: "\<And>i. F i \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
501 |
shows "borel = sigma UNIV (range F)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
502 |
using assms by (intro borel_eq_sigmaI1[where X="(\<lambda>(i, j). G i j) ` A" and F=F]) auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
503 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
504 |
lemma borel_eq_sigmaI5: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
505 |
fixes F :: "'i \<Rightarrow> 'j \<Rightarrow> 'a::topological_space set" and G :: "'l \<Rightarrow> 'a::topological_space set" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
506 |
assumes borel_eq: "borel = sigma UNIV (range G)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
507 |
assumes X: "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
508 |
assumes F: "\<And>i j. F i j \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
509 |
shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
510 |
using assms by (intro borel_eq_sigmaI1[where X="range G" and F="(\<lambda>(i, j). F i j)"]) auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
511 |
|
69722
b5163b2132c5
minor tagging updates in 13 theories
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69683
diff
changeset
|
512 |
theorem second_countable_borel_measurable: |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
513 |
fixes X :: "'a::second_countable_topology set set" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
514 |
assumes eq: "open = generate_topology X" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
515 |
shows "borel = sigma UNIV X" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
516 |
unfolding borel_def |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
517 |
proof (intro sigma_eqI sigma_sets_eqI) |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
518 |
interpret X: sigma_algebra UNIV "sigma_sets UNIV X" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
519 |
by (rule sigma_algebra_sigma_sets) simp |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
520 |
|
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
521 |
fix S :: "'a set" assume "S \<in> Collect open" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
522 |
then have "generate_topology X S" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
523 |
by (auto simp: eq) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
524 |
then show "S \<in> sigma_sets UNIV X" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
525 |
proof induction |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
526 |
case (UN K) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
527 |
then have K: "\<And>k. k \<in> K \<Longrightarrow> open k" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
528 |
unfolding eq by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
529 |
from ex_countable_basis obtain B :: "'a set set" where |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
530 |
B: "\<And>b. b \<in> B \<Longrightarrow> open b" "\<And>X. open X \<Longrightarrow> \<exists>b\<subseteq>B. (\<Union>b) = X" and "countable B" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
531 |
by (auto simp: topological_basis_def) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
532 |
from B(2)[OF K] obtain m where m: "\<And>k. k \<in> K \<Longrightarrow> m k \<subseteq> B" "\<And>k. k \<in> K \<Longrightarrow> (\<Union>m k) = k" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
533 |
by metis |
63040 | 534 |
define U where "U = (\<Union>k\<in>K. m k)" |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
535 |
with m have "countable U" |
61808 | 536 |
by (intro countable_subset[OF _ \<open>countable B\<close>]) auto |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
537 |
have "\<Union>U = (\<Union>A\<in>U. A)" by simp |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
538 |
also have "\<dots> = \<Union>K" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
539 |
unfolding U_def UN_simps by (simp add: m) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
540 |
finally have "\<Union>U = \<Union>K" . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
541 |
|
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
542 |
have "\<forall>b\<in>U. \<exists>k\<in>K. b \<subseteq> k" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
543 |
using m by (auto simp: U_def) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
544 |
then obtain u where u: "\<And>b. b \<in> U \<Longrightarrow> u b \<in> K" and "\<And>b. b \<in> U \<Longrightarrow> b \<subseteq> u b" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
545 |
by metis |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
546 |
then have "(\<Union>b\<in>U. u b) \<subseteq> \<Union>K" "\<Union>U \<subseteq> (\<Union>b\<in>U. u b)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
547 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
548 |
then have "\<Union>K = (\<Union>b\<in>U. u b)" |
61808 | 549 |
unfolding \<open>\<Union>U = \<Union>K\<close> by auto |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
550 |
also have "\<dots> \<in> sigma_sets UNIV X" |
61808 | 551 |
using u UN by (intro X.countable_UN' \<open>countable U\<close>) auto |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
552 |
finally show "\<Union>K \<in> sigma_sets UNIV X" . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
553 |
qed auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
554 |
qed (auto simp: eq intro: generate_topology.Basis) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
555 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
556 |
lemma borel_eq_closed: "borel = sigma UNIV (Collect closed)" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
557 |
unfolding borel_def |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
558 |
proof (intro sigma_eqI sigma_sets_eqI, safe) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
559 |
fix x :: "'a set" assume "open x" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
560 |
hence "x = UNIV - (UNIV - x)" by auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
561 |
also have "\<dots> \<in> sigma_sets UNIV (Collect closed)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
562 |
by (force intro: sigma_sets.Compl simp: \<open>open x\<close>) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
563 |
finally show "x \<in> sigma_sets UNIV (Collect closed)" by simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
564 |
next |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
565 |
fix x :: "'a set" assume "closed x" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
566 |
hence "x = UNIV - (UNIV - x)" by auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
567 |
also have "\<dots> \<in> sigma_sets UNIV (Collect open)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
568 |
by (force intro: sigma_sets.Compl simp: \<open>closed x\<close>) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
569 |
finally show "x \<in> sigma_sets UNIV (Collect open)" by simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
570 |
qed simp_all |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
571 |
|
69722
b5163b2132c5
minor tagging updates in 13 theories
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69683
diff
changeset
|
572 |
proposition borel_eq_countable_basis: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
573 |
fixes B::"'a::topological_space set set" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
574 |
assumes "countable B" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
575 |
assumes "topological_basis B" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
576 |
shows "borel = sigma UNIV B" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
577 |
unfolding borel_def |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
578 |
proof (intro sigma_eqI sigma_sets_eqI, safe) |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
579 |
interpret countable_basis using assms by unfold_locales |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
580 |
fix X::"'a set" assume "open X" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
581 |
from open_countable_basisE[OF this] guess B' . note B' = this |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
582 |
then show "X \<in> sigma_sets UNIV B" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
583 |
by (blast intro: sigma_sets_UNION \<open>countable B\<close> countable_subset) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
584 |
next |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
585 |
fix b assume "b \<in> B" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
586 |
hence "open b" by (rule topological_basis_open[OF assms(2)]) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
587 |
thus "b \<in> sigma_sets UNIV (Collect open)" by auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
588 |
qed simp_all |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
589 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
590 |
lemma borel_measurable_continuous_on_restrict: |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
591 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space" |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
592 |
assumes f: "continuous_on A f" |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
593 |
shows "f \<in> borel_measurable (restrict_space borel A)" |
57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
594 |
proof (rule borel_measurableI) |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
595 |
fix S :: "'b set" assume "open S" |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
596 |
with f obtain T where "f -` S \<inter> A = T \<inter> A" "open T" |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
597 |
by (metis continuous_on_open_invariant) |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
598 |
then show "f -` S \<inter> space (restrict_space borel A) \<in> sets (restrict_space borel A)" |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
599 |
by (force simp add: sets_restrict_space space_restrict_space) |
57137 | 600 |
qed |
601 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
602 |
lemma borel_measurable_continuous_on1: "continuous_on UNIV f \<Longrightarrow> f \<in> borel_measurable borel" |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
603 |
by (drule borel_measurable_continuous_on_restrict) simp |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
604 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
605 |
lemma borel_measurable_continuous_on_if: |
59415 | 606 |
"A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> continuous_on (- A) g \<Longrightarrow> |
607 |
(\<lambda>x. if x \<in> A then f x else g x) \<in> borel_measurable borel" |
|
608 |
by (auto simp add: measurable_If_restrict_space_iff Collect_neg_eq |
|
609 |
intro!: borel_measurable_continuous_on_restrict) |
|
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
610 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
611 |
lemma borel_measurable_continuous_countable_exceptions: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
612 |
fixes f :: "'a::t1_space \<Rightarrow> 'b::topological_space" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
613 |
assumes X: "countable X" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
614 |
assumes "continuous_on (- X) f" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
615 |
shows "f \<in> borel_measurable borel" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
616 |
proof (rule measurable_discrete_difference[OF _ X]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
617 |
have "X \<in> sets borel" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
618 |
by (rule sets.countable[OF _ X]) auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
619 |
then show "(\<lambda>x. if x \<in> X then undefined else f x) \<in> borel_measurable borel" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
620 |
by (intro borel_measurable_continuous_on_if assms continuous_intros) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
621 |
qed auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
622 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
623 |
lemma borel_measurable_continuous_on: |
57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
624 |
assumes f: "continuous_on UNIV f" and g: "g \<in> borel_measurable M" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
625 |
shows "(\<lambda>x. f (g x)) \<in> borel_measurable M" |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
626 |
using measurable_comp[OF g borel_measurable_continuous_on1[OF f]] by (simp add: comp_def) |
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
627 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
628 |
lemma borel_measurable_continuous_on_indicator: |
57138
7b3146180291
generalizd measurability on restricted space; rule for integrability on compact sets
hoelzl
parents:
57137
diff
changeset
|
629 |
fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" |
59415 | 630 |
shows "A \<in> sets borel \<Longrightarrow> continuous_on A f \<Longrightarrow> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable borel" |
631 |
by (subst borel_measurable_restrict_space_iff[symmetric]) |
|
632 |
(auto intro: borel_measurable_continuous_on_restrict) |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
633 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
634 |
lemma borel_measurable_Pair[measurable (raw)]: |
50881
ae630bab13da
renamed countable_basis_space to second_countable_topology
hoelzl
parents:
50526
diff
changeset
|
635 |
fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
636 |
assumes f[measurable]: "f \<in> borel_measurable M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
637 |
assumes g[measurable]: "g \<in> borel_measurable M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
638 |
shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
639 |
proof (subst borel_eq_countable_basis) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
640 |
let ?B = "SOME B::'b set set. countable B \<and> topological_basis B" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
641 |
let ?C = "SOME B::'c set set. countable B \<and> topological_basis B" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
642 |
let ?P = "(\<lambda>(b, c). b \<times> c) ` (?B \<times> ?C)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
643 |
show "countable ?P" "topological_basis ?P" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
644 |
by (auto intro!: countable_basis topological_basis_prod is_basis) |
38656 | 645 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
646 |
show "(\<lambda>x. (f x, g x)) \<in> measurable M (sigma UNIV ?P)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
647 |
proof (rule measurable_measure_of) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
648 |
fix S assume "S \<in> ?P" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
649 |
then obtain b c where "b \<in> ?B" "c \<in> ?C" and S: "S = b \<times> c" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
650 |
then have borel: "open b" "open c" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
651 |
by (auto intro: is_basis topological_basis_open) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
652 |
have "(\<lambda>x. (f x, g x)) -` S \<inter> space M = (f -` b \<inter> space M) \<inter> (g -` c \<inter> space M)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
653 |
unfolding S by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
654 |
also have "\<dots> \<in> sets M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
655 |
using borel by simp |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
656 |
finally show "(\<lambda>x. (f x, g x)) -` S \<inter> space M \<in> sets M" . |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
657 |
qed auto |
39087
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
658 |
qed |
96984bf6fa5b
Measurable on euclidean space is equiv. to measurable components
hoelzl
parents:
39083
diff
changeset
|
659 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
660 |
lemma borel_measurable_continuous_Pair: |
50881
ae630bab13da
renamed countable_basis_space to second_countable_topology
hoelzl
parents:
50526
diff
changeset
|
661 |
fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology" |
50003 | 662 |
assumes [measurable]: "f \<in> borel_measurable M" |
663 |
assumes [measurable]: "g \<in> borel_measurable M" |
|
49774 | 664 |
assumes H: "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))" |
665 |
shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M" |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
666 |
proof - |
49774 | 667 |
have eq: "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. (\<lambda>x. H (fst x) (snd x)) (f x, g x))" by auto |
668 |
show ?thesis |
|
669 |
unfolding eq by (rule borel_measurable_continuous_on[OF H]) auto |
|
670 |
qed |
|
671 |
||
69683 | 672 |
subsection \<open>Borel spaces on order topologies\<close> |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
673 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
674 |
lemma [measurable]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
675 |
fixes a b :: "'a::linorder_topology" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
676 |
shows lessThan_borel: "{..< a} \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
677 |
and greaterThan_borel: "{a <..} \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
678 |
and greaterThanLessThan_borel: "{a<..<b} \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
679 |
and atMost_borel: "{..a} \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
680 |
and atLeast_borel: "{a..} \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
681 |
and atLeastAtMost_borel: "{a..b} \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
682 |
and greaterThanAtMost_borel: "{a<..b} \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
683 |
and atLeastLessThan_borel: "{a..<b} \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
684 |
unfolding greaterThanAtMost_def atLeastLessThan_def |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
685 |
by (blast intro: borel_open borel_closed open_lessThan open_greaterThan open_greaterThanLessThan |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
686 |
closed_atMost closed_atLeast closed_atLeastAtMost)+ |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
687 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
688 |
lemma borel_Iio: |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
689 |
"borel = sigma UNIV (range lessThan :: 'a::{linorder_topology, second_countable_topology} set set)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
690 |
unfolding second_countable_borel_measurable[OF open_generated_order] |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
691 |
proof (intro sigma_eqI sigma_sets_eqI) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
692 |
from countable_dense_setE guess D :: "'a set" . note D = this |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
693 |
|
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
694 |
interpret L: sigma_algebra UNIV "sigma_sets UNIV (range lessThan)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
695 |
by (rule sigma_algebra_sigma_sets) simp |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
696 |
|
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
697 |
fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
698 |
then obtain y where "A = {y <..} \<or> A = {..< y}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
699 |
