author | hoelzl |
Wed, 06 Jan 2016 12:18:53 +0100 | |
changeset 62083 | 7582b39f51ed |
parent 61969 | e01015e49041 |
child 62343 | 24106dc44def |
permissions | -rw-r--r-- |
42067 | 1 |
(* Title: HOL/Probability/Radon_Nikodym.thy |
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Author: Johannes Hölzl, TU München |
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*) |
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61808 | 5 |
section \<open>Radon-Nikod{\'y}m derivative\<close> |
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38656 | 7 |
theory Radon_Nikodym |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
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imports Bochner_Integration |
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begin |
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||
47694 | 11 |
definition "diff_measure M N = |
12 |
measure_of (space M) (sets M) (\<lambda>A. emeasure M A - emeasure N A)" |
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||
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:
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|
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lemma |
47694 | 15 |
shows space_diff_measure[simp]: "space (diff_measure M N) = space M" |
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and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M" |
|
17 |
by (auto simp: diff_measure_def) |
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||
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lemma emeasure_diff_measure: |
|
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assumes fin: "finite_measure M" "finite_measure N" and sets_eq: "sets M = sets N" |
|
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assumes pos: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure N A \<le> emeasure M A" and A: "A \<in> sets M" |
|
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shows "emeasure (diff_measure M N) A = emeasure M A - emeasure N A" (is "_ = ?\<mu> A") |
|
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unfolding diff_measure_def |
|
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proof (rule emeasure_measure_of_sigma) |
|
25 |
show "sigma_algebra (space M) (sets M)" .. |
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show "positive (sets M) ?\<mu>" |
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using pos by (simp add: positive_def ereal_diff_positive) |
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show "countably_additive (sets M) ?\<mu>" |
|
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proof (rule countably_additiveI) |
|
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fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> sets M" and "disjoint_family A" |
|
31 |
then have suminf: |
|
32 |
"(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)" |
|
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"(\<Sum>i. emeasure N (A i)) = emeasure N (\<Union>i. A i)" |
|
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by (simp_all add: suminf_emeasure sets_eq) |
|
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with A have "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) = |
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(\<Sum>i. emeasure M (A i)) - (\<Sum>i. emeasure N (A i))" |
|
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using fin |
|
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by (intro suminf_ereal_minus pos emeasure_nonneg) |
|
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(auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure) |
|
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then show "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) = |
|
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emeasure M (\<Union>i. A i) - emeasure N (\<Union>i. A i) " |
|
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by (simp add: suminf) |
|
43 |
qed |
|
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qed fact |
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||
38656 | 46 |
lemma (in sigma_finite_measure) Ex_finite_integrable_function: |
56996 | 47 |
shows "\<exists>h\<in>borel_measurable M. integral\<^sup>N M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>) \<and> (\<forall>x. 0 \<le> h x)" |
38656 | 48 |
proof - |
49 |
obtain A :: "nat \<Rightarrow> 'a set" where |
|
50003 | 50 |
range[measurable]: "range A \<subseteq> sets M" and |
38656 | 51 |
space: "(\<Union>i. A i) = space M" and |
47694 | 52 |
measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and |
38656 | 53 |
disjoint: "disjoint_family A" |
47694 | 54 |
using sigma_finite_disjoint by auto |
55 |
let ?B = "\<lambda>i. 2^Suc i * emeasure M (A i)" |
|
38656 | 56 |
have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)" |
57 |
proof |
|
47694 | 58 |
fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)" |
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using measure[of i] emeasure_nonneg[of M "A i"] |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
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60 |
by (auto intro!: dense simp: ereal_0_gt_inverse ereal_zero_le_0_iff) |
38656 | 61 |
qed |
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from choice[OF this] obtain n where n: "\<And>i. 0 < n i" |
|
47694 | 63 |
"\<And>i. n i < inverse (2^Suc i * emeasure M (A i))" by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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|
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{ fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this |
46731 | 65 |
let ?h = "\<lambda>x. \<Sum>i. n i * indicator (A i) x" |
38656 | 66 |
show ?thesis |
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proof (safe intro!: bexI[of _ ?h] del: notI) |
|
39092 | 68 |
have "\<And>i. A i \<in> sets M" |
44890
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new fastforce replacing fastsimp - less confusing name
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parents:
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|
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using range by fastforce+ |
56996 | 70 |
then have "integral\<^sup>N M ?h = (\<Sum>i. n i * emeasure M (A i))" using pos |
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by (simp add: nn_integral_suminf nn_integral_cmult_indicator) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset
|
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also have "\<dots> \<le> (\<Sum>i. (1 / 2)^Suc i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset
|
73 |
proof (rule suminf_le_pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