by blast |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
700 |
then show "A \<in> sigma_sets UNIV (range lessThan)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
701 |
proof |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
702 |
assume A: "A = {y <..}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
703 |
show ?thesis |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
704 |
proof cases |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
705 |
assume "\<forall>x>y. \<exists>d. y < d \<and> d < x" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
706 |
with D(2)[of "{y <..< x}" for x] have "\<forall>x>y. \<exists>d\<in>D. y < d \<and> d < x" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
707 |
by (auto simp: set_eq_iff) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
708 |
then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. y < d}. {..< d})" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
709 |
by (auto simp: A) (metis less_asym) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
710 |
also have "\<dots> \<in> sigma_sets UNIV (range lessThan)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
711 |
using D(1) by (intro L.Diff L.top L.countable_INT'') auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
712 |
finally show ?thesis . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
713 |
next |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
714 |
assume "\<not> (\<forall>x>y. \<exists>d. y < d \<and> d < x)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
715 |
then obtain x where "y < x" "\<And>d. y < d \<Longrightarrow> \<not> d < x" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
716 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
717 |
then have "A = UNIV - {..< x}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
718 |
unfolding A by (auto simp: not_less[symmetric]) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
719 |
also have "\<dots> \<in> sigma_sets UNIV (range lessThan)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
720 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
721 |
finally show ?thesis . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
722 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
723 |
qed auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
724 |
qed auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
725 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
726 |
lemma borel_Ioi: |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
727 |
"borel = sigma UNIV (range greaterThan :: 'a::{linorder_topology, second_countable_topology} set set)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
728 |
unfolding second_countable_borel_measurable[OF open_generated_order] |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
729 |
proof (intro sigma_eqI sigma_sets_eqI) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
730 |
from countable_dense_setE guess D :: "'a set" . note D = this |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
731 |
|
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
732 |
interpret L: sigma_algebra UNIV "sigma_sets UNIV (range greaterThan)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
733 |
by (rule sigma_algebra_sigma_sets) simp |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
734 |
|
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
735 |
fix A :: "'a set" assume "A \<in> range lessThan \<union> range greaterThan" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
736 |
then obtain y where "A = {y <..} \<or> A = {..< y}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
737 |
by blast |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
738 |
then show "A \<in> sigma_sets UNIV (range greaterThan)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
739 |
proof |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
740 |
assume A: "A = {..< y}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
741 |
show ?thesis |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
742 |
proof cases |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
743 |
assume "\<forall>x<y. \<exists>d. x < d \<and> d < y" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
744 |
with D(2)[of "{x <..< y}" for x] have "\<forall>x<y. \<exists>d\<in>D. x < d \<and> d < y" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
745 |
by (auto simp: set_eq_iff) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
746 |
then have "A = UNIV - (\<Inter>d\<in>{d\<in>D. d < y}. {d <..})" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
747 |
by (auto simp: A) (metis less_asym) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
748 |
also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
749 |
using D(1) by (intro L.Diff L.top L.countable_INT'') auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
750 |
finally show ?thesis . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
751 |
next |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
752 |
assume "\<not> (\<forall>x<y. \<exists>d. x < d \<and> d < y)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
753 |
then obtain x where "x < y" "\<And>d. y > d \<Longrightarrow> x \<ge> d" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
754 |
by (auto simp: not_less[symmetric]) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
755 |
then have "A = UNIV - {x <..}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
756 |
unfolding A Compl_eq_Diff_UNIV[symmetric] by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
757 |
also have "\<dots> \<in> sigma_sets UNIV (range greaterThan)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
758 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
759 |
finally show ?thesis . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
760 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
761 |
qed auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
762 |
qed auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
763 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
764 |
lemma borel_measurableI_less: |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
765 |
fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
766 |
shows "(\<And>y. {x\<in>space M. f x < y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
767 |
unfolding borel_Iio |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
768 |
by (rule measurable_measure_of) (auto simp: Int_def conj_commute) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
769 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
770 |
lemma borel_measurableI_greater: |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
771 |
fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
772 |
shows "(\<And>y. {x\<in>space M. y < f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
773 |
unfolding borel_Ioi |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
774 |
by (rule measurable_measure_of) (auto simp: Int_def conj_commute) |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
775 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
776 |
lemma borel_measurableI_le: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
777 |
fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
778 |
shows "(\<And>y. {x\<in>space M. f x \<le> y} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
779 |
by (rule borel_measurableI_greater) (auto simp: not_le[symmetric]) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
780 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
781 |
lemma borel_measurableI_ge: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
782 |
fixes f :: "'a \<Rightarrow> 'b::{linorder_topology, second_countable_topology}" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
783 |
shows "(\<And>y. {x\<in>space M. y \<le> f x} \<in> sets M) \<Longrightarrow> f \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
784 |
by (rule borel_measurableI_less) (auto simp: not_le[symmetric]) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
785 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
786 |
lemma borel_measurable_less[measurable]: |
63332 | 787 |
fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
788 |
assumes "f \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
789 |
assumes "g \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
790 |
shows "{w \<in> space M. f w < g w} \<in> sets M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
791 |
proof - |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
792 |
have "{w \<in> space M. f w < g w} = (\<lambda>x. (f x, g x)) -` {x. fst x < snd x} \<inter> space M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
793 |
by auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
794 |
also have "\<dots> \<in> sets M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
795 |
by (intro measurable_sets[OF borel_measurable_Pair borel_open, OF assms open_Collect_less] |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
796 |
continuous_intros) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
797 |
finally show ?thesis . |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
798 |
qed |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
799 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
800 |
lemma (* FIX ME needs name *) |
63332 | 801 |
fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
802 |
assumes f[measurable]: "f \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
803 |
assumes g[measurable]: "g \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
804 |
shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
805 |
and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
806 |
and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
807 |
unfolding eq_iff not_less[symmetric] |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
808 |
by measurable |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
809 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
810 |
lemma borel_measurable_SUP[measurable (raw)]: |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
811 |
fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
812 |
assumes [simp]: "countable I" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
813 |
assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M" |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
814 |
shows "(\<lambda>x. SUP i\<in>I. F i x) \<in> borel_measurable M" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
815 |
by (rule borel_measurableI_greater) (simp add: less_SUP_iff) |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
816 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
817 |
lemma borel_measurable_INF[measurable (raw)]: |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
818 |
fixes F :: "_ \<Rightarrow> _ \<Rightarrow> _::{complete_linorder, linorder_topology, second_countable_topology}" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
819 |
assumes [simp]: "countable I" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
820 |
assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M" |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
821 |
shows "(\<lambda>x. INF i\<in>I. F i x) \<in> borel_measurable M" |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
822 |
by (rule borel_measurableI_less) (simp add: INF_less_iff) |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
823 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
824 |
lemma borel_measurable_cSUP[measurable (raw)]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
825 |
fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
826 |
assumes [simp]: "countable I" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
827 |
assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
828 |
assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_above ((\<lambda>i. F i x) ` I)" |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
829 |
shows "(\<lambda>x. SUP i\<in>I. F i x) \<in> borel_measurable M" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
830 |
proof cases |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
831 |
assume "I = {}" then show ?thesis |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
832 |
unfolding \<open>I = {}\<close> image_empty by simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
833 |
next |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
834 |
assume "I \<noteq> {}" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
835 |
show ?thesis |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
836 |
proof (rule borel_measurableI_le) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
837 |
fix y |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
838 |
have "{x \<in> space M. \<forall>i\<in>I. F i x \<le> y} \<in> sets M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
839 |
by measurable |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
840 |
also have "{x \<in> space M. \<forall>i\<in>I. F i x \<le> y} = {x \<in> space M. (SUP i\<in>I. F i x) \<le> y}" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
841 |
by (simp add: cSUP_le_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong) |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
842 |
finally show "{x \<in> space M. (SUP i\<in>I. F i x) \<le> y} \<in> sets M" . |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
843 |
qed |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
844 |
qed |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
845 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
846 |
lemma borel_measurable_cINF[measurable (raw)]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
847 |
fixes F :: "_ \<Rightarrow> _ \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology, second_countable_topology}" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
848 |
assumes [simp]: "countable I" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
849 |
assumes [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
850 |
assumes bdd: "\<And>x. x \<in> space M \<Longrightarrow> bdd_below ((\<lambda>i. F i x) ` I)" |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
851 |
shows "(\<lambda>x. INF i\<in>I. F i x) \<in> borel_measurable M" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
852 |
proof cases |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
853 |
assume "I = {}" then show ?thesis |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
854 |
unfolding \<open>I = {}\<close> image_empty by simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
855 |
next |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
856 |
assume "I \<noteq> {}" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
857 |
show ?thesis |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
858 |
proof (rule borel_measurableI_ge) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
859 |
fix y |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
860 |
have "{x \<in> space M. \<forall>i\<in>I. y \<le> F i x} \<in> sets M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
861 |
by measurable |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
862 |
also have "{x \<in> space M. \<forall>i\<in>I. y \<le> F i x} = {x \<in> space M. y \<le> (INF i\<in>I. F i x)}" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
863 |
by (simp add: le_cINF_iff \<open>I \<noteq> {}\<close> bdd cong: conj_cong) |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
864 |
finally show "{x \<in> space M. y \<le> (INF i\<in>I. F i x)} \<in> sets M" . |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
865 |
qed |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
866 |
qed |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
867 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
868 |
lemma borel_measurable_lfp[consumes 1, case_names continuity step]: |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
869 |
fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})" |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60150
diff
changeset
|
870 |
assumes "sup_continuous F" |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
871 |
assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
872 |
shows "lfp F \<in> borel_measurable M" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
873 |
proof - |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
874 |
{ fix i have "((F ^^ i) bot) \<in> borel_measurable M" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
875 |
by (induct i) (auto intro!: *) } |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
876 |
then have "(\<lambda>x. SUP i. (F ^^ i) bot x) \<in> borel_measurable M" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
877 |
by measurable |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
878 |
also have "(\<lambda>x. SUP i. (F ^^ i) bot x) = (SUP i. (F ^^ i) bot)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
879 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
880 |
also have "(SUP i. (F ^^ i) bot) = lfp F" |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60150
diff
changeset
|
881 |
by (rule sup_continuous_lfp[symmetric]) fact |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
882 |
finally show ?thesis . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
883 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
884 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
885 |
lemma borel_measurable_gfp[consumes 1, case_names continuity step]: |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
886 |
fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b::{complete_linorder, linorder_topology, second_countable_topology})" |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60150
diff
changeset
|
887 |
assumes "inf_continuous F" |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
888 |
assumes *: "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
889 |
shows "gfp F \<in> borel_measurable M" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
890 |
proof - |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
891 |
{ fix i have "((F ^^ i) top) \<in> borel_measurable M" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
892 |
by (induct i) (auto intro!: * simp: bot_fun_def) } |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
893 |
then have "(\<lambda>x. INF i. (F ^^ i) top x) \<in> borel_measurable M" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
894 |
by measurable |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
895 |
also have "(\<lambda>x. INF i. (F ^^ i) top x) = (INF i. (F ^^ i) top)" |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
896 |
by auto |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
897 |
also have "\<dots> = gfp F" |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60150
diff
changeset
|
898 |
by (rule inf_continuous_gfp[symmetric]) fact |
59088
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
899 |
finally show ?thesis . |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
900 |
qed |
ff2bd4a14ddb
generalized (borel_)measurable_SUP/INF/lfp/gfp; tuned proofs for sigma-closure of product spaces
hoelzl
parents:
59000
diff
changeset
|
901 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
902 |
lemma borel_measurable_max[measurable (raw)]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
903 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. max (g x) (f x) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
904 |
by (rule borel_measurableI_less) simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
905 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
906 |
lemma borel_measurable_min[measurable (raw)]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