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|
74 |
fix N |
47694 | 75 |
have "n N * emeasure M (A N) \<le> inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)" |
76 |
using n[of N] |
|
43920 | 77 |
by (intro ereal_mult_right_mono) auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
78 |
also have "\<dots> \<le> (1 / 2) ^ Suc N" |
38656 | 79 |
using measure[of N] n[of N] |
47694 | 80 |
by (cases rule: ereal2_cases[of "n N" "emeasure M (A N)"]) |
43920 | 81 |
(simp_all add: inverse_eq_divide power_divide one_ereal_def ereal_power_divide) |
47694 | 82 |
finally show "n N * emeasure M (A N) \<le> (1 / 2) ^ Suc N" . |
61808 | 83 |
show "0 \<le> n N * emeasure M (A N)" using n[of N] \<open>A N \<in> sets M\<close> by (simp add: emeasure_nonneg) |
38656 | 84 |
qed |
56996 | 85 |
finally show "integral\<^sup>N M ?h \<noteq> \<infinity>" unfolding suminf_half_series_ereal by auto |
38656 | 86 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
87 |
{ fix x assume "x \<in> space M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
88 |
then obtain i where "x \<in> A i" using space[symmetric] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
89 |
with disjoint n have "?h x = n i" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
90 |
by (auto intro!: suminf_cmult_indicator intro: less_imp_le) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
91 |
then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by auto } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
92 |
note pos = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
93 |
fix x show "0 \<le> ?h x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
94 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
95 |
assume "x \<in> space M" then show "0 \<le> ?h x" using pos by (auto intro: less_imp_le) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
96 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
97 |
assume "x \<notin> space M" then have "\<And>i. x \<notin> A i" using space by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
98 |
then show "0 \<le> ?h x" by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
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|
99 |
qed |
50003 | 100 |
qed measurable |
38656 | 101 |
qed |
102 |
||
40871 | 103 |
subsection "Absolutely continuous" |
104 |
||
47694 | 105 |
definition absolutely_continuous :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where |
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"absolutely_continuous M N \<longleftrightarrow> null_sets M \<subseteq> null_sets N" |
|
107 |
||
108 |
lemma absolutely_continuousI_count_space: "absolutely_continuous (count_space A) M" |
|
109 |
unfolding absolutely_continuous_def by (auto simp: null_sets_count_space) |
|
38656 | 110 |
|
47694 | 111 |
lemma absolutely_continuousI_density: |
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"f \<in> borel_measurable M \<Longrightarrow> absolutely_continuous M (density M f)" |
|
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by (force simp add: absolutely_continuous_def null_sets_density_iff dest: AE_not_in) |
|
114 |
||
115 |
lemma absolutely_continuousI_point_measure_finite: |
|
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"(\<And>x. \<lbrakk> x \<in> A ; f x \<le> 0 \<rbrakk> \<Longrightarrow> g x \<le> 0) \<Longrightarrow> absolutely_continuous (point_measure A f) (point_measure A g)" |
|
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unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff) |
|
118 |
||
119 |
lemma absolutely_continuous_AE: |
|
120 |
assumes sets_eq: "sets M' = sets M" |
|
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and "absolutely_continuous M M'" "AE x in M. P x" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
122 |
shows "AE x in M'. P x" |
40859 | 123 |
proof - |
61808 | 124 |
from \<open>AE x in M. P x\<close> obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N" |
47694 | 125 |
unfolding eventually_ae_filter by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
changeset
|
126 |
show "AE x in M'. P x" |
47694 | 127 |
proof (rule AE_I') |
128 |
show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp |
|
61808 | 129 |
from \<open>absolutely_continuous M M'\<close> show "N \<in> null_sets M'" |
47694 | 130 |
using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto |
40859 | 131 |
qed |
132 |
qed |
|
133 |
||
40871 | 134 |
subsection "Existence of the Radon-Nikodym derivative" |
135 |
||
38656 | 136 |
lemma (in finite_measure) Radon_Nikodym_aux_epsilon: |
137 |
fixes e :: real assumes "0 < e" |
|
47694 | 138 |
assumes "finite_measure N" and sets_eq: "sets N = sets M" |
139 |
shows "\<exists>A\<in>sets M. measure M (space M) - measure N (space M) \<le> measure M A - measure N A \<and> |
|
140 |
(\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < measure M B - measure N B)" |
|
38656 | 141 |
proof - |
47694 | 142 |
interpret M': finite_measure N by fact |
143 |
let ?d = "\<lambda>A. measure M A - measure N A" |
|
46731 | 144 |
let ?A = "\<lambda>A. if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B) |
38656 | 145 |
then {} |
146 |
else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)" |
|
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def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}" |
|
148 |
have A_simps[simp]: |
|
149 |
"A 0 = {}" |
|
150 |
"\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all |
|
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{ fix A assume "A \<in> sets M" |
|
152 |
have "?A A \<in> sets M" |
|
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by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) } |
|
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note A'_in_sets = this |
|
155 |
{ fix n have "A n \<in> sets M" |
|
156 |
proof (induct n) |
|
157 |
case (Suc n) thus "A (Suc n) \<in> sets M" |
|
158 |
using A'_in_sets[of "A n"] by (auto split: split_if_asm) |
|
159 |
qed (simp add: A_def) } |
|
160 |
note A_in_sets = this |
|
161 |
hence "range A \<subseteq> sets M" by auto |
|
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{ fix n B |
|
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assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> -e" |
|
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hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B)" by (auto simp: not_less) |
|
165 |
have "?d (A (Suc n)) \<le> ?d (A n) - e" unfolding A_simps if_not_P[OF False] |
|
166 |
proof (rule someI2_ex[OF Ex]) |
|
167 |
fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e" |
|
168 |
hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto |
|
169 |
hence "?d (A n \<union> B) = ?d (A n) + ?d B" |
|