907 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. min (g x) (f x) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
908 |
by (rule borel_measurableI_greater) simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
909 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
910 |
lemma borel_measurable_Min[measurable (raw)]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
911 |
"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M) \<Longrightarrow> (\<lambda>x. Min ((\<lambda>i. f i x)`I) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
912 |
proof (induct I rule: finite_induct) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
913 |
case (insert i I) then show ?case |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
914 |
by (cases "I = {}") auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
915 |
qed auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
916 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
917 |
lemma borel_measurable_Max[measurable (raw)]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
918 |
"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M) \<Longrightarrow> (\<lambda>x. Max ((\<lambda>i. f i x)`I) :: 'b::{second_countable_topology, linorder_topology}) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
919 |
proof (induct I rule: finite_induct) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
920 |
case (insert i I) then show ?case |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
921 |
by (cases "I = {}") auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
922 |
qed auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
923 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
924 |
lemma borel_measurable_sup[measurable (raw)]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
925 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. sup (g x) (f x) :: 'b::{lattice, second_countable_topology, linorder_topology}) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
926 |
unfolding sup_max by measurable |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
927 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
928 |
lemma borel_measurable_inf[measurable (raw)]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
929 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. inf (g x) (f x) :: 'b::{lattice, second_countable_topology, linorder_topology}) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
930 |
unfolding inf_min by measurable |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
931 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
932 |
lemma [measurable (raw)]: (*FIXME needs name *) |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
933 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
934 |
assumes "\<And>i. f i \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
935 |
shows borel_measurable_liminf: "(\<lambda>x. liminf (\<lambda>i. f i x)) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
936 |
and borel_measurable_limsup: "(\<lambda>x. limsup (\<lambda>i. f i x)) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
937 |
unfolding liminf_SUP_INF limsup_INF_SUP using assms by auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
938 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
939 |
lemma measurable_convergent[measurable (raw)]: |
63332 | 940 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
941 |
assumes [measurable]: "\<And>i. f i \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
942 |
shows "Measurable.pred M (\<lambda>x. convergent (\<lambda>i. f i x))" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
943 |
unfolding convergent_ereal by measurable |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
944 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
945 |
lemma sets_Collect_convergent[measurable]: |
63332 | 946 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
947 |
assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
948 |
shows "{x\<in>space M. convergent (\<lambda>i. f i x)} \<in> sets M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
949 |
by measurable |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
950 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
951 |
lemma borel_measurable_lim[measurable (raw)]: |
63332 | 952 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
953 |
assumes [measurable]: "\<And>i. f i \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
954 |
shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
955 |
proof - |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
956 |
have "\<And>x. lim (\<lambda>i. f i x) = (if convergent (\<lambda>i. f i x) then limsup (\<lambda>i. f i x) else (THE i. False))" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
957 |
by (simp add: lim_def convergent_def convergent_limsup_cl) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
958 |
then show ?thesis |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
959 |
by simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
960 |
qed |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
961 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
962 |
lemma borel_measurable_LIMSEQ_order: |
63332 | 963 |
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology}" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
964 |
assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
965 |
and u: "\<And>i. u i \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
966 |
shows "u' \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
967 |
proof - |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
968 |
have "\<And>x. x \<in> space M \<Longrightarrow> u' x = liminf (\<lambda>n. u n x)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
969 |
using u' by (simp add: lim_imp_Liminf[symmetric]) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
970 |
with u show ?thesis by (simp cong: measurable_cong) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
971 |
qed |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
972 |
|
69683 | 973 |
subsection \<open>Borel spaces on topological monoids\<close> |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
974 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
975 |
lemma borel_measurable_add[measurable (raw)]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
976 |
fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, topological_monoid_add}" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
977 |
assumes f: "f \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
978 |
assumes g: "g \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
979 |
shows "(\<lambda>x. f x + g x) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
980 |
using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
981 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
982 |
lemma borel_measurable_sum[measurable (raw)]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
983 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, topological_comm_monoid_add}" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
984 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
985 |
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
986 |
proof cases |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
987 |
assume "finite S" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
988 |
thus ?thesis using assms by induct auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
989 |
qed simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
990 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
991 |
lemma borel_measurable_suminf_order[measurable (raw)]: |
63332 | 992 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{complete_linorder, second_countable_topology, linorder_topology, topological_comm_monoid_add}" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
993 |
assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
994 |
shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
995 |
unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
996 |
|
69683 | 997 |
subsection \<open>Borel spaces on Euclidean spaces\<close> |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
998 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
999 |
lemma borel_measurable_inner[measurable (raw)]: |
50881
ae630bab13da
renamed countable_basis_space to second_countable_topology
hoelzl
parents:
50526
diff
changeset
|
1000 |
fixes f g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_inner}" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1001 |
assumes "f \<in> borel_measurable M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1002 |
assumes "g \<in> borel_measurable M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1003 |
shows "(\<lambda>x. f x \<bullet> g x) \<in> borel_measurable M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1004 |
using assms |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1005 |
by (rule borel_measurable_continuous_Pair) (intro continuous_intros) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1006 |
|
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1007 |
notation |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1008 |
eucl_less (infix "<e" 50) |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1009 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1010 |
lemma box_oc: "{x. a <e x \<and> x \<le> b} = {x. a <e x} \<inter> {..b}" |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1011 |
and box_co: "{x. a \<le> x \<and> x <e b} = {a..} \<inter> {x. x <e b}" |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1012 |
by auto |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1013 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1014 |
lemma eucl_ivals[measurable]: |
61076 | 1015 |
fixes a b :: "'a::ordered_euclidean_space" |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1016 |
shows "{x. x <e a} \<in> sets borel" |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1017 |
and "{x. a <e x} \<in> sets borel" |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1018 |
and "{..a} \<in> sets borel" |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1019 |
and "{a..} \<in> sets borel" |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1020 |
and "{a..b} \<in> sets borel" |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1021 |
and "{x. a <e x \<and> x \<le> b} \<in> sets borel" |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1022 |
and "{x. a \<le> x \<and> x <e b} \<in> sets borel" |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1023 |
unfolding box_oc box_co |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1024 |
by (auto intro: borel_open borel_closed) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1025 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1026 |
lemma (*FIX ME this has no name *) |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1027 |
fixes i :: "'a::{second_countable_topology, real_inner}" |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1028 |
shows hafspace_less_borel: "{x. a < x \<bullet> i} \<in> sets borel" |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1029 |
and hafspace_greater_borel: "{x. x \<bullet> i < a} \<in> sets borel" |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1030 |
and hafspace_less_eq_borel: "{x. a \<le> x \<bullet> i} \<in> sets borel" |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1031 |
and hafspace_greater_eq_borel: "{x. x \<bullet> i \<le> a} \<in> sets borel" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1032 |
by simp_all |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1033 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1034 |
lemma borel_eq_box: |
61076 | 1035 |
"borel = sigma UNIV (range (\<lambda> (a, b). box a b :: 'a :: euclidean_space set))" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1036 |
(is "_ = ?SIGMA") |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1037 |
proof (rule borel_eq_sigmaI1[OF borel_def]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1038 |
fix M :: "'a set" assume "M \<in> {S. open S}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1039 |
then have "open M" by simp |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1040 |
show "M \<in> ?SIGMA" |
61808 | 1041 |
apply (subst open_UNION_box[OF \<open>open M\<close>]) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1042 |
apply (safe intro!: sets.countable_UN' countable_PiE countable_Collect) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1043 |
apply (auto intro: countable_rat) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1044 |
done |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1045 |
qed (auto simp: box_def) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1046 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1047 |
lemma halfspace_gt_in_halfspace: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1048 |
assumes i: "i \<in> A" |
62372 | 1049 |
shows "{x::'a. a < x \<bullet> i} \<in> |
61076 | 1050 |
sigma_sets UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> A))" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1051 |
(is "?set \<in> ?SIGMA") |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1052 |
proof - |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1053 |
interpret sigma_algebra UNIV ?SIGMA |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1054 |
by (intro sigma_algebra_sigma_sets) simp_all |
61076 | 1055 |
have *: "?set = (\<Union>n. UNIV - {x::'a. x \<bullet> i < a + 1 / real (Suc n)})" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1056 |
proof (safe, simp_all add: not_less del: of_nat_Suc) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1057 |
fix x :: 'a assume "a < x \<bullet> i" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1058 |
with reals_Archimedean[of "x \<bullet> i - a"] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1059 |
obtain n where "a + 1 / real (Suc n) < x \<bullet> i" |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1060 |
by (auto simp: field_simps) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1061 |
then show "\<exists>n. a + 1 / real (Suc n) \<le> x \<bullet> i" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1062 |
by (blast intro: less_imp_le) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1063 |
next |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1064 |
fix x n |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1065 |
have "a < a + 1 / real (Suc n)" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1066 |
also assume "\<dots> \<le> x" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1067 |
finally show "a < x" . |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1068 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1069 |
show "?set \<in> ?SIGMA" unfolding * |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61284
diff
changeset
|
1070 |
by (auto intro!: Diff sigma_sets_Inter i) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1071 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1072 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1073 |
lemma borel_eq_halfspace_less: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1074 |
"borel = sigma UNIV ((\<lambda>(a, i). {x::'a::euclidean_space. x \<bullet> i < a}) ` (UNIV \<times> Basis))" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1075 |
(is "_ = ?SIGMA") |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1076 |
proof (rule borel_eq_sigmaI2[OF borel_eq_box]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1077 |
fix a b :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1078 |
have "box a b = {x\<in>space ?SIGMA. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1079 |
by (auto simp: box_def) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1080 |
also have "\<dots> \<in> sets ?SIGMA" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1081 |
by (intro sets.sets_Collect_conj sets.sets_Collect_finite_All sets.sets_Collect_const) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1082 |
(auto intro!: halfspace_gt_in_halfspace countable_PiE countable_rat) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1083 |
finally show "box a b \<in> sets ?SIGMA" . |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1084 |
qed auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1085 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1086 |
lemma borel_eq_halfspace_le: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1087 |
"borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. x \<bullet> i \<le> a}) ` (UNIV \<times> Basis))" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1088 |
(is "_ = ?SIGMA") |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1089 |
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1090 |
fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1091 |
then have i: "i \<in> Basis" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1092 |
have *: "{x::'a. x\<bullet>i < a} = (\<Union>n. {x. x\<bullet>i \<le> a - 1/real (Suc n)})" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1093 |
proof (safe, simp_all del: of_nat_Suc) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1094 |
fix x::'a assume *: "x\<bullet>i < a" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1095 |
with reals_Archimedean[of "a - x\<bullet>i"] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1096 |
obtain n where "x \<bullet> i < a - 1 / (real (Suc n))" |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1097 |
by (auto simp: field_simps) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1098 |
then show "\<exists>n. x \<bullet> i \<le> a - 1 / (real (Suc n))" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1099 |
by (blast intro: less_imp_le) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1100 |
next |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1101 |
fix x::'a and n |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1102 |
assume "x\<bullet>i \<le> a - 1 / real (Suc n)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1103 |
also have "\<dots> < a" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1104 |
finally show "x\<bullet>i < a" . |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1105 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1106 |
show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding * |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1107 |
by (intro sets.countable_UN) (auto intro: i) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1108 |
qed auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1109 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1110 |
lemma borel_eq_halfspace_ge: |
61076 | 1111 |
"borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a \<le> x \<bullet> i}) ` (UNIV \<times> Basis))" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1112 |
(is "_ = ?SIGMA") |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1113 |
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_less]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1114 |
fix a :: real and i :: 'a assume i: "(a, i) \<in> UNIV \<times> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1115 |
have *: "{x::'a. x\<bullet>i < a} = space ?SIGMA - {x::'a. a \<le> x\<bullet>i}" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1116 |
show "{x. x\<bullet>i < a} \<in> ?SIGMA" unfolding * |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1117 |
using i by (intro sets.compl_sets) auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1118 |
qed auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1119 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1120 |