61808 | 170 |
using \<open>A n \<in> sets M\<close> finite_measure_Union M'.finite_measure_Union by (simp add: sets_eq) |
38656 | 171 |
also have "\<dots> \<le> ?d (A n) - e" using dB by simp |
172 |
finally show "?d (A n \<union> B) \<le> ?d (A n) - e" . |
|
173 |
qed } |
|
174 |
note dA_epsilon = this |
|
175 |
{ fix n have "?d (A (Suc n)) \<le> ?d (A n)" |
|
176 |
proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e") |
|
61808 | 177 |
case True from dA_epsilon[OF this] show ?thesis using \<open>0 < e\<close> by simp |
38656 | 178 |
next |
179 |
case False |
|
180 |
hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le) |
|
181 |
thus ?thesis by simp |
|
182 |
qed } |
|
183 |
note dA_mono = this |
|
184 |
show ?thesis |
|
185 |
proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B") |
|
186 |
case True then obtain n where B: "\<And>B. \<lbrakk> B \<in> sets M; B \<subseteq> space M - A n\<rbrakk> \<Longrightarrow> -e < ?d B" by blast |
|
187 |
show ?thesis |
|
188 |
proof (safe intro!: bexI[of _ "space M - A n"]) |
|
189 |
fix B assume "B \<in> sets M" "B \<subseteq> space M - A n" |
|
190 |
from B[OF this] show "-e < ?d B" . |
|
191 |
next |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
192 |
show "space M - A n \<in> sets M" by (rule sets.compl_sets) fact |
38656 | 193 |
next |
194 |
show "?d (space M) \<le> ?d (space M - A n)" |
|
195 |
proof (induct n) |
|
196 |
fix n assume "?d (space M) \<le> ?d (space M - A n)" |
|
197 |
also have "\<dots> \<le> ?d (space M - A (Suc n))" |
|
50244
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qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
198 |
using A_in_sets sets.sets_into_space dA_mono[of n] finite_measure_compl M'.finite_measure_compl |
47694 | 199 |
by (simp del: A_simps add: sets_eq sets_eq_imp_space_eq[OF sets_eq]) |
38656 | 200 |
finally show "?d (space M) \<le> ?d (space M - A (Suc n))" . |
201 |
qed simp |
|
202 |
qed |
|
203 |
next |
|
204 |
case False hence B: "\<And>n. \<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e" |
|
205 |
by (auto simp add: not_less) |
|
206 |
{ fix n have "?d (A n) \<le> - real n * e" |
|
207 |
proof (induct 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:
61169
diff
changeset
|
208 |
case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: of_nat_Suc field_simps) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
209 |
next |
47694 | 210 |
case 0 with measure_empty show ?case by (simp add: zero_ereal_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
211 |
qed } note dA_less = this |
38656 | 212 |
have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq |
213 |
proof (rule incseq_SucI) |
|
214 |
fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto |
|
215 |
qed |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
216 |
have A: "incseq A" by (auto intro!: incseq_SucI) |
61808 | 217 |
from finite_Lim_measure_incseq[OF _ A] \<open>range A \<subseteq> sets M\<close> |
47694 | 218 |
M'.finite_Lim_measure_incseq[OF _ A] |
61969 | 219 |
have convergent: "(\<lambda>i. ?d (A i)) \<longlonglongrightarrow> ?d (\<Union>i. A i)" |
47694 | 220 |
by (auto intro!: tendsto_diff simp: sets_eq) |
38656 | 221 |
obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto |
222 |
moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less] |
|
61808 | 223 |
have "real n \<le> - ?d (\<Union>i. A i) / e" using \<open>0<e\<close> by (simp add: field_simps) |
38656 | 224 |
ultimately show ?thesis by auto |
225 |
qed |
|
226 |
qed |
|
227 |
||
47694 | 228 |
lemma (in finite_measure) Radon_Nikodym_aux: |
229 |
assumes "finite_measure N" and sets_eq: "sets N = sets M" |
|
230 |
shows "\<exists>A\<in>sets M. measure M (space M) - measure N (space M) \<le> |
|
231 |
measure M A - measure N A \<and> |
|
232 |
(\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> measure M B - measure N B)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
233 |
proof - |
47694 | 234 |
interpret N: finite_measure N by fact |
235 |
let ?d = "\<lambda>A. measure M A - measure N A" |
|
58184
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
236 |
let ?P = "\<lambda>A n. if n = 0 then A = space M else (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow> - 1 / real (Suc n) < ?d C)" |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
237 |
let ?Q = "\<lambda>A B. A \<subseteq> B \<and> ?d B \<le> ?d A" |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
238 |
|
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
239 |
have "\<exists>A. \<forall>n. (A n \<in> sets M \<and> ?P (A n) n) \<and> ?Q (A (Suc n)) (A n)" |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
240 |
proof (rule dependent_nat_choice) |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
241 |
show "\<exists>A. A \<in> sets M \<and> ?P A 0" |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
242 |
by auto |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
243 |
next |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
244 |
fix A n assume "A \<in> sets M \<and> ?P A n" |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
245 |
then have A: "A \<in> sets M" by auto |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
246 |
then have "finite_measure (density M (indicator A))" "0 < 1 / real (Suc (Suc n))" |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
247 |
"finite_measure (density N (indicator A))" "sets (density N (indicator A)) = sets (density M (indicator A))" |
47694 | 248 |
by (auto simp: finite_measure_restricted N.finite_measure_restricted sets_eq) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
249 |
from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. note X = this |
58184
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
250 |
with A have "A \<inter> X \<in> sets M \<and> ?P (A \<inter> X) (Suc n) \<and> ?Q (A \<inter> X) A" |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56409
diff
changeset
|
251 |
by (simp add: measure_restricted sets_eq sets.Int) (metis inf_absorb2) |
58184
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
252 |
then show "\<exists>B. (B \<in> sets M \<and> ?P B (Suc n)) \<and> ?Q B A" |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
253 |
by blast |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
254 |
qed |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
255 |
then obtain A where A: "\<And>n. A n \<in> sets M" "\<And>n. ?P (A n) n" "\<And>n. ?Q (A (Suc n)) (A n)" |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
256 |
by metis |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
257 |
then have mono_dA: "mono (\<lambda>i. ?d (A i))" and A_0[simp]: "A 0 = space M" |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