lemma borel_eq_halfspace_greater: |
61076 | 1121 |
"borel = sigma UNIV ((\<lambda> (a, i). {x::'a::euclidean_space. a < x \<bullet> i}) ` (UNIV \<times> Basis))" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1122 |
(is "_ = ?SIGMA") |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1123 |
proof (rule borel_eq_sigmaI2[OF borel_eq_halfspace_le]) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1124 |
fix a :: real and i :: 'a assume "(a, i) \<in> (UNIV \<times> Basis)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1125 |
then have i: "i \<in> Basis" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1126 |
have *: "{x::'a. x\<bullet>i \<le> a} = space ?SIGMA - {x::'a. a < x\<bullet>i}" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1127 |
show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding * |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1128 |
by (intro sets.compl_sets) (auto intro: i) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1129 |
qed auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1130 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1131 |
lemma borel_eq_atMost: |
61076 | 1132 |
"borel = sigma UNIV (range (\<lambda>a. {..a::'a::ordered_euclidean_space}))" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1133 |
(is "_ = ?SIGMA") |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1134 |
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1135 |
fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1136 |
then have "i \<in> Basis" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1137 |
then have *: "{x::'a. x\<bullet>i \<le> a} = (\<Union>k::nat. {.. (\<Sum>n\<in>Basis. (if n = i then a else real k)*\<^sub>R n)})" |
62390 | 1138 |
proof (safe, simp_all add: eucl_le[where 'a='a] split: if_split_asm) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1139 |
fix x :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1140 |
from real_arch_simple[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"] guess k::nat .. |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1141 |
then have "\<And>i. i \<in> Basis \<Longrightarrow> x\<bullet>i \<le> real k" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1142 |
by (subst (asm) Max_le_iff) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1143 |
then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia \<le> real k" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1144 |
by (auto intro!: exI[of _ k]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1145 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1146 |
show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" unfolding * |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1147 |
by (intro sets.countable_UN) auto |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1148 |
qed auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1149 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1150 |
lemma borel_eq_greaterThan: |
61076 | 1151 |
"borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. a <e x}))" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1152 |
(is "_ = ?SIGMA") |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1153 |
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_le]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1154 |
fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1155 |
then have i: "i \<in> Basis" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1156 |
have "{x::'a. x\<bullet>i \<le> a} = UNIV - {x::'a. a < x\<bullet>i}" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1157 |
also have *: "{x::'a. a < x\<bullet>i} = |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1158 |
(\<Union>k::nat. {x. (\<Sum>n\<in>Basis. (if n = i then a else -real k) *\<^sub>R n) <e x})" using i |
62390 | 1159 |
proof (safe, simp_all add: eucl_less_def split: if_split_asm) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1160 |
fix x :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1161 |
from reals_Archimedean2[of "Max ((\<lambda>i. -x\<bullet>i)`Basis)"] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1162 |
guess k::nat .. note k = this |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1163 |
{ fix i :: 'a assume "i \<in> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1164 |
then have "-x\<bullet>i < real k" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1165 |
using k by (subst (asm) Max_less_iff) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1166 |
then have "- real k < x\<bullet>i" by simp } |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1167 |
then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> -real k < x \<bullet> ia" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1168 |
by (auto intro!: exI[of _ k]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1169 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1170 |
finally show "{x. x\<bullet>i \<le> a} \<in> ?SIGMA" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1171 |
apply (simp only:) |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1172 |
apply (intro sets.countable_UN sets.Diff) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1173 |
apply (auto intro: sigma_sets_top) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1174 |
done |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1175 |
qed auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1176 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1177 |
lemma borel_eq_lessThan: |
61076 | 1178 |
"borel = sigma UNIV (range (\<lambda>a::'a::ordered_euclidean_space. {x. x <e a}))" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1179 |
(is "_ = ?SIGMA") |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1180 |
proof (rule borel_eq_sigmaI4[OF borel_eq_halfspace_ge]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1181 |
fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1182 |
then have i: "i \<in> Basis" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1183 |
have "{x::'a. a \<le> x\<bullet>i} = UNIV - {x::'a. x\<bullet>i < a}" by auto |
61808 | 1184 |
also have *: "{x::'a. x\<bullet>i < a} = (\<Union>k::nat. {x. x <e (\<Sum>n\<in>Basis. (if n = i then a else real k) *\<^sub>R n)})" using \<open>i\<in> Basis\<close> |
62390 | 1185 |
proof (safe, simp_all add: eucl_less_def split: if_split_asm) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1186 |
fix x :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1187 |
from reals_Archimedean2[of "Max ((\<lambda>i. x\<bullet>i)`Basis)"] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1188 |
guess k::nat .. note k = this |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1189 |
{ fix i :: 'a assume "i \<in> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1190 |
then have "x\<bullet>i < real k" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1191 |
using k by (subst (asm) Max_less_iff) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1192 |
then have "x\<bullet>i < real k" by simp } |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1193 |
then show "\<exists>k::nat. \<forall>ia\<in>Basis. ia \<noteq> i \<longrightarrow> x \<bullet> ia < real k" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1194 |
by (auto intro!: exI[of _ k]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1195 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1196 |
finally show "{x. a \<le> x\<bullet>i} \<in> ?SIGMA" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1197 |
apply (simp only:) |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1198 |
apply (intro sets.countable_UN sets.Diff) |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1199 |
apply (auto intro: sigma_sets_top ) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1200 |
done |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1201 |
qed auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1202 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1203 |
lemma borel_eq_atLeastAtMost: |
61076 | 1204 |
"borel = sigma UNIV (range (\<lambda>(a,b). {a..b} ::'a::ordered_euclidean_space set))" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1205 |
(is "_ = ?SIGMA") |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1206 |
proof (rule borel_eq_sigmaI5[OF borel_eq_atMost]) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1207 |
fix a::'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1208 |
have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1209 |
proof (safe, simp_all add: eucl_le[where 'a='a]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1210 |
fix x :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1211 |
from real_arch_simple[of "Max ((\<lambda>i. - x\<bullet>i)`Basis)"] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1212 |
guess k::nat .. note k = this |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1213 |
{ fix i :: 'a assume "i \<in> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1214 |
with k have "- x\<bullet>i \<le> real k" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1215 |
by (subst (asm) Max_le_iff) (auto simp: field_simps) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1216 |
then have "- real k \<le> x\<bullet>i" by simp } |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1217 |
then show "\<exists>n::nat. \<forall>i\<in>Basis. - real n \<le> x \<bullet> i" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1218 |
by (auto intro!: exI[of _ k]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1219 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1220 |
show "{..a} \<in> ?SIGMA" unfolding * |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1221 |
by (intro sets.countable_UN) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1222 |
(auto intro!: sigma_sets_top) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1223 |
qed auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1224 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1225 |
lemma borel_set_induct[consumes 1, case_names empty interval compl union]: |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1226 |
assumes "A \<in> sets borel" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1227 |
assumes empty: "P {}" and int: "\<And>a b. a \<le> b \<Longrightarrow> P {a..b}" and compl: "\<And>A. A \<in> sets borel \<Longrightarrow> P A \<Longrightarrow> P (-A)" and |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1228 |
un: "\<And>f. disjoint_family f \<Longrightarrow> (\<And>i. f i \<in> sets borel) \<Longrightarrow> (\<And>i. P (f i)) \<Longrightarrow> P (\<Union>i::nat. f i)" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1229 |
shows "P (A::real set)" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1230 |
proof - |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1231 |
let ?G = "range (\<lambda>(a,b). {a..b::real})" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1232 |
have "Int_stable ?G" "?G \<subseteq> Pow UNIV" "A \<in> sigma_sets UNIV ?G" |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1233 |
using assms(1) by (auto simp add: borel_eq_atLeastAtMost Int_stable_def) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1234 |
thus ?thesis |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1235 |
proof (induction rule: sigma_sets_induct_disjoint) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1236 |
case (union f) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1237 |
from union.hyps(2) have "\<And>i. f i \<in> sets borel" by (auto simp: borel_eq_atLeastAtMost) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1238 |
with union show ?case by (auto intro: un) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1239 |
next |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1240 |
case (basic A) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1241 |
then obtain a b where "A = {a .. b}" by auto |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1242 |
then show ?case |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1243 |
by (cases "a \<le> b") (auto intro: int empty) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1244 |
qed (auto intro: empty compl simp: Compl_eq_Diff_UNIV[symmetric] borel_eq_atLeastAtMost) |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1245 |
qed |
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1246 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1247 |
lemma borel_sigma_sets_Ioc: "borel = sigma UNIV (range (\<lambda>(a, b). {a <.. b::real}))" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1248 |
proof (rule borel_eq_sigmaI5[OF borel_eq_atMost]) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1249 |
fix i :: real |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1250 |
have "{..i} = (\<Union>j::nat. {-j <.. i})" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1251 |
by (auto simp: minus_less_iff reals_Archimedean2) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1252 |
also have "\<dots> \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))" |
62372 | 1253 |
by (intro sets.countable_nat_UN) auto |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1254 |
finally show "{..i} \<in> sets (sigma UNIV (range (\<lambda>(i, j). {i<..j})))" . |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1255 |
qed simp |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1256 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1257 |
lemma eucl_lessThan: "{x::real. x <e a} = lessThan a" |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1258 |
by (simp add: eucl_less_def lessThan_def) |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1259 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1260 |
lemma borel_eq_atLeastLessThan: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1261 |
"borel = sigma UNIV (range (\<lambda>(a, b). {a ..< b :: real}))" (is "_ = ?SIGMA") |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1262 |
proof (rule borel_eq_sigmaI5[OF borel_eq_lessThan]) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1263 |
have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1264 |
fix x :: real |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1265 |
have "{..<x} = (\<Union>i::nat. {-real i ..< x})" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1266 |
by (auto simp: move_uminus real_arch_simple) |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1267 |
then show "{y. y <e x} \<in> ?SIGMA" |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1268 |
by (auto intro: sigma_sets.intros(2-) simp: eucl_lessThan) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1269 |
qed auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1270 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1271 |
lemma borel_measurable_halfspacesI: |
61076 | 1272 |
fixes f :: "'a \<Rightarrow> 'c::euclidean_space" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1273 |
assumes F: "borel = sigma UNIV (F ` (UNIV \<times> Basis))" |
62372 | 1274 |
and S_eq: "\<And>a i. S a i = f -` F (a,i) \<inter> space M" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1275 |
shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a::real. S a i \<in> sets M)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1276 |
proof safe |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1277 |
fix a :: real and i :: 'b assume i: "i \<in> Basis" and f: "f \<in> borel_measurable M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1278 |
then show "S a i \<in> sets M" unfolding assms |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1279 |
by (auto intro!: measurable_sets simp: assms(1)) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1280 |
next |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1281 |
assume a: "\<forall>i\<in>Basis. \<forall>a. S a i \<in> sets M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1282 |
then show "f \<in> borel_measurable M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1283 |
by (auto intro!: measurable_measure_of simp: S_eq F) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1284 |
qed |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1285 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1286 |
lemma borel_measurable_iff_halfspace_le: |
61076 | 1287 |
fixes f :: "'a \<Rightarrow> 'c::euclidean_space" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1288 |
shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i \<le> a} \<in> sets M)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1289 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1290 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1291 |
lemma borel_measurable_iff_halfspace_less: |
61076 | 1292 |
fixes f :: "'a \<Rightarrow> 'c::euclidean_space" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1293 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i < a} \<in> sets M)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1294 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1295 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1296 |
lemma borel_measurable_iff_halfspace_ge: |
61076 | 1297 |
fixes f :: "'a \<Rightarrow> 'c::euclidean_space" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1298 |
shows "f \<in> borel_measurable M = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a \<le> f w \<bullet> i} \<in> sets M)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1299 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1300 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1301 |
lemma borel_measurable_iff_halfspace_greater: |
61076 | 1302 |
fixes f :: "'a \<Rightarrow> 'c::euclidean_space" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1303 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a < f w \<bullet> i} \<in> sets M)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1304 |
by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_greater]) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1305 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1306 |
lemma borel_measurable_iff_le: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1307 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1308 |
using borel_measurable_iff_halfspace_le[where 'c=real] by simp |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1309 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1310 |
lemma borel_measurable_iff_less: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1311 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1312 |
using borel_measurable_iff_halfspace_less[where 'c=real] by simp |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1313 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1314 |
lemma borel_measurable_iff_ge: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1315 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1316 |
using borel_measurable_iff_halfspace_ge[where 'c=real] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1317 |
by simp |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1318 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1319 |
lemma borel_measurable_iff_greater: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1320 |
"(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1321 |
using borel_measurable_iff_halfspace_greater[where 'c=real] by simp |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1322 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1323 |
lemma borel_measurable_euclidean_space: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1324 |
fixes f :: "'a \<Rightarrow> 'c::euclidean_space" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1325 |
shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1326 |