258 |
by (auto simp add: mono_iff_le_Suc) |
38656 | 259 |
show ?thesis |
260 |
proof (safe intro!: bexI[of _ "\<Inter>i. A i"]) |
|
61808 | 261 |
show "(\<Inter>i. A i) \<in> sets M" using \<open>\<And>n. A n \<in> sets M\<close> by auto |
58184
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
262 |
have "decseq A" using A by (auto intro!: decseq_SucI) |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
263 |
from A(1) finite_Lim_measure_decseq[OF _ this] N.finite_Lim_measure_decseq[OF _ this] |
61969 | 264 |
have "(\<lambda>i. ?d (A i)) \<longlonglongrightarrow> ?d (\<Inter>i. A i)" by (auto intro!: tendsto_diff simp: sets_eq) |
38656 | 265 |
thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _] |
47694 | 266 |
by (rule_tac LIMSEQ_le_const) auto |
38656 | 267 |
next |
268 |
fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)" |
|
269 |
show "0 \<le> ?d B" |
|
270 |
proof (rule ccontr) |
|
271 |
assume "\<not> 0 \<le> ?d B" |
|
272 |
hence "0 < - ?d B" by auto |
|
273 |
from ex_inverse_of_nat_Suc_less[OF this] |
|
274 |
obtain n where *: "?d B < - 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:
61169
diff
changeset
|
275 |
by (auto simp: field_simps) |
58184
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
276 |
also have "\<dots> \<le> - 1 / real (Suc (Suc n))" |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
277 |
by (auto simp: field_simps) |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
278 |
finally show False |
db1381d811ab
cleanup Wfrec; introduce dependent_wf/wellorder_choice
hoelzl
parents:
57447
diff
changeset
|
279 |
using * A(2)[of "Suc n"] B by (auto elim!: ballE[of _ _ B]) |
38656 | 280 |
qed |
281 |
qed |
|
282 |
qed |
|
283 |
||
284 |
lemma (in finite_measure) Radon_Nikodym_finite_measure: |
|
47694 | 285 |
assumes "finite_measure N" and sets_eq: "sets N = sets M" |
286 |
assumes "absolutely_continuous M N" |
|
287 |
shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N" |
|
38656 | 288 |
proof - |
47694 | 289 |
interpret N: finite_measure N by fact |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
290 |
def G \<equiv> "{g \<in> borel_measurable M. (\<forall>x. 0 \<le> g x) \<and> (\<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A)}" |
50003 | 291 |
{ fix f have "f \<in> G \<Longrightarrow> f \<in> borel_measurable M" by (auto simp: G_def) } |
292 |
note this[measurable_dest] |
|
38656 | 293 |
have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto |
294 |
hence "G \<noteq> {}" by auto |
|
295 |
{ fix f g assume f: "f \<in> G" and g: "g \<in> G" |
|
296 |
have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def |
|
297 |
proof safe |
|
298 |
show "?max \<in> borel_measurable M" using f g unfolding G_def by auto |
|
299 |
let ?A = "{x \<in> space M. f x \<le> g x}" |
|
300 |
have "?A \<in> sets M" using f g unfolding G_def by auto |
|
301 |
fix A assume "A \<in> sets M" |
|
61808 | 302 |
hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using \<open>?A \<in> sets M\<close> by auto |
47694 | 303 |
hence sets': "?A \<inter> A \<in> sets N" "(space M - ?A) \<inter> A \<in> sets N" by (auto simp: sets_eq) |
38656 | 304 |
have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A" |
61808 | 305 |
using sets.sets_into_space[OF \<open>A \<in> sets M\<close>] by auto |
38656 | 306 |
have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x = |
307 |
g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x" |
|
308 |
by (auto simp: indicator_def max_def) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
309 |
hence "(\<integral>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
310 |
(\<integral>\<^sup>+x. g x * indicator (?A \<inter> A) x \<partial>M) + |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
311 |
(\<integral>\<^sup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)" |
38656 | 312 |
using f g sets unfolding G_def |
56996 | 313 |
by (auto cong: nn_integral_cong intro!: nn_integral_add) |
47694 | 314 |
also have "\<dots> \<le> N (?A \<inter> A) + N ((space M - ?A) \<inter> A)" |
38656 | 315 |
using f g sets unfolding G_def by (auto intro!: add_mono) |
47694 | 316 |
also have "\<dots> = N A" |
317 |
using plus_emeasure[OF sets'] union by auto |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
318 |
finally show "(\<integral>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> N A" . |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
319 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
320 |
fix x show "0 \<le> max (g x) (f x)" using f g by (auto simp: G_def split: split_max) |
38656 | 321 |
qed } |
322 |
note max_in_G = this |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
323 |
{ fix f assume "incseq f" and f: "\<And>i. f i \<in> G" |
50003 | 324 |
then have [measurable]: "\<And>i. f i \<in> borel_measurable M" by (auto simp: G_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
325 |
have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def |
38656 | 326 |
proof safe |
50003 | 327 |
show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M" by measurable |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
328 |
{ fix x show "0 \<le> (SUP i. f i x)" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44918
diff
changeset
|
329 |
using f by (auto simp: G_def intro: SUP_upper2) } |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
330 |
next |
38656 | 331 |
fix A assume "A \<in> sets M" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
332 |
have "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
333 |
(\<integral>\<^sup>+x. (SUP i. f i x * indicator A x) \<partial>M)" |
56996 | 334 |
by (intro nn_integral_cong) (simp split: split_indicator) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
335 |
also have "\<dots> = (SUP i. (\<integral>\<^sup>+x. f i x * indicator A x \<partial>M))" |
61808 | 336 |
using \<open>incseq f\<close> f \<open>A \<in> sets M\<close> |
56996 | 337 |
by (intro nn_integral_monotone_convergence_SUP) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
338 |
(auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
339 |
finally show "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A" |
61808 | 340 |
using f \<open>A \<in> sets M\<close> by (auto intro!: SUP_least simp: G_def) |
38656 | 341 |
qed } |
342 |
note SUP_in_G = this |
|
56996 | 343 |
let ?y = "SUP g : G. integral\<^sup>N M g" |
47694 | 344 |
have y_le: "?y \<le> N (space M)" unfolding G_def |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44918
diff
changeset
|
345 |
proof (safe intro!: SUP_least) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
346 |
fix g assume "\<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A" |
56996 | 347 |
from this[THEN bspec, OF sets.top] show "integral\<^sup>N M g \<le> N (space M)" |
348 |
by (simp cong: nn_integral_cong) |
|
38656 | 349 |
qed |
61808 | 350 |
from SUP_countable_SUP [OF \<open>G \<noteq> {}\<close>, of "integral\<^sup>N M"] guess ys .. note ys = this |
56996 | 351 |
then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n" |
38656 | 352 |
proof safe |
56996 | 353 |
fix n assume "range ys \<subseteq> integral\<^sup>N M ` G" |
354 |
hence "ys n \<in> integral\<^sup>N M ` G" by auto |
|
355 |
thus "\<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n" by auto |
|
38656 | 356 |
qed |
56996 | 357 |
from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^sup>N M (gs n) = ys n" by auto |
358 |
hence y_eq: "?y = (SUP i. integral\<^sup>N M (gs i))" using ys by auto |
|
46731 | 359 |
let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
360 |
def f \<equiv> "\<lambda>x. SUP i. ?g i x" |
46731 | 361 |
let ?F = "\<lambda>A x. f x * indicator A x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
362 |
have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto |
38656 | 363 |
{ fix i have "?g i \<in> G" |
364 |
proof (induct i) |
|
365 |
case 0 thus ?case by simp fact |
|
366 |
next |
|
367 |
case (Suc i) |
|
61808 | 368 |
with Suc gs_not_empty \<open>gs (Suc i) \<in> G\<close> show ?case |
38656 | 369 |
by (auto simp add: atMost_Suc intro!: max_in_G) |
370 |
qed } |
|
371 |
note g_in_G = this |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
372 |
have "incseq ?g" using gs_not_empty |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
373 |
by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc) |
50003 | 374 |
from SUP_in_G[OF this g_in_G] have [measurable]: "f \<in> G" unfolding f_def . |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
375 |
then have [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto |
56996 | 376 |
have "integral\<^sup>N M f = (SUP i. integral\<^sup>N M (?g i))" unfolding f_def |
61808 | 377 |
using g_in_G \<open>incseq ?g\<close> |
56996 | 378 |
by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def) |
38656 | 379 |
also have "\<dots> = ?y" |
380 |
proof (rule antisym) |
|
56996 | 381 |
show "(SUP i. integral\<^sup>N M (?g i)) \<le> ?y" |
56166 | 382 |
using g_in_G by (auto intro: SUP_mono) |
56996 | 383 |
show "?y \<le> (SUP i. integral\<^sup>N M (?g i))" unfolding y_eq |
384 |
by (auto intro!: SUP_mono nn_integral_mono Max_ge) |
|
38656 | 385 |
qed |
56996 | 386 |
finally have int_f_eq_y: "integral\<^sup>N M f = ?y" . |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
387 |
have "\<And>x. 0 \<le> f x" |
61808 | 388 |
unfolding f_def using \<open>\<And>i. gs i \<in> G\<close> |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44918
diff
changeset
|
389 |
by (auto intro!: SUP_upper2 Max_ge_iff[THEN iffD2] simp: G_def) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
390 |
let ?t = "\<lambda>A. N A - (\<integral>\<^sup>+x. ?F A x \<partial>M)" |
47694 | 391 |
let ?M = "diff_measure N (density M f)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
392 |
have f_le_N: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. ?F A x \<partial>M) \<le> N A" |
61808 | 393 |
using \<open>f \<in> G\<close> unfolding G_def by auto |
47694 | 394 |
have emeasure_M: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure ?M A = ?t A" |
395 |
proof (subst emeasure_diff_measure) |
|
396 |
from f_le_N[of "space M"] show "finite_measure N" "finite_measure (density M f)" |
|
56996 | 397 |
by (auto intro!: finite_measureI simp: emeasure_density cong: nn_integral_cong) |
47694 | 398 |
next |
399 |
fix B assume "B \<in> sets N" with f_le_N[of B] show "emeasure (density M f) B \<le> emeasure N B" |
|
56996 | 400 |
by (auto simp: sets_eq emeasure_density cong: nn_integral_cong) |
47694 | 401 |
qed (auto simp: sets_eq emeasure_density) |
56996 | 402 |
from emeasure_M[of "space M"] N.finite_emeasure_space nn_integral_nonneg[of M "?F (space M)"] |
47694 | 403 |
interpret M': finite_measure ?M |
404 |
by (auto intro!: finite_measureI simp: sets_eq_imp_space_eq[OF sets_eq] N.emeasure_eq_measure ereal_minus_eq_PInfty_iff) |
|
405 |
||
406 |
have ac: "absolutely_continuous M ?M" unfolding absolutely_continuous_def |
|
45777
c36637603821
remove unnecessary sublocale instantiations in HOL-Probability (for clarity and speedup); remove Infinite_Product_Measure.product_prob_space which was a duplicate of Probability_Measure.product_prob_space
hoelzl
parents:
45769
diff
changeset
|
407 |
proof |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
408 |
fix A assume A_M: "A \<in> null_sets M" |
61808 | 409 |
with \<open>absolutely_continuous M N\<close> have A_N: "A \<in> null_sets N" |
47694 | 410 |
unfolding absolutely_continuous_def by auto |
61808 | 411 |
moreover from A_M A_N have "(\<integral>\<^sup>+ x. ?F A x \<partial>M) \<le> N A" using \<open>f \<in> G\<close> by (auto simp: G_def) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
412 |
ultimately have "N A - (\<integral>\<^sup>+ x. ?F A x \<partial>M) = 0" |
56996 | 413 |
using nn_integral_nonneg[of M] by (auto intro!: antisym) |
47694 | 414 |
then show "A \<in> null_sets ?M" |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
415 |
using A_M by (simp add: emeasure_M null_sets_def sets_eq) |
38656 | 416 |
qed |
47694 | 417 |
have upper_bound: "\<forall>A\<in>sets M. ?M A \<le> 0" |
38656 | 418 |
proof (rule ccontr) |
419 |
assume "\<not> ?thesis" |
|
47694 | 420 |
then obtain A where A: "A \<in> sets M" and pos: "0 < ?M A" |
38656 | 421 |
by (auto simp: not_le) |
422 |
note pos |
|
47694 | 423 |
also have "?M A \<le> ?M (space M)" |
424 |
using emeasure_space[of ?M A] by (simp add: sets_eq[THEN sets_eq_imp_space_eq]) |
|
425 |
finally have pos_t: "0 < ?M (space M)" by simp |
|
38656 | 426 |
moreover |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
427 |
from pos_t have "emeasure M (space M) \<noteq> 0" |
47694 | 428 |
using ac unfolding absolutely_continuous_def by (auto simp: null_sets_def) |
429 |
then have pos_M: "0 < emeasure M (space M)" |
|
430 |
using emeasure_nonneg[of M "space M"] by (simp add: le_less) |
|
38656 | 431 |
moreover |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
432 |
have "(\<integral>\<^sup>+x. f x * indicator (space M) x \<partial>M) \<le> N (space M)" |
61808 | 433 |
using \<open>f \<in> G\<close> unfolding G_def by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
434 |
hence "(\<integral>\<^sup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>" |
47694 | 435 |
using M'.finite_emeasure_space by auto |
38656 | 436 |
moreover |
47694 | 437 |
def b \<equiv> "?M (space M) / emeasure M (space M) / 2" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
438 |