proof safe |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1327 |
assume f: "\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1328 |
then show "f \<in> borel_measurable M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1329 |
by (subst borel_measurable_iff_halfspace_le) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1330 |
qed auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1331 |
|
69683 | 1332 |
subsection "Borel measurable operators" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1333 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1334 |
lemma borel_measurable_norm[measurable]: "norm \<in> borel_measurable borel" |
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1335 |
by (intro borel_measurable_continuous_on1 continuous_intros) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1336 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1337 |
lemma borel_measurable_sgn [measurable]: "(sgn::'a::real_normed_vector \<Rightarrow> 'a) \<in> borel_measurable borel" |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1338 |
by (rule borel_measurable_continuous_countable_exceptions[where X="{0}"]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1339 |
(auto intro!: continuous_on_sgn continuous_on_id) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1340 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1341 |
lemma borel_measurable_uminus[measurable (raw)]: |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1342 |
fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1343 |
assumes g: "g \<in> borel_measurable M" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1344 |
shows "(\<lambda>x. - g x) \<in> borel_measurable M" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1345 |
by (rule borel_measurable_continuous_on[OF _ g]) (intro continuous_intros) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1346 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1347 |
lemma borel_measurable_diff[measurable (raw)]: |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1348 |
fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}" |
49774 | 1349 |
assumes f: "f \<in> borel_measurable M" |
1350 |
assumes g: "g \<in> borel_measurable M" |
|
1351 |
shows "(\<lambda>x. f x - g x) \<in> borel_measurable M" |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1352 |
using borel_measurable_add [of f M "- g"] assms by (simp add: fun_Compl_def) |
49774 | 1353 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1354 |
lemma borel_measurable_times[measurable (raw)]: |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1355 |
fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_algebra}" |
49774 | 1356 |
assumes f: "f \<in> borel_measurable M" |
1357 |
assumes g: "g \<in> borel_measurable M" |
|
1358 |
shows "(\<lambda>x. f x * g x) \<in> borel_measurable M" |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56212
diff
changeset
|
1359 |
using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros) |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1360 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1361 |
lemma borel_measurable_prod[measurable (raw)]: |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1362 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b::{second_countable_topology, real_normed_field}" |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1363 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1364 |
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1365 |
proof cases |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1366 |
assume "finite S" |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1367 |
thus ?thesis using assms by induct auto |
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1368 |
qed simp |
49774 | 1369 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1370 |
lemma borel_measurable_dist[measurable (raw)]: |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1371 |
fixes g f :: "'a \<Rightarrow> 'b::{second_countable_topology, metric_space}" |
49774 | 1372 |
assumes f: "f \<in> borel_measurable M" |
1373 |
assumes g: "g \<in> borel_measurable M" |
|
1374 |
shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M" |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1375 |
using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros) |
62372 | 1376 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1377 |
lemma borel_measurable_scaleR[measurable (raw)]: |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1378 |
fixes g :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1379 |
assumes f: "f \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1380 |
assumes g: "g \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1381 |
shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M" |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56212
diff
changeset
|
1382 |
using f g by (rule borel_measurable_continuous_Pair) (intro continuous_intros) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1383 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1384 |
lemma borel_measurable_uminus_eq [simp]: |
66164
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1385 |
fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, real_normed_vector}" |
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1386 |
shows "(\<lambda>x. - f x) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r") |
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1387 |
proof |
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1388 |
assume ?l from borel_measurable_uminus[OF this] show ?r by simp |
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1389 |
qed auto |
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1390 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1391 |
lemma affine_borel_measurable_vector: |
38656 | 1392 |
fixes f :: "'a \<Rightarrow> 'x::real_normed_vector" |
1393 |
assumes "f \<in> borel_measurable M" |
|
1394 |
shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M" |
|
1395 |
proof (rule borel_measurableI) |
|
1396 |
fix S :: "'x set" assume "open S" |
|
1397 |
show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M" |
|
1398 |
proof cases |
|
1399 |
assume "b \<noteq> 0" |
|
61808 | 1400 |
with \<open>open S\<close> have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S") |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53216
diff
changeset
|
1401 |
using open_affinity [of S "inverse b" "- a /\<^sub>R b"] |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53216
diff
changeset
|
1402 |
by (auto simp: algebra_simps) |
47694 | 1403 |
hence "?S \<in> sets borel" by auto |
38656 | 1404 |
moreover |
61808 | 1405 |
from \<open>b \<noteq> 0\<close> have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S" |
38656 | 1406 |
apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all) |
40859 | 1407 |
ultimately show ?thesis using assms unfolding in_borel_measurable_borel |
38656 | 1408 |
by auto |
1409 |
qed simp |
|
1410 |
qed |
|
1411 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1412 |
lemma borel_measurable_const_scaleR[measurable (raw)]: |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1413 |
"f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. b *\<^sub>R f x ::'a::real_normed_vector) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1414 |
using affine_borel_measurable_vector[of f M 0 b] by simp |
38656 | 1415 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1416 |
lemma borel_measurable_const_add[measurable (raw)]: |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1417 |
"f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. a + f x ::'a::real_normed_vector) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1418 |
using affine_borel_measurable_vector[of f M a 1] by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1419 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1420 |
lemma borel_measurable_inverse[measurable (raw)]: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1421 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra" |
49774 | 1422 |
assumes f: "f \<in> borel_measurable M" |
35692 | 1423 |
shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M" |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1424 |
apply (rule measurable_compose[OF f]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1425 |
apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1426 |
apply (auto intro!: continuous_on_inverse continuous_on_id) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1427 |
done |
35692 | 1428 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1429 |
lemma borel_measurable_divide[measurable (raw)]: |
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
1430 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1431 |
(\<lambda>x. f x / g x::'b::{second_countable_topology, real_normed_div_algebra}) \<in> borel_measurable M" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1432 |
by (simp add: divide_inverse) |
38656 | 1433 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1434 |
lemma borel_measurable_abs[measurable (raw)]: |
50003 | 1435 |
"f \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M" |
1436 |
unfolding abs_real_def by simp |
|
38656 | 1437 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1438 |
lemma borel_measurable_nth[measurable (raw)]: |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
1439 |
"(\<lambda>x::real^'n. x $ i) \<in> borel_measurable borel" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50419
diff
changeset
|
1440 |
by (simp add: cart_eq_inner_axis) |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41025
diff
changeset
|
1441 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1442 |
lemma convex_measurable: |
59415 | 1443 |
fixes A :: "'a :: euclidean_space set" |
62372 | 1444 |
shows "X \<in> borel_measurable M \<Longrightarrow> X ` space M \<subseteq> A \<Longrightarrow> open A \<Longrightarrow> convex_on A q \<Longrightarrow> |
59415 | 1445 |
(\<lambda>x. q (X x)) \<in> borel_measurable M" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1446 |
by (rule measurable_compose[where f=X and N="restrict_space borel A"]) |
59415 | 1447 |
(auto intro!: borel_measurable_continuous_on_restrict convex_on_continuous measurable_restrict_space2) |
41830 | 1448 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1449 |
lemma borel_measurable_ln[measurable (raw)]: |
49774 | 1450 |
assumes f: "f \<in> borel_measurable M" |
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1451 |
shows "(\<lambda>x. ln (f x :: real)) \<in> borel_measurable M" |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1452 |
apply (rule measurable_compose[OF f]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1453 |
apply (rule borel_measurable_continuous_countable_exceptions[of "{0}"]) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1454 |
apply (auto intro!: continuous_on_ln continuous_on_id) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57259
diff
changeset
|
1455 |
done |
41830 | 1456 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1457 |
lemma borel_measurable_log[measurable (raw)]: |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1458 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<lambda>x. log (g x) (f x)) \<in> borel_measurable M" |
49774 | 1459 |
unfolding log_def by auto |
41830 | 1460 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1461 |
lemma borel_measurable_exp[measurable]: |
58656 | 1462 |
"(exp::'a::{real_normed_field,banach}\<Rightarrow>'a) \<in> borel_measurable borel" |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51351
diff
changeset
|
1463 |
by (intro borel_measurable_continuous_on1 continuous_at_imp_continuous_on ballI isCont_exp) |
50419 | 1464 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1465 |
lemma measurable_real_floor[measurable]: |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1466 |
"(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)" |
47761 | 1467 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1468 |
have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real_of_int a \<le> x \<and> x < real_of_int (a + 1))" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1469 |
by (auto intro: floor_eq2) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1470 |
then show ?thesis |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1471 |
by (auto simp: vimage_def measurable_count_space_eq2_countable) |
47761 | 1472 |
qed |
1473 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1474 |
lemma measurable_real_ceiling[measurable]: |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1475 |
"(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1476 |
unfolding ceiling_def[abs_def] by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1477 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1478 |
lemma borel_measurable_real_floor: "(\<lambda>x::real. real_of_int \<lfloor>x\<rfloor>) \<in> borel_measurable borel" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1479 |
by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1480 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1481 |
lemma borel_measurable_root [measurable]: "root n \<in> borel_measurable borel" |
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1482 |
by (intro borel_measurable_continuous_on1 continuous_intros) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1483 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1484 |
lemma borel_measurable_sqrt [measurable]: "sqrt \<in> borel_measurable borel" |
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1485 |
by (intro borel_measurable_continuous_on1 continuous_intros) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1486 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1487 |
lemma borel_measurable_power [measurable (raw)]: |
59415 | 1488 |
fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}" |
1489 |
assumes f: "f \<in> borel_measurable M" |
|
1490 |
shows "(\<lambda>x. (f x) ^ n) \<in> borel_measurable M" |
|
1491 |
by (intro borel_measurable_continuous_on [OF _ f] continuous_intros) |
|
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1492 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1493 |
lemma borel_measurable_Re [measurable]: "Re \<in> borel_measurable borel" |
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1494 |
by (intro borel_measurable_continuous_on1 continuous_intros) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1495 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1496 |
lemma borel_measurable_Im [measurable]: "Im \<in> borel_measurable borel" |
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1497 |
by (intro borel_measurable_continuous_on1 continuous_intros) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1498 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1499 |
lemma borel_measurable_of_real [measurable]: "(of_real :: _ \<Rightarrow> (_::real_normed_algebra)) \<in> borel_measurable borel" |
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1500 |
by (intro borel_measurable_continuous_on1 continuous_intros) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1501 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1502 |
lemma borel_measurable_sin [measurable]: "(sin :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel" |
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1503 |
by (intro borel_measurable_continuous_on1 continuous_intros) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1504 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1505 |
lemma borel_measurable_cos [measurable]: "(cos :: _ \<Rightarrow> (_::{real_normed_field,banach})) \<in> borel_measurable borel" |
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1506 |
by (intro borel_measurable_continuous_on1 continuous_intros) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1507 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1508 |
lemma borel_measurable_arctan [measurable]: "arctan \<in> borel_measurable borel" |
57235
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1509 |
by (intro borel_measurable_continuous_on1 continuous_intros) |
b0b9a10e4bf4
properties of Erlang and exponentially distributed random variables (by Sudeep Kanav)
hoelzl
parents:
57138
diff
changeset
|
1510 |
|
68833
fde093888c16
tagged 21 theories in the Analysis library for the manual
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
68635
diff
changeset
|
1511 |
lemma%important borel_measurable_complex_iff: |
57259
3a448982a74a
add more derivative and continuity rules for complex-values functions
hoelzl
parents:
57235
diff
changeset
|
1512 |
"f \<in> borel_measurable M \<longleftrightarrow> |
3a448982a74a
add more derivative and continuity rules for complex-values functions
hoelzl
parents:
57235
diff
changeset
|
1513 |
(\<lambda>x. Re (f x)) \<in> borel_measurable M \<and> (\<lambda>x. Im (f x)) \<in> borel_measurable M" |
3a448982a74a
add more derivative and continuity rules for complex-values functions
hoelzl
parents:
57235
diff
changeset
|
1514 |
apply auto |
3a448982a74a
add more derivative and continuity rules for complex-values functions
hoelzl
parents:
57235
diff
changeset
|
1515 |
apply (subst fun_complex_eq) |
3a448982a74a
add more derivative and continuity rules for complex-values functions
hoelzl
parents:
57235
diff
changeset
|
1516 |
apply (intro borel_measurable_add) |
3a448982a74a
add more derivative and continuity rules for complex-values functions
hoelzl
parents:
57235
diff
changeset
|
1517 |
apply auto |
3a448982a74a
add more derivative and continuity rules for complex-values functions
hoelzl
parents:
57235
diff
changeset
|
1518 |
done |
3a448982a74a
add more derivative and continuity rules for complex-values functions
hoelzl
parents:
57235
diff
changeset
|
1519 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1520 |
lemma powr_real_measurable [measurable]: |
67278
c60e3d615b8c
Removed Analysis/ex/Circle_Area; replaced by more general Analysis/Ball_Volume
eberlm <eberlm@in.tum.de>
parents:
66164
diff
changeset
|
1521 |
assumes "f \<in> measurable M borel" "g \<in> measurable M borel" |
c60e3d615b8c
Removed Analysis/ex/Circle_Area; replaced by more general Analysis/Ball_Volume
eberlm <eberlm@in.tum.de>
parents:
66164
diff
changeset
|
1522 |
shows "(\<lambda>x. f x powr g x :: real) \<in> measurable M borel" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1523 |
using assms by (simp_all add: powr_def) |
67278
c60e3d615b8c
Removed Analysis/ex/Circle_Area; replaced by more general Analysis/Ball_Volume
eberlm <eberlm@in.tum.de>
parents:
66164
diff
changeset
|
1524 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1525 |
lemma measurable_of_bool[measurable]: "of_bool \<in> count_space UNIV \<rightarrow>\<^sub>M borel" |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1526 |
by simp |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1527 |
|
69683 | 1528 |
subsection "Borel space on the extended reals" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1529 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1530 |
lemma borel_measurable_ereal[measurable (raw)]: |
43920 | 1531 |
assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M" |
60771 | 1532 |
using continuous_on_ereal f by (rule borel_measurable_continuous_on) (rule continuous_on_id) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1533 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1534 |
lemma borel_measurable_real_of_ereal[measurable (raw)]: |
62372 | 1535 |
fixes f :: "'a \<Rightarrow> ereal" |
49774 | 1536 |
assumes f: "f \<in> borel_measurable M" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1537 |