ultimately have b: "b \<noteq> 0 \<and> 0 \<le> b \<and> b \<noteq> \<infinity>" |
47694 | 439 |
by (auto simp: ereal_divide_eq) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
440 |
then have b: "b \<noteq> 0" "0 \<le> b" "0 < b" "b \<noteq> \<infinity>" by auto |
47694 | 441 |
let ?Mb = "density M (\<lambda>_. b)" |
442 |
have Mb: "finite_measure ?Mb" "sets ?Mb = sets ?M" |
|
443 |
using b by (auto simp: emeasure_density_const sets_eq intro!: finite_measureI) |
|
444 |
from M'.Radon_Nikodym_aux[OF this] guess A0 .. |
|
445 |
then have "A0 \<in> sets 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:
61169
diff
changeset
|
446 |
and space_less_A0: "measure ?M (space M) - real_of_ereal b * measure M (space M) \<le> measure ?M A0 - real_of_ereal b * measure M A0" |
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:
61169
diff
changeset
|
447 |
and *: "\<And>B. B \<in> sets M \<Longrightarrow> B \<subseteq> A0 \<Longrightarrow> 0 \<le> measure ?M B - real_of_ereal b * measure M B" |
47694 | 448 |
using b by (simp_all add: measure_density_const sets_eq_imp_space_eq[OF sets_eq] sets_eq) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
449 |
{ fix B assume B: "B \<in> sets M" "B \<subseteq> A0" |
47694 | 450 |
with *[OF this] have "b * emeasure M B \<le> ?M B" |
451 |
using b unfolding M'.emeasure_eq_measure emeasure_eq_measure by (cases b) auto } |
|
38656 | 452 |
note bM_le_t = this |
46731 | 453 |
let ?f0 = "\<lambda>x. f x + b * indicator A0 x" |
38656 | 454 |
{ fix A assume A: "A \<in> sets M" |
61808 | 455 |
hence "A \<inter> A0 \<in> sets M" using \<open>A0 \<in> sets M\<close> by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
456 |
have "(\<integral>\<^sup>+x. ?f0 x * indicator A x \<partial>M) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
457 |
(\<integral>\<^sup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)" |
56996 | 458 |
by (auto intro!: nn_integral_cong split: split_indicator) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
459 |
hence "(\<integral>\<^sup>+x. ?f0 x * indicator A x \<partial>M) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
460 |
(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) + b * emeasure M (A \<inter> A0)" |
61808 | 461 |
using \<open>A0 \<in> sets M\<close> \<open>A \<inter> A0 \<in> sets M\<close> A b \<open>f \<in> G\<close> |
56996 | 462 |
by (simp add: nn_integral_add nn_integral_cmult_indicator G_def) } |
38656 | 463 |
note f0_eq = this |
464 |
{ fix A assume A: "A \<in> sets M" |
|
61808 | 465 |
hence "A \<inter> A0 \<in> sets M" using \<open>A0 \<in> sets M\<close> by auto |
466 |
have f_le_v: "(\<integral>\<^sup>+x. ?F A x \<partial>M) \<le> N A" using \<open>f \<in> G\<close> A unfolding G_def by auto |
|
38656 | 467 |
note f0_eq[OF A] |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
468 |
also have "(\<integral>\<^sup>+x. ?F A x \<partial>M) + b * emeasure M (A \<inter> A0) \<le> (\<integral>\<^sup>+x. ?F A x \<partial>M) + ?M (A \<inter> A0)" |
61808 | 469 |
using bM_le_t[OF \<open>A \<inter> A0 \<in> sets M\<close>] \<open>A \<in> sets M\<close> \<open>A0 \<in> sets M\<close> |
38656 | 470 |
by (auto intro!: add_left_mono) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
471 |
also have "\<dots> \<le> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) + ?M A" |
61808 | 472 |
using emeasure_mono[of "A \<inter> A0" A ?M] \<open>A \<in> sets M\<close> \<open>A0 \<in> sets M\<close> |
47694 | 473 |
by (auto intro!: add_left_mono simp: sets_eq) |
474 |
also have "\<dots> \<le> N A" |
|
61808 | 475 |
unfolding emeasure_M[OF \<open>A \<in> sets M\<close>] |
56996 | 476 |
using f_le_v N.emeasure_eq_measure[of A] nn_integral_nonneg[of M "?F A"] |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
477 |
by (cases "\<integral>\<^sup>+x. ?F A x \<partial>M", cases "N A") auto |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
478 |
finally have "(\<integral>\<^sup>+x. ?f0 x * indicator A x \<partial>M) \<le> N A" . } |
61808 | 479 |
hence "?f0 \<in> G" using \<open>A0 \<in> sets M\<close> b \<open>f \<in> G\<close> by (auto simp: G_def) |
56996 | 480 |
have int_f_finite: "integral\<^sup>N M f \<noteq> \<infinity>" |
47694 | 481 |
by (metis N.emeasure_finite ereal_infty_less_eq2(1) int_f_eq_y y_le) |
482 |
have "0 < ?M (space M) - emeasure ?Mb (space M)" |
|
483 |
using pos_t |
|
484 |
by (simp add: b emeasure_density_const) |
|
485 |
(simp add: M'.emeasure_eq_measure emeasure_eq_measure pos_M b_def) |
|
486 |
also have "\<dots> \<le> ?M A0 - b * emeasure M A0" |
|
61808 | 487 |
using space_less_A0 \<open>A0 \<in> sets M\<close> b |
47694 | 488 |
by (cases b) (auto simp add: b emeasure_density_const sets_eq M'.emeasure_eq_measure emeasure_eq_measure) |
489 |
finally have 1: "b * emeasure M A0 < ?M A0" |
|
61808 | 490 |
by (metis M'.emeasure_real \<open>A0 \<in> sets M\<close> bM_le_t diff_self ereal_less(1) ereal_minus(1) |
47694 | 491 |
less_eq_ereal_def mult_zero_left not_square_less_zero subset_refl zero_ereal_def) |
492 |
with b have "0 < ?M A0" |
|
493 |
by (metis M'.emeasure_real MInfty_neq_PInfty(1) emeasure_real ereal_less_eq(5) ereal_zero_times |
|
494 |
ereal_mult_eq_MInfty ereal_mult_eq_PInfty ereal_zero_less_0_iff less_eq_ereal_def) |
|
61808 | 495 |
then have "emeasure M A0 \<noteq> 0" using ac \<open>A0 \<in> sets M\<close> |
47694 | 496 |
by (auto simp: absolutely_continuous_def null_sets_def) |
497 |
then have "0 < emeasure M A0" using emeasure_nonneg[of M A0] by auto |
|
498 |
hence "0 < b * emeasure M A0" using b by (auto simp: ereal_zero_less_0_iff) |
|
56996 | 499 |
with int_f_finite have "?y + 0 < integral\<^sup>N M f + b * emeasure M A0" unfolding int_f_eq_y |
61808 | 500 |
using \<open>f \<in> G\<close> |
56996 | 501 |
by (intro ereal_add_strict_mono) (auto intro!: SUP_upper2 nn_integral_nonneg) |
61808 | 502 |
also have "\<dots> = integral\<^sup>N M ?f0" using f0_eq[OF sets.top] \<open>A0 \<in> sets M\<close> sets.sets_into_space |
56996 | 503 |
by (simp cong: nn_integral_cong) |
504 |
finally have "?y < integral\<^sup>N M ?f0" by simp |
|
61808 | 505 |
moreover from \<open>?f0 \<in> G\<close> have "integral\<^sup>N M ?f0 \<le> ?y" by (auto intro!: SUP_upper) |
38656 | 506 |
ultimately show False by auto |
507 |
qed |
|
47694 | 508 |
let ?f = "\<lambda>x. max 0 (f x)" |
38656 | 509 |
show ?thesis |
47694 | 510 |
proof (intro bexI[of _ ?f] measure_eqI conjI) |
511 |
show "sets (density M ?f) = sets N" |
|
512 |
by (simp add: sets_eq) |
|
513 |
fix A assume A: "A\<in>sets (density M ?f)" |
|
514 |
then show "emeasure (density M ?f) A = emeasure N A" |
|
61808 | 515 |
using \<open>f \<in> G\<close> A upper_bound[THEN bspec, of A] N.emeasure_eq_measure[of A] |
56996 | 516 |
by (cases "integral\<^sup>N M (?F A)") |
47694 | 517 |
(auto intro!: antisym simp add: emeasure_density G_def emeasure_M density_max_0[symmetric]) |
518 |
qed auto |
|
38656 | 519 |
qed |
520 |
||
40859 | 521 |
lemma (in finite_measure) split_space_into_finite_sets_and_rest: |
47694 | 522 |
assumes ac: "absolutely_continuous M N" and sets_eq: "sets N = sets M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
523 |
shows "\<exists>A0\<in>sets M. \<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> A0 = space M - (\<Union>i. B i) \<and> |
47694 | 524 |
(\<forall>A\<in>sets M. A \<subseteq> A0 \<longrightarrow> (emeasure M A = 0 \<and> N A = 0) \<or> (emeasure M A > 0 \<and> N A = \<infinity>)) \<and> |
525 |
(\<forall>i. N (B i) \<noteq> \<infinity>)" |
|
38656 | 526 |
proof - |
47694 | 527 |
let ?Q = "{Q\<in>sets M. N Q \<noteq> \<infinity>}" |
528 |
let ?a = "SUP Q:?Q. emeasure M Q" |
|
529 |
have "{} \<in> ?Q" by auto |
|
38656 | 530 |
then have Q_not_empty: "?Q \<noteq> {}" by blast |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
531 |
have "?a \<le> emeasure M (space M)" using sets.sets_into_space |
47694 | 532 |
by (auto intro!: SUP_least emeasure_mono) |
533 |
then have "?a \<noteq> \<infinity>" using finite_emeasure_space |
|
38656 | 534 |
by auto |
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56166
diff
changeset
|
535 |
from SUP_countable_SUP [OF Q_not_empty, of "emeasure M"] |
47694 | 536 |
obtain Q'' where "range Q'' \<subseteq> emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)" |
38656 | 537 |
by auto |
47694 | 538 |
then have "\<forall>i. \<exists>Q'. Q'' i = emeasure M Q' \<and> Q' \<in> ?Q" by auto |
539 |
from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = emeasure M (Q' i)" "\<And>i. Q' i \<in> ?Q" |
|
38656 | 540 |
by auto |
47694 | 541 |
then have a_Lim: "?a = (SUP i::nat. emeasure M (Q' i))" using a by simp |
46731 | 542 |
let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i" |
47694 | 543 |
have Union: "(SUP i. emeasure M (?O i)) = emeasure M (\<Union>i. ?O i)" |
544 |
proof (rule SUP_emeasure_incseq[of ?O]) |
|
545 |
show "range ?O \<subseteq> sets M" using Q' by auto |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44568
diff
changeset
|
546 |
show "incseq ?O" by (fastforce intro!: incseq_SucI) |
38656 | 547 |
qed |
548 |
have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto |
|
47694 | 549 |
have O_sets: "\<And>i. ?O i \<in> sets M" using Q' by auto |
38656 | 550 |
then have O_in_G: "\<And>i. ?O i \<in> ?Q" |
551 |
proof (safe del: notI) |
|
47694 | 552 |
fix i have "Q' ` {..i} \<subseteq> sets M" using Q' by auto |
553 |
then have "N (?O i) \<le> (\<Sum>i\<le>i. N (Q' i))" |
|
554 |
by (simp add: sets_eq emeasure_subadditive_finite) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
555 |
also have "\<dots> < \<infinity>" using Q' by (simp add: setsum_Pinfty) |
47694 | 556 |
finally show "N (?O i) \<noteq> \<infinity>" by simp |
38656 | 557 |
qed auto |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44568
diff
changeset
|
558 |
have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce |
47694 | 559 |
have a_eq: "?a = emeasure M (\<Union>i. ?O i)" unfolding Union[symmetric] |
38656 | 560 |
proof (rule antisym) |
47694 | 561 |
show "?a \<le> (SUP i. emeasure M (?O i))" unfolding a_Lim |
562 |
using Q' by (auto intro!: SUP_mono emeasure_mono) |
|
563 |
show "(SUP i. emeasure M (?O i)) \<le> ?a" unfolding SUP_def |
|
38656 | 564 |
proof (safe intro!: Sup_mono, unfold bex_simps) |
565 |
fix i |
|
52141
eff000cab70f
weaker precendence of syntax for big intersection and union on sets
haftmann
parents:
51329
diff
changeset
|
566 |
have *: "(\<Union>(Q' ` {..i})) = ?O i" by auto |
47694 | 567 |
then show "\<exists>x. (x \<in> sets M \<and> N x \<noteq> \<infinity>) \<and> |
52141
eff000cab70f
weaker precendence of syntax for big intersection and union on sets
haftmann
parents:
51329
diff
changeset
|
568 |
emeasure M (\<Union>(Q' ` {..i})) \<le> emeasure M x" |
38656 | 569 |
using O_in_G[of i] by (auto intro!: exI[of _ "?O i"]) |
570 |
qed |
|
571 |
qed |
|
46731 | 572 |
let ?O_0 = "(\<Union>i. ?O i)" |
38656 | 573 |
have "?O_0 \<in> sets M" using Q' by auto |
40859 | 574 |
def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n" |
38656 | 575 |
{ fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) } |
576 |
note Q_sets = this |
|
40859 | 577 |
show ?thesis |
578 |
proof (intro bexI exI conjI ballI impI allI) |
|
579 |
show "disjoint_family Q" |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44568
diff
changeset
|
580 |
by (fastforce simp: disjoint_family_on_def Q_def |
40859 | 581 |
split: nat.split_asm) |
582 |
show "range Q \<subseteq> sets M" |
|
583 |
using Q_sets by auto |
|
584 |
{ fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0" |
|
47694 | 585 |
show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>" |
40859 | 586 |
proof (rule disjCI, simp) |
47694 | 587 |
assume *: "0 < emeasure M A \<longrightarrow> N A \<noteq> \<infinity>" |
588 |
show "emeasure M A = 0 \<and> N A = 0" |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
589 |
proof (cases "emeasure M A = 0") |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
590 |
case True |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
591 |
with ac A have "N A = 0" |
40859 | 592 |
unfolding absolutely_continuous_def by auto |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
593 |
with True show ?thesis by simp |
40859 | 594 |
next |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
595 |
case False |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
596 |
with * have "N A \<noteq> \<infinity>" using emeasure_nonneg[of M A] by auto |
47694 | 597 |
with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
598 |
using Q' by (auto intro!: plus_emeasure sets.countable_UN) |
47694 | 599 |
also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))" |
600 |
proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified]) |
|
40859 | 601 |
show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M" |
61808 | 602 |
using \<open>N A \<noteq> \<infinity>\<close> O_sets A by auto |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44568
diff
changeset
|
603 |
qed (fastforce intro!: incseq_SucI) |
40859 | 604 |
also have "\<dots> \<le> ?a" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44918
diff
changeset
|
605 |
proof (safe intro!: SUP_least) |
40859 | 606 |
fix i have "?O i \<union> A \<in> ?Q" |
607 |
proof (safe del: notI) |
|
608 |
show "?O i \<union> A \<in> sets M" using O_sets A by auto |
|
47694 | 609 |
from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A" |
610 |
using emeasure_subadditive[of "?O i" N A] A O_sets by (auto simp: sets_eq) |