shows "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M" |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1538 |
apply (rule measurable_compose[OF f]) |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1539 |
apply (rule borel_measurable_continuous_countable_exceptions[of "{\<infinity>, -\<infinity> }"]) |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1540 |
apply (auto intro: continuous_on_real simp: Compl_eq_Diff_UNIV) |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1541 |
done |
49774 | 1542 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1543 |
lemma borel_measurable_ereal_cases: |
62372 | 1544 |
fixes f :: "'a \<Rightarrow> ereal" |
49774 | 1545 |
assumes f: "f \<in> borel_measurable M" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1546 |
assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x)))) \<in> borel_measurable M" |
49774 | 1547 |
shows "(\<lambda>x. H (f x)) \<in> borel_measurable M" |
1548 |
proof - |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1549 |
let ?F = "\<lambda>x. if f x = \<infinity> then H \<infinity> else if f x = - \<infinity> then H (-\<infinity>) else H (ereal (real_of_ereal (f x)))" |
49774 | 1550 |
{ fix x have "H (f x) = ?F x" by (cases "f x") auto } |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1551 |
with f H show ?thesis by simp |
47694 | 1552 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1553 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1554 |
lemma (*FIX ME needs a name *) |
50003 | 1555 |
fixes f :: "'a \<Rightarrow> ereal" assumes f[measurable]: "f \<in> borel_measurable M" |
1556 |
shows borel_measurable_ereal_abs[measurable(raw)]: "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M" |
|
1557 |
and borel_measurable_ereal_inverse[measurable(raw)]: "(\<lambda>x. inverse (f x) :: ereal) \<in> borel_measurable M" |
|
1558 |
and borel_measurable_uminus_ereal[measurable(raw)]: "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M" |
|
49774 | 1559 |
by (auto simp del: abs_real_of_ereal simp: borel_measurable_ereal_cases[OF f] measurable_If) |
1560 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1561 |
lemma borel_measurable_uminus_eq_ereal[simp]: |
49774 | 1562 |
"(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r") |
1563 |
proof |
|
1564 |
assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp |
|
1565 |
qed auto |
|
1566 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1567 |
lemma set_Collect_ereal2: |
62372 | 1568 |
fixes f g :: "'a \<Rightarrow> ereal" |
49774 | 1569 |
assumes f: "f \<in> borel_measurable M" |
1570 |
assumes g: "g \<in> borel_measurable M" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1571 |
assumes H: "{x \<in> space M. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))} \<in> sets M" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1572 |
"{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1573 |
"{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1574 |
"{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1575 |
"{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel" |
49774 | 1576 |
shows "{x \<in> space M. H (f x) (g x)} \<in> sets M" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1577 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1578 |
let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = -\<infinity> then H y (-\<infinity>) else H y (ereal (real_of_ereal (g x)))" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1579 |
let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = -\<infinity> then ?G (-\<infinity>) x else ?G (ereal (real_of_ereal (f x))) x" |
49774 | 1580 |
{ fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto } |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1581 |
note * = this |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1582 |
from assms show ?thesis |
62390 | 1583 |
by (subst *) (simp del: space_borel split del: if_split) |
49774 | 1584 |
qed |
1585 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1586 |
lemma borel_measurable_ereal_iff: |
43920 | 1587 |
shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1588 |
proof |
43920 | 1589 |
assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M" |
1590 |
from borel_measurable_real_of_ereal[OF this] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1591 |
show "f \<in> borel_measurable M" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1592 |
qed auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1593 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1594 |
lemma borel_measurable_erealD[measurable_dest]: |
59353
f0707dc3d9aa
measurability prover: removed app splitting, replaced by more powerful destruction rules
hoelzl
parents:
59088
diff
changeset
|
1595 |
"(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<Longrightarrow> g \<in> measurable N M \<Longrightarrow> (\<lambda>x. f (g x)) \<in> borel_measurable N" |
f0707dc3d9aa
measurability prover: removed app splitting, replaced by more powerful destruction rules
hoelzl
parents:
59088
diff
changeset
|
1596 |
unfolding borel_measurable_ereal_iff by simp |
f0707dc3d9aa
measurability prover: removed app splitting, replaced by more powerful destruction rules
hoelzl
parents:
59088
diff
changeset
|
1597 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1598 |
theorem borel_measurable_ereal_iff_real: |
43923 | 1599 |
fixes f :: "'a \<Rightarrow> ereal" |
1600 |
shows "f \<in> borel_measurable M \<longleftrightarrow> |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1601 |
((\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1602 |
proof safe |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1603 |
assume *: "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1604 |
have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1605 |
with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1606 |
let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real_of_ereal (f x))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1607 |
have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto |
43920 | 1608 |
also have "?f = f" by (auto simp: fun_eq_iff ereal_real) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1609 |
finally show "f \<in> borel_measurable M" . |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1610 |
qed simp_all |
41830 | 1611 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1612 |
lemma borel_measurable_ereal_iff_Iio: |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1613 |
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)" |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1614 |
by (auto simp: borel_Iio measurable_iff_measure_of) |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1615 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1616 |
lemma borel_measurable_ereal_iff_Ioi: |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1617 |
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)" |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1618 |
by (auto simp: borel_Ioi measurable_iff_measure_of) |
35582 | 1619 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1620 |
lemma vimage_sets_compl_iff: |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1621 |
"f -` A \<inter> space M \<in> sets M \<longleftrightarrow> f -` (- A) \<inter> space M \<in> sets M" |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1622 |
proof - |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1623 |
{ fix A assume "f -` A \<inter> space M \<in> sets M" |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1624 |
moreover have "f -` (- A) \<inter> space M = space M - f -` A \<inter> space M" by auto |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1625 |
ultimately have "f -` (- A) \<inter> space M \<in> sets M" by auto } |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1626 |
from this[of A] this[of "-A"] show ?thesis |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1627 |
by (metis double_complement) |
49774 | 1628 |
qed |
1629 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1630 |
lemma borel_measurable_iff_Iic_ereal: |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1631 |
"(f::'a\<Rightarrow>ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)" |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1632 |
unfolding borel_measurable_ereal_iff_Ioi vimage_sets_compl_iff[where A="{a <..}" for a] by simp |
38656 | 1633 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1634 |
lemma borel_measurable_iff_Ici_ereal: |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1635 |
"(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)" |
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1636 |
unfolding borel_measurable_ereal_iff_Iio vimage_sets_compl_iff[where A="{..< a}" for a] by simp |
38656 | 1637 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1638 |
lemma borel_measurable_ereal2: |
62372 | 1639 |
fixes f g :: "'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1640 |
assumes f: "f \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1641 |
assumes g: "g \<in> borel_measurable M" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1642 |
assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1643 |
"(\<lambda>x. H (-\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1644 |
"(\<lambda>x. H (\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1645 |
"(\<lambda>x. H (ereal (real_of_ereal (f x))) (-\<infinity>)) \<in> borel_measurable M" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1646 |
"(\<lambda>x. H (ereal (real_of_ereal (f x))) (\<infinity>)) \<in> borel_measurable M" |
49774 | 1647 |
shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1648 |
proof - |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1649 |
let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = - \<infinity> then H y (-\<infinity>) else H y (ereal (real_of_ereal (g x)))" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1650 |
let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = - \<infinity> then ?G (-\<infinity>) x else ?G (ereal (real_of_ereal (f x))) x" |
49774 | 1651 |
{ fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto } |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1652 |
note * = this |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1653 |
from assms show ?thesis unfolding * by simp |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1654 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1655 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1656 |
lemma [measurable(raw)]: |
43920 | 1657 |
fixes f :: "'a \<Rightarrow> ereal" |
50003 | 1658 |
assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1659 |
shows borel_measurable_ereal_add: "(\<lambda>x. f x + g x) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1660 |
and borel_measurable_ereal_times: "(\<lambda>x. f x * g x) \<in> borel_measurable M" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1661 |
by (simp_all add: borel_measurable_ereal2) |
49774 | 1662 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1663 |
lemma [measurable(raw)]: |
49774 | 1664 |
fixes f g :: "'a \<Rightarrow> ereal" |
1665 |
assumes "f \<in> borel_measurable M" |
|
1666 |
assumes "g \<in> borel_measurable M" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1667 |
shows borel_measurable_ereal_diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1668 |
and borel_measurable_ereal_divide: "(\<lambda>x. f x / g x) \<in> borel_measurable M" |
50003 | 1669 |
using assms by (simp_all add: minus_ereal_def divide_ereal_def) |
38656 | 1670 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1671 |
lemma borel_measurable_ereal_sum[measurable (raw)]: |
43920 | 1672 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal" |
41096 | 1673 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
1674 |
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M" |
|
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1675 |
using assms by (induction S rule: infinite_finite_induct) auto |
38656 | 1676 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1677 |
lemma borel_measurable_ereal_prod[measurable (raw)]: |
43920 | 1678 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ereal" |
38656 | 1679 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
41096 | 1680 |
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M" |
59361
fd5da2434be4
piecewise measurability using restrict_space; cleanup Borel_Space
hoelzl
parents:
59353
diff
changeset
|
1681 |
using assms by (induction S rule: infinite_finite_induct) auto |
38656 | 1682 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1683 |
lemma borel_measurable_extreal_suminf[measurable (raw)]: |
43920 | 1684 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
50003 | 1685 |
assumes [measurable]: "\<And>i. f i \<in> borel_measurable M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41969
diff
changeset
|
1686 |
shows "(\<lambda>x. (\<Sum>i. f i x)) \<in> borel_measurable M" |
50003 | 1687 |
unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp |
39092 | 1688 |
|
69683 | 1689 |
subsection "Borel space on the extended non-negative reals" |
62625 | 1690 |
|
69597 | 1691 |
text \<open> \<^type>\<open>ennreal\<close> is a topological monoid, so no rules for plus are required, also all order |
62625 | 1692 |
statements are usually done on type classes. \<close> |
1693 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1694 |
lemma measurable_enn2ereal[measurable]: "enn2ereal \<in> borel \<rightarrow>\<^sub>M borel" |
62625 | 1695 |
by (intro borel_measurable_continuous_on1 continuous_on_enn2ereal) |
1696 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1697 |
lemma measurable_e2ennreal[measurable]: "e2ennreal \<in> borel \<rightarrow>\<^sub>M borel" |
62625 | 1698 |
by (intro borel_measurable_continuous_on1 continuous_on_e2ennreal) |
1699 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1700 |
lemma borel_measurable_enn2real[measurable (raw)]: |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1701 |
"f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. enn2real (f x)) \<in> M \<rightarrow>\<^sub>M borel" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1702 |
unfolding enn2real_def[abs_def] by measurable |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1703 |
|
68833
fde093888c16
tagged 21 theories in the Analysis library for the manual
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
68635
diff
changeset
|
1704 |
definition%important [simp]: "is_borel f M \<longleftrightarrow> f \<in> borel_measurable M" |
62625 | 1705 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1706 |
lemma is_borel_transfer[transfer_rule]: "rel_fun (rel_fun (=) pcr_ennreal) (=) is_borel is_borel" |
62625 | 1707 |
unfolding is_borel_def[abs_def] |
1708 |
proof (safe intro!: rel_funI ext dest!: rel_fun_eq_pcr_ennreal[THEN iffD1]) |
|
1709 |
fix f and M :: "'a measure" |
|
1710 |
show "f \<in> borel_measurable M" if f: "enn2ereal \<circ> f \<in> borel_measurable M" |
|
1711 |
using measurable_compose[OF f measurable_e2ennreal] by simp |
|
1712 |
qed simp |
|
1713 |
||
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1714 |
context |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1715 |
includes ennreal.lifting |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1716 |
begin |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1717 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1718 |
lemma measurable_ennreal[measurable]: "ennreal \<in> borel \<rightarrow>\<^sub>M borel" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1719 |
unfolding is_borel_def[symmetric] |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1720 |
by transfer simp |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1721 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1722 |
lemma borel_measurable_ennreal_iff[simp]: |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1723 |
assumes [simp]: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1724 |
shows "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel \<longleftrightarrow> f \<in> M \<rightarrow>\<^sub>M borel" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1725 |
proof safe |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1726 |
assume "(\<lambda>x. ennreal (f x)) \<in> M \<rightarrow>\<^sub>M borel" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1727 |
then have "(\<lambda>x. enn2real (ennreal (f x))) \<in> M \<rightarrow>\<^sub>M borel" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1728 |
by measurable |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1729 |
then show "f \<in> M \<rightarrow>\<^sub>M borel" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1730 |
by (rule measurable_cong[THEN iffD1, rotated]) auto |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1731 |
qed measurable |
62625 | 1732 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1733 |
lemma borel_measurable_times_ennreal[measurable (raw)]: |
62625 | 1734 |
fixes f g :: "'a \<Rightarrow> ennreal" |
1735 |
shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. f x * g x) \<in> M \<rightarrow>\<^sub>M borel" |
|
1736 |
unfolding is_borel_def[symmetric] by transfer simp |
|
1737 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1738 |
lemma borel_measurable_inverse_ennreal[measurable (raw)]: |
62625 | 1739 |
fixes f :: "'a \<Rightarrow> ennreal" |
1740 |
shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. inverse (f x)) \<in> M \<rightarrow>\<^sub>M borel" |
|
1741 |
unfolding is_borel_def[symmetric] by transfer simp |
|
1742 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1743 |
lemma borel_measurable_divide_ennreal[measurable (raw)]: |
62625 | 1744 |
fixes f :: "'a \<Rightarrow> ennreal" |
1745 |
shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. f x / g x) \<in> M \<rightarrow>\<^sub>M borel" |
|
1746 |
unfolding divide_ennreal_def by simp |
|
1747 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1748 |
lemma borel_measurable_minus_ennreal[measurable (raw)]: |
62625 | 1749 |
fixes f :: "'a \<Rightarrow> ennreal" |
1750 |
shows "f \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> g \<in> M \<rightarrow>\<^sub>M borel \<Longrightarrow> (\<lambda>x. f x - g x) \<in> M \<rightarrow>\<^sub>M borel" |
|
1751 |
unfolding is_borel_def[symmetric] by transfer simp |
|
1752 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1753 |
lemma borel_measurable_prod_ennreal[measurable (raw)]: |
62625 | 1754 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ennreal" |
1755 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" |
|
1756 |
shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M" |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1757 |
using assms by (induction S rule: infinite_finite_induct) auto |
62625 | 1758 |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1759 |
end |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62625
diff
changeset
|
1760 |
|
62625 | 1761 |
hide_const (open) is_borel |
1762 |
||
69683 | 1763 |
subsection \<open>LIMSEQ is borel measurable\<close> |
39092 | 1764 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1765 |
lemma borel_measurable_LIMSEQ_real: |
39092 | 1766 |
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" |
61969 | 1767 |
assumes u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x" |
39092 | 1768 |
and u: "\<And>i. u i \<in> borel_measurable M" |
1769 |
shows "u' \<in> borel_measurable M" |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1770 |
proof - |
43920 | 1771 |
have "\<And>x. x \<in> space M \<Longrightarrow> liminf (\<lambda>n. ereal (u n x)) = ereal (u' x)" |