|
611 |
with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>" |
|
61808 | 612 |
using \<open>N A \<noteq> \<infinity>\<close> by auto |
40859 | 613 |
qed |
47694 | 614 |
then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper) |
40859 | 615 |
qed |
47694 | 616 |
finally have "emeasure M A = 0" |
617 |
unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure) |
|
61808 | 618 |
with \<open>emeasure M A \<noteq> 0\<close> show ?thesis by auto |
40859 | 619 |
qed |
620 |
qed } |
|
47694 | 621 |
{ fix i show "N (Q i) \<noteq> \<infinity>" |
40859 | 622 |
proof (cases i) |
623 |
case 0 then show ?thesis |
|
624 |
unfolding Q_def using Q'[of 0] by simp |
|
625 |
next |
|
626 |
case (Suc n) |
|
61808 | 627 |
with \<open>?O n \<in> ?Q\<close> \<open>?O (Suc n) \<in> ?Q\<close> |
60585 | 628 |
emeasure_Diff[OF _ _ _ O_mono, of N n] emeasure_nonneg[of N "(\<Union>x\<le>n. Q' x)"] |
47694 | 629 |
show ?thesis |
630 |
by (auto simp: sets_eq ereal_minus_eq_PInfty_iff Q_def) |
|
40859 | 631 |
qed } |
632 |
show "space M - ?O_0 \<in> sets M" using Q'_sets by auto |
|
633 |
{ fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)" |
|
634 |
proof (induct j) |
|
635 |
case 0 then show ?case by (simp add: Q_def) |
|
636 |
next |
|
637 |
case (Suc j) |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44568
diff
changeset
|
638 |
have eq: "\<And>j. (\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q' i)" by fastforce |
40859 | 639 |
have "{..j} \<union> {..Suc j} = {..Suc j}" by auto |
640 |
then have "(\<Union>i\<le>Suc j. Q' i) = (\<Union>i\<le>j. Q' i) \<union> Q (Suc j)" |
|
641 |
by (simp add: UN_Un[symmetric] Q_def del: UN_Un) |
|
642 |
then show ?case using Suc by (auto simp add: eq atMost_Suc) |
|
643 |
qed } |
|
644 |
then have "(\<Union>j. (\<Union>i\<le>j. ?O i)) = (\<Union>j. (\<Union>i\<le>j. Q i))" by simp |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44568
diff
changeset
|
645 |
then show "space M - ?O_0 = space M - (\<Union>i. Q i)" by fastforce |
40859 | 646 |
qed |
647 |
qed |
|
648 |
||
649 |
lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite: |
|
47694 | 650 |
assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M" |
651 |
shows "\<exists>f\<in>borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N" |
|
40859 | 652 |
proof - |
653 |
from split_space_into_finite_sets_and_rest[OF assms] |
|
654 |
obtain Q0 and Q :: "nat \<Rightarrow> 'a set" |
|
655 |
where Q: "disjoint_family Q" "range Q \<subseteq> sets M" |
|
656 |
and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)" |
|
47694 | 657 |
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>" |
658 |
and Q_fin: "\<And>i. N (Q i) \<noteq> \<infinity>" by force |
|
40859 | 659 |
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto |
47694 | 660 |
let ?N = "\<lambda>i. density N (indicator (Q i))" and ?M = "\<lambda>i. density M (indicator (Q i))" |
661 |
have "\<forall>i. \<exists>f\<in>borel_measurable (?M i). (\<forall>x. 0 \<le> f x) \<and> density (?M i) f = ?N i" |
|
662 |
proof (intro allI finite_measure.Radon_Nikodym_finite_measure) |
|
38656 | 663 |
fix i |
47694 | 664 |
from Q show "finite_measure (?M i)" |
56996 | 665 |
by (auto intro!: finite_measureI cong: nn_integral_cong |
47694 | 666 |
simp add: emeasure_density subset_eq sets_eq) |
667 |
from Q have "emeasure (?N i) (space N) = emeasure N (Q i)" |
|
56996 | 668 |
by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: nn_integral_cong) |
47694 | 669 |
with Q_fin show "finite_measure (?N i)" |
670 |
by (auto intro!: finite_measureI) |
|
671 |
show "sets (?N i) = sets (?M i)" by (simp add: sets_eq) |
|
50003 | 672 |
have [measurable]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq) |
47694 | 673 |
show "absolutely_continuous (?M i) (?N i)" |
61808 | 674 |
using \<open>absolutely_continuous M N\<close> \<open>Q i \<in> sets M\<close> |
47694 | 675 |
by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq |
676 |
intro!: absolutely_continuous_AE[OF sets_eq]) |
|
38656 | 677 |
qed |
47694 | 678 |
from choice[OF this[unfolded Bex_def]] |
679 |
obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x" |
|
680 |
and f_density: "\<And>i. density (?M i) (f i) = ?N i" |
|
54776
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
53374
diff
changeset
|
681 |
by force |
47694 | 682 |
{ fix A i assume A: "A \<in> sets M" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
683 |
with Q borel have "(\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M) = emeasure (density (?M i) (f i)) A" |
56996 | 684 |
by (auto simp add: emeasure_density nn_integral_density subset_eq |
685 |
intro!: nn_integral_cong split: split_indicator) |
|
47694 | 686 |
also have "\<dots> = emeasure N (Q i \<inter> A)" |
687 |
using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
688 |
finally have "emeasure N (Q i \<inter> A) = (\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)" .. } |
47694 | 689 |
note integral_eq = this |
46731 | 690 |
let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator Q0 x" |
38656 | 691 |
show ?thesis |
692 |
proof (safe intro!: bexI[of _ ?f]) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
693 |
show "?f \<in> borel_measurable M" using Q0 borel Q_sets |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
694 |
by (auto intro!: measurable_If) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
695 |
show "\<And>x. 0 \<le> ?f x" using borel by (auto intro!: suminf_0_le simp: indicator_def) |
47694 | 696 |
show "density M ?f = N" |
697 |
proof (rule measure_eqI) |
|
698 |
fix A assume "A \<in> sets (density M ?f)" |
|
699 |
then have "A \<in> sets M" by simp |
|
700 |
have Qi: "\<And>i. Q i \<in> sets M" using Q by auto |
|
701 |
have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M" |
|
702 |
"\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x" |
|
61808 | 703 |
using borel Qi Q0(1) \<open>A \<in> sets M\<close> by (auto intro!: borel_measurable_ereal_times) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
704 |
have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator (Q0 \<inter> A) x \<partial>M)" |
56996 | 705 |
using borel by (intro nn_integral_cong) (auto simp: indicator_def) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
706 |
also have "\<dots> = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M (Q0 \<inter> A)" |
61808 | 707 |
using borel Qi Q0(1) \<open>A \<in> sets M\<close> |
56996 | 708 |
by (subst nn_integral_add) (auto simp del: ereal_infty_mult |
709 |
simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le) |
|
47694 | 710 |
also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)" |
61808 | 711 |
by (subst integral_eq[OF \<open>A \<in> sets