46731 | 1772 |
using u' by (simp add: lim_imp_Liminf) |
43920 | 1773 |
moreover from u have "(\<lambda>x. liminf (\<lambda>n. ereal (u n x))) \<in> borel_measurable M" |
39092 | 1774 |
by auto |
43920 | 1775 |
ultimately show ?thesis by (simp cong: measurable_cong add: borel_measurable_ereal_iff) |
39092 | 1776 |
qed |
1777 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1778 |
lemma borel_measurable_LIMSEQ_metric: |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1779 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b :: metric_space" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1780 |
assumes [measurable]: "\<And>i. f i \<in> borel_measurable M" |
61969 | 1781 |
assumes lim: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) \<longlonglongrightarrow> g x" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1782 |
shows "g \<in> borel_measurable M" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1783 |
unfolding borel_eq_closed |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1784 |
proof (safe intro!: measurable_measure_of) |
62372 | 1785 |
fix A :: "'b set" assume "closed A" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1786 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1787 |
have [measurable]: "(\<lambda>x. infdist (g x) A) \<in> borel_measurable M" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62390
diff
changeset
|
1788 |
proof (rule borel_measurable_LIMSEQ_real) |
61969 | 1789 |
show "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. infdist (f i x) A) \<longlonglongrightarrow> infdist (g x) A" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1790 |
by (intro tendsto_infdist lim) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1791 |
show "\<And>i. (\<lambda>x. infdist (f i x) A) \<in> borel_measurable M" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1792 |
by (intro borel_measurable_continuous_on[where f="\<lambda>x. infdist x A"] |
60150
bd773c47ad0b
New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents:
60017
diff
changeset
|
1793 |
continuous_at_imp_continuous_on ballI continuous_infdist continuous_ident) auto |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1794 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1795 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1796 |
show "g -` A \<inter> space M \<in> sets M" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1797 |
proof cases |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1798 |
assume "A \<noteq> {}" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1799 |
then have "\<And>x. infdist x A = 0 \<longleftrightarrow> x \<in> A" |
61808 | 1800 |
using \<open>closed A\<close> by (simp add: in_closed_iff_infdist_zero) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1801 |
then have "g -` A \<inter> space M = {x\<in>space M. infdist (g x) A = 0}" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1802 |
by auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1803 |
also have "\<dots> \<in> sets M" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1804 |
by measurable |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1805 |
finally show ?thesis . |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1806 |
qed simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1807 |
qed auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56371
diff
changeset
|
1808 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1809 |
lemma sets_Collect_Cauchy[measurable]: |
57036 | 1810 |
fixes f :: "nat \<Rightarrow> 'a => 'b::{metric_space, second_countable_topology}" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1811 |
assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M" |
49774 | 1812 |
shows "{x\<in>space M. Cauchy (\<lambda>i. f i x)} \<in> sets M" |
57036 | 1813 |
unfolding metric_Cauchy_iff2 using f by auto |
49774 | 1814 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1815 |
lemma borel_measurable_lim_metric[measurable (raw)]: |
57036 | 1816 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1817 |
assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M" |
49774 | 1818 |
shows "(\<lambda>x. lim (\<lambda>i. f i x)) \<in> borel_measurable M" |
1819 |
proof - |
|
63040 | 1820 |
define u' where "u' x = lim (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)" for x |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1821 |
then have *: "\<And>x. lim (\<lambda>i. f i x) = (if Cauchy (\<lambda>i. f i x) then u' x else (THE x. False))" |
64287 | 1822 |
by (auto simp: lim_def convergent_eq_Cauchy[symmetric]) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1823 |
have "u' \<in> borel_measurable M" |
57036 | 1824 |
proof (rule borel_measurable_LIMSEQ_metric) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1825 |
fix x |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1826 |
have "convergent (\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0)" |
49774 | 1827 |
by (cases "Cauchy (\<lambda>i. f i x)") |
64287 | 1828 |
(auto simp add: convergent_eq_Cauchy[symmetric] convergent_def) |
61969 | 1829 |
then show "(\<lambda>i. if Cauchy (\<lambda>i. f i x) then f i x else 0) \<longlonglongrightarrow> u' x" |
62372 | 1830 |
unfolding u'_def |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1831 |
by (rule convergent_LIMSEQ_iff[THEN iffD1]) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1832 |
qed measurable |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1833 |
then show ?thesis |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1834 |
unfolding * by measurable |
49774 | 1835 |
qed |
1836 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1837 |
lemma borel_measurable_suminf[measurable (raw)]: |
57036 | 1838 |
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1839 |
assumes f[measurable]: "\<And>i. f i \<in> borel_measurable M" |
49774 | 1840 |
shows "(\<lambda>x. suminf (\<lambda>i. f i x)) \<in> borel_measurable M" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1841 |
unfolding suminf_def sums_def[abs_def] lim_def[symmetric] by simp |
49774 | 1842 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1843 |
lemma Collect_closed_imp_pred_borel: "closed {x. P x} \<Longrightarrow> Measurable.pred borel P" |
63389 | 1844 |
by (simp add: pred_def) |
1845 |
||
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1846 |
(* Proof by Jeremy Avigad and Luke Serafin *) |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1847 |
lemma isCont_borel_pred[measurable]: |
63389 | 1848 |
fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space" |
1849 |
shows "Measurable.pred borel (isCont f)" |
|
1850 |
proof (subst measurable_cong) |
|
1851 |
let ?I = "\<lambda>j. inverse(real (Suc j))" |
|
1852 |
show "isCont f x = (\<forall>i. \<exists>j. \<forall>y z. dist x y < ?I j \<and> dist x z < ?I j \<longrightarrow> dist (f y) (f z) \<le> ?I i)" for x |
|
1853 |
unfolding continuous_at_eps_delta |
|
1854 |
proof safe |
|
1855 |
fix i assume "\<forall>e>0. \<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e" |
|
1856 |
moreover have "0 < ?I i / 2" |
|
1857 |
by simp |
|
1858 |
ultimately obtain d where d: "0 < d" "\<And>y. dist x y < d \<Longrightarrow> dist (f y) (f x) < ?I i / 2" |
|
1859 |
by (metis dist_commute) |
|
1860 |
then obtain j where j: "?I j < d" |
|
1861 |
by (metis reals_Archimedean) |
|
1862 |
||
1863 |
show "\<exists>j. \<forall>y z. dist x y < ?I j \<and> dist x z < ?I j \<longrightarrow> dist (f y) (f z) \<le> ?I i" |
|
1864 |
proof (safe intro!: exI[where x=j]) |
|
1865 |
fix y z assume *: "dist x y < ?I j" "dist x z < ?I j" |
|
1866 |
have "dist (f y) (f z) \<le> dist (f y) (f x) + dist (f z) (f x)" |
|
1867 |
by (rule dist_triangle2) |
|
1868 |
also have "\<dots> < ?I i / 2 + ?I i / 2" |
|
1869 |
by (intro add_strict_mono d less_trans[OF _ j] *) |
|
1870 |
also have "\<dots> \<le> ?I i" |
|
1871 |
by (simp add: field_simps of_nat_Suc) |
|
1872 |
finally show "dist (f y) (f z) \<le> ?I i" |
|
1873 |
by simp |
|
1874 |
qed |
|
1875 |
next |
|
1876 |
fix e::real assume "0 < e" |
|
1877 |
then obtain n where n: "?I n < e" |
|
1878 |
by (metis reals_Archimedean) |
|
1879 |
assume "\<forall>i. \<exists>j. \<forall>y z. dist x y < ?I j \<and> dist x z < ?I j \<longrightarrow> dist (f y) (f z) \<le> ?I i" |
|
1880 |
from this[THEN spec, of "Suc n"] |
|
1881 |
obtain j where j: "\<And>y z. dist x y < ?I j \<Longrightarrow> dist x z < ?I j \<Longrightarrow> dist (f y) (f z) \<le> ?I (Suc n)" |
|
1882 |
by auto |
|
1883 |
||
1884 |
show "\<exists>d>0. \<forall>y. dist y x < d \<longrightarrow> dist (f y) (f x) < e" |
|
1885 |
proof (safe intro!: exI[of _ "?I j"]) |
|
1886 |
fix y assume "dist y x < ?I j" |
|
1887 |
then have "dist (f y) (f x) \<le> ?I (Suc n)" |
|
1888 |
by (intro j) (auto simp: dist_commute) |
|
1889 |
also have "?I (Suc n) < ?I n" |
|
1890 |
by simp |
|
1891 |
also note n |
|
1892 |
finally show "dist (f y) (f x) < e" . |
|
1893 |
qed simp |
|
1894 |
qed |
|
1895 |
qed (intro pred_intros_countable closed_Collect_all closed_Collect_le open_Collect_less |
|
1896 |
Collect_closed_imp_pred_borel closed_Collect_imp open_Collect_conj continuous_intros) |
|
1897 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1898 |
lemma isCont_borel: |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1899 |
fixes f :: "'b::metric_space \<Rightarrow> 'a::metric_space" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1900 |
shows "{x. isCont f x} \<in> sets borel" |
63389 | 1901 |
by simp |
62083 | 1902 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1903 |
lemma is_real_interval: |
61880
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1904 |
assumes S: "is_interval S" |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1905 |
shows "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or> S = {a<..} \<or> S = {a..} \<or> |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1906 |
S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}" |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1907 |
using S unfolding is_interval_1 by (blast intro: interval_cases) |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1908 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1909 |
lemma real_interval_borel_measurable: |
61880
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1910 |
assumes "is_interval (S::real set)" |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1911 |
shows "S \<in> sets borel" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1912 |
proof - |
61880
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1913 |
from assms is_real_interval have "\<exists>a b::real. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or> |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1914 |
S = {a<..} \<or> S = {a..} \<or> S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}" by auto |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1915 |
then guess a .. |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1916 |
then guess b .. |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1917 |
thus ?thesis |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1918 |
by auto |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1919 |
qed |
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1920 |
|
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
64272
diff
changeset
|
1921 |
text \<open>The next lemmas hold in any second countable linorder (including ennreal or ereal for instance), |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
64272
diff
changeset
|
1922 |
but in the current state they are restricted to reals.\<close> |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
64272
diff
changeset
|
1923 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1924 |
lemma borel_measurable_mono_on_fnc: |
62083 | 1925 |
fixes f :: "real \<Rightarrow> real" and A :: "real set" |
1926 |
assumes "mono_on f A" |
|
1927 |
shows "f \<in> borel_measurable (restrict_space borel A)" |
|
1928 |
apply (rule measurable_restrict_countable[OF mono_on_ctble_discont[OF assms]]) |
|
1929 |
apply (auto intro!: image_eqI[where x="{x}" for x] simp: sets_restrict_space) |
|
1930 |
apply (auto simp add: sets_restrict_restrict_space continuous_on_eq_continuous_within |
|
62372 | 1931 |
cong: measurable_cong_sets |
62083 | 1932 |
intro!: borel_measurable_continuous_on_restrict intro: continuous_within_subset) |
1933 |
done |
|
1934 |
||
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1935 |
lemma borel_measurable_piecewise_mono: |
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
64272
diff
changeset
|
1936 |
fixes f::"real \<Rightarrow> real" and C::"real set set" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
64272
diff
changeset
|
1937 |
assumes "countable C" "\<And>c. c \<in> C \<Longrightarrow> c \<in> sets borel" "\<And>c. c \<in> C \<Longrightarrow> mono_on f c" "(\<Union>C) = UNIV" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
64272
diff
changeset
|
1938 |
shows "f \<in> borel_measurable borel" |
68833
fde093888c16
tagged 21 theories in the Analysis library for the manual
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
68635
diff
changeset
|
1939 |
by (rule measurable_piecewise_restrict[of C], auto intro: borel_measurable_mono_on_fnc simp: assms) |
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
64272
diff
changeset
|
1940 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1941 |
lemma borel_measurable_mono: |
61880
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1942 |
fixes f :: "real \<Rightarrow> real" |
62083 | 1943 |
shows "mono f \<Longrightarrow> f \<in> borel_measurable borel" |
1944 |
using borel_measurable_mono_on_fnc[of f UNIV] by (simp add: mono_def mono_on_def) |
|
61880
ff4d33058566
moved some theorems from the CLT proof; reordered some theorems / notation
hoelzl
parents:
61808
diff
changeset
|
1945 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1946 |
lemma measurable_bdd_below_real[measurable (raw)]: |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1947 |
fixes F :: "'a \<Rightarrow> 'i \<Rightarrow> real" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1948 |
assumes [simp]: "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> M \<rightarrow>\<^sub>M borel" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1949 |
shows "Measurable.pred M (\<lambda>x. bdd_below ((\<lambda>i. F i x)`I))" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1950 |
proof (subst measurable_cong) |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1951 |
show "bdd_below ((\<lambda>i. F i x)`I) \<longleftrightarrow> (\<exists>q\<in>\<int>. \<forall>i\<in>I. q \<le> F i x)" for x |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1952 |
by (auto simp: bdd_below_def intro!: bexI[of _ "of_int (floor _)"] intro: order_trans of_int_floor_le) |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1953 |
show "Measurable.pred M (\<lambda>w. \<exists>q\<in>\<int>. \<forall>i\<in>I. q \<le> F i w)" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1954 |
using countable_int by measurable |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1955 |
qed |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1956 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1957 |
lemma borel_measurable_cINF_real[measurable (raw)]: |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1958 |
fixes F :: "_ \<Rightarrow> _ \<Rightarrow> real" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1959 |
assumes [simp]: "countable I" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1960 |
assumes F[measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M" |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
1961 |
shows "(\<lambda>x. INF i\<in>I. F i x) \<in> borel_measurable M" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1962 |
proof (rule measurable_piecewise_restrict) |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1963 |
let ?\<Omega> = "{x\<in>space M. bdd_below ((\<lambda>i. F i x)`I)}" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1964 |
show "countable {?\<Omega>, - ?\<Omega>}" "space M \<subseteq> \<Union>{?\<Omega>, - ?\<Omega>}" "\<And>X. X \<in> {?\<Omega>, - ?\<Omega>} \<Longrightarrow> X \<inter> space M \<in> sets M" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1965 |
by auto |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
1966 |
fix X assume "X \<in> {?\<Omega>, - ?\<Omega>}" then show "(\<lambda>x. INF i\<in>I. F i x) \<in> borel_measurable (restrict_space M X)" |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1967 |
proof safe |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
1968 |
show "(\<lambda>x. INF i\<in>I. F i x) \<in> borel_measurable (restrict_space M ?\<Omega>)" |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1969 |
by (intro borel_measurable_cINF measurable_restrict_space1 F) |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1970 |
(auto simp: space_restrict_space) |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
1971 |
show "(\<lambda>x. INF i\<in>I. F i x) \<in> borel_measurable (restrict_space M (-?\<Omega>))" |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1972 |
proof (subst measurable_cong) |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1973 |
fix x assume "x \<in> space (restrict_space M (-?\<Omega>))" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1974 |
then have "\<not> (\<forall>i\<in>I. - F i x \<le> y)" for y |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1975 |
by (auto simp: space_restrict_space bdd_above_def bdd_above_uminus[symmetric]) |
69260
0a9688695a1b
removed relics of ASCII syntax for indexed big operators
haftmann
parents:
69022
diff
changeset
|
1976 |
then show "(INF i\<in>I. F i x) = - (THE x. False)" |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1977 |
by (auto simp: space_restrict_space Inf_real_def Sup_real_def Least_def simp del: Set.ball_simps(10)) |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1978 |
qed simp |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1979 |
qed |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1980 |
qed |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1981 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1982 |
lemma borel_Ici: "borel = sigma UNIV (range (\<lambda>x::real. {x ..}))" |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1983 |
proof (safe intro!: borel_eq_sigmaI1[OF borel_Iio]) |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1984 |
fix x :: real |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1985 |
have eq: "{..<x} = space (sigma UNIV (range atLeast)) - {x ..}" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1986 |
by auto |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1987 |
show "{..<x} \<in> sets (sigma UNIV (range atLeast))" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1988 |
unfolding eq by (intro sets.compl_sets) auto |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1989 |
qed auto |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1990 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1991 |
lemma borel_measurable_pred_less[measurable (raw)]: |
64008
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1992 |
fixes f :: "'a \<Rightarrow> 'b::{second_countable_topology, linorder_topology}" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1993 |
shows "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> Measurable.pred M (\<lambda>w. f w < g w)" |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1994 |
unfolding Measurable.pred_def by (rule borel_measurable_less) |
17a20ca86d62
HOL-Probability: more about probability, prepare for Markov processes in the AFP
hoelzl
parents:
63952
diff
changeset
|
1995 |
|
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1996 |
no_notation |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1997 |
eucl_less (infix "<e" 50) |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54230
diff
changeset
|
1998 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
1999 |
lemma borel_measurable_Max2[measurable (raw)]: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2000 |
fixes f::"_ \<Rightarrow> _ \<Rightarrow> 'a::{second_countable_topology, dense_linorder, linorder_topology}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2001 |
assumes "finite I" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2002 |
and [measurable]: "\<And>i. f i \<in> borel_measurable M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2003 |
shows "(\<lambda>x. Max{f i x |i. i \<in> I}) \<in> borel_measurable M" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2004 |
by (simp add: borel_measurable_Max[OF assms(1), where ?f=f and ?M=M] Setcompr_eq_image) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2005 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2006 |
lemma measurable_compose_n [measurable (raw)]: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2007 |
assumes "T \<in> measurable M M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2008 |
shows "(T^^n) \<in> measurable M M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2009 |
by (induction n, auto simp add: measurable_compose[OF _ assms]) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2010 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2011 |
lemma measurable_real_imp_nat: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2012 |
fixes f::"'a \<Rightarrow> nat" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2013 |
assumes [measurable]: "(\<lambda>x. real(f x)) \<in> borel_measurable M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2014 |
shows "f \<in> measurable M (count_space UNIV)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2015 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2016 |
let ?g = "(\<lambda>x. real(f x))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2017 |
have "\<And>(n::nat). ?g-`({real n}) \<inter> space M = f-`{n} \<inter> space M" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2018 |
moreover have "\<And>(n::nat). ?g-`({real n}) \<inter> space M \<in> sets M" using assms by measurable |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2019 |
ultimately have "\<And>(n::nat). f-`{n} \<inter> space M \<in> sets M" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2020 |
then show ?thesis using measurable_count_space_eq2_countable by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2021 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2022 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2023 |
lemma measurable_equality_set [measurable]: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2024 |
fixes f g::"_\<Rightarrow> 'a::{second_countable_topology, t2_space}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2025 |
assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2026 |
shows "{x \<in> space M. f x = g x} \<in> sets M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2027 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2028 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2029 |
define A where "A = {x \<in> space M. f x = g x}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2030 |
define B where "B = {y. \<exists>x::'a. y = (x,x)}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2031 |
have "A = (\<lambda>x. (f x, g x))-`B \<inter> space M" unfolding A_def B_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2032 |
moreover have "(\<lambda>x. (f x, g x)) \<in> borel_measurable M" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2033 |
moreover have "B \<in> sets borel" unfolding B_def by (simp add: closed_diagonal) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2034 |
ultimately have "A \<in> sets M" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2035 |
then show ?thesis unfolding A_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2036 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2037 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2038 |
lemma measurable_inequality_set [measurable]: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2039 |
fixes f g::"_ \<Rightarrow> 'a::{second_countable_topology, linorder_topology}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2040 |
assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2041 |
shows "{x \<in> space M. f x \<le> g x} \<in> sets M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2042 |
"{x \<in> space M. f x < g x} \<in> sets M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2043 |
"{x \<in> space M. f x \<ge> g x} \<in> sets M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2044 |
"{x \<in> space M. f x > g x} \<in> sets M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2045 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2046 |
define F where "F = (\<lambda>x. (f x, g x))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2047 |
have * [measurable]: "F \<in> borel_measurable M" unfolding F_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2048 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2049 |
have "{x \<in> space M. f x \<le> g x} = F-`{(x, y) | x y. x \<le> y} \<inter> space M" unfolding F_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2050 |
moreover have "{(x, y) | x y. x \<le> (y::'a)} \<in> sets borel" using closed_subdiagonal borel_closed by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2051 |
ultimately show "{x \<in> space M. f x \<le> g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2052 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2053 |
have "{x \<in> space M. f x < g x} = F-`{(x, y) | x y. x < y} \<inter> space M" unfolding F_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2054 |
moreover have "{(x, y) | x y. x < (y::'a)} \<in> sets borel" using open_subdiagonal borel_open by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2055 |
ultimately show "{x \<in> space M. f x < g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2056 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2057 |
have "{x \<in> space M. f x \<ge> g x} = F-`{(x, y) | x y. x \<ge> y} \<inter> space M" unfolding F_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2058 |
moreover have "{(x, y) | x y. x \<ge> (y::'a)} \<in> sets borel" using closed_superdiagonal borel_closed by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2059 |
ultimately show "{x \<in> space M. f x \<ge> g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2060 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2061 |
have "{x \<in> space M. f x > g x} = F-`{(x, y) | x y. x > y} \<inter> space M" unfolding F_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2062 |
moreover have "{(x, y) | x y. x > (y::'a)} \<in> sets borel" using open_superdiagonal borel_open by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2063 |
ultimately show "{x \<in> space M. f x > g x} \<in> sets M" using * by (metis (mono_tags, lifting) measurable_sets) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2064 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2065 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2066 |
proposition measurable_limit [measurable]: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2067 |
fixes f::"nat \<Rightarrow> 'a \<Rightarrow> 'b::first_countable_topology" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2068 |
assumes [measurable]: "\<And>n::nat. f n \<in> borel_measurable M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2069 |
shows "Measurable.pred M (\<lambda>x. (\<lambda>n. f n x) \<longlonglongrightarrow> c)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2070 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2071 |
obtain A :: "nat \<Rightarrow> 'b set" where A: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2072 |
"\<And>i. open (A i)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2073 |
"\<And>i. c \<in> A i" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2074 |
"\<And>S. open S \<Longrightarrow> c \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2075 |
by (rule countable_basis_at_decseq) blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2076 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2077 |
have [measurable]: "\<And>N i. (f N)-`(A i) \<inter> space M \<in> sets M" using A(1) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2078 |
then have mes: "(\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i) \<inter> space M) \<in> sets M" by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2079 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2080 |
have "(u \<longlonglongrightarrow> c) \<longleftrightarrow> (\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" for u::"nat \<Rightarrow> 'b" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2081 |
proof |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2082 |
assume "u \<longlonglongrightarrow> c" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2083 |
then have "eventually (\<lambda>n. u n \<in> A i) sequentially" for i using A(1)[of i] A(2)[of i] |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2084 |
by (simp add: topological_tendstoD) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2085 |
then show "(\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2086 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2087 |
assume H: "(\<forall>i. eventually (\<lambda>n. u n \<in> A i) sequentially)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2088 |
show "(u \<longlonglongrightarrow> c)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2089 |
proof (rule topological_tendstoI) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2090 |
fix S assume "open S" "c \<in> S" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2091 |
with A(3)[OF this] obtain i where "A i \<subseteq> S" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2092 |
using eventually_False_sequentially eventually_mono by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2093 |
moreover have "eventually (\<lambda>n. u n \<in> A i) sequentially" using H by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2094 |
ultimately show "\<forall>\<^sub>F n in sequentially. u n \<in> S" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2095 |
by (simp add: eventually_mono subset_eq) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2096 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2097 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2098 |
then have "{x. (\<lambda>n. f n x) \<longlonglongrightarrow> c} = (\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2099 |
by (auto simp add: atLeast_def eventually_at_top_linorder) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2100 |
then have "{x \<in> space M. (\<lambda>n. f n x) \<longlonglongrightarrow> c} = (\<Inter>i. \<Union>n. \<Inter>N\<in>{n..}. (f N)-`(A i) \<inter> space M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2101 |
by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2102 |
then have "{x \<in> space M. (\<lambda>n. f n x) \<longlonglongrightarrow> c} \<in> sets M" using mes by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2103 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2104 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2105 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2106 |
lemma measurable_limit2 [measurable]: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2107 |
fixes u::"nat \<Rightarrow> 'a \<Rightarrow> real" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2108 |
assumes [measurable]: "\<And>n. u n \<in> borel_measurable M" "v \<in> borel_measurable M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2109 |
shows "Measurable.pred M (\<lambda>x. (\<lambda>n. u n x) \<longlonglongrightarrow> v x)" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2110 |
proof - |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2111 |
define w where "w = (\<lambda>n x. u n x - v x)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2112 |
have [measurable]: "w n \<in> borel_measurable M" for n unfolding w_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2113 |
have "((\<lambda>n. u n x) \<longlonglongrightarrow> v x) \<longleftrightarrow> ((\<lambda>n. w n x) \<longlonglongrightarrow> 0)" for x |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2114 |
unfolding w_def using Lim_null by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2115 |
then show ?thesis using measurable_limit by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2116 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2117 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2118 |
lemma measurable_P_restriction [measurable (raw)]: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2119 |
assumes [measurable]: "Measurable.pred M P" "A \<in> sets M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2120 |
shows "{x \<in> A. P x} \<in> sets M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2121 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2122 |
have "A \<subseteq> space M" using sets.sets_into_space[OF assms(2)]. |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2123 |
then have "{x \<in> A. P x} = A \<inter> {x \<in> space M. P x}" by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2124 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2125 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2126 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2127 |
lemma measurable_sum_nat [measurable (raw)]: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2128 |
fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> nat" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2129 |
assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> measurable M (count_space UNIV)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2130 |
shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> measurable M (count_space UNIV)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2131 |
proof cases |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2132 |
assume "finite S" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2133 |
then show ?thesis using assms by induct auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2134 |
qed simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2135 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2136 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2137 |
lemma measurable_abs_powr [measurable]: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2138 |
fixes p::real |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2139 |
assumes [measurable]: "f \<in> borel_measurable M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2140 |
shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> borel_measurable M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2141 |
unfolding powr_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2142 |
|
69566 | 2143 |
text \<open>The next one is a variation around \<open>measurable_restrict_space\<close>.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2144 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2145 |
lemma measurable_restrict_space3: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2146 |
assumes "f \<in> measurable M N" and |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2147 |
"f \<in> A \<rightarrow> B" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2148 |
shows "f \<in> measurable (restrict_space M A) (restrict_space N B)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2149 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2150 |
have "f \<in> measurable (restrict_space M A) N" using assms(1) measurable_restrict_space1 by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2151 |
then show ?thesis by (metis Int_iff funcsetI funcset_mem |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2152 |
measurable_restrict_space2[of f, of "restrict_space M A", of B, of N] assms(2) space_restrict_space) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2153 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2154 |
|
69566 | 2155 |
text \<open>The next one is a variation around \<open>measurable_piecewise_restrict\<close>.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2156 |
|
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2157 |
lemma measurable_piecewise_restrict2: |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2158 |
assumes [measurable]: "\<And>n. A n \<in> sets M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2159 |
and "space M = (\<Union>(n::nat). A n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2160 |
"\<And>n. \<exists>h \<in> measurable M N. (\<forall>x \<in> A n. f x = h x)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2161 |
shows "f \<in> measurable M N" |
69652
3417a8f154eb
updated tagging first 5
Angeliki KoutsoukouArgyraki <ak2110@cam.ac.uk>
parents:
69597
diff
changeset
|
2162 |
proof (rule measurableI) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2163 |
fix B assume [measurable]: "B \<in> sets N" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2164 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2165 |
fix n::nat |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2166 |
obtain h where [measurable]: "h \<in> measurable M N" and "\<forall>x \<in> A n. f x = h x" using assms(3) by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2167 |
then have *: "f-`B \<inter> A n = h-`B \<inter> A n" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2168 |
have "h-`B \<inter> A n = h-`B \<inter> space M \<inter> A n" using assms(2) sets.sets_into_space by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2169 |
then have "h-`B \<inter> A n \<in> sets M" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2170 |
then have "f-`B \<inter> A n \<in> sets M" using * by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2171 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2172 |
then have "(\<Union>n. f-`B \<inter> A n) \<in> sets M" by measurable |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2173 |
moreover have "f-`B \<inter> space M = (\<Union>n. f-`B \<inter> A n)" using assms(2) by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2174 |
ultimately show "f-`B \<inter> space M \<in> sets M" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2175 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2176 |
fix x assume "x \<in> space M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2177 |
then obtain n where "x \<in> A n" using assms(2) by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2178 |
obtain h where [measurable]: "h \<in> measurable M N" and "\<forall>x \<in> A n. f x = h x" using assms(3) by blast |
64911 | 2179 |
then have "f x = h x" using \<open>x \<in> A n\<close> by blast |
2180 |
moreover have "h x \<in> space N" by (metis measurable_space \<open>x \<in> space M\<close> \<open>h \<in> measurable M N\<close>) |
|
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2181 |
ultimately show "f x \<in> space N" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2182 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2183 |
|
51683
baefa3b461c2
generalize Borel-set properties from real/ereal/ordered_euclidean_spaces to order_topology and real_normed_vector
hoelzl
parents:
51478
diff
changeset
|
2184 |
end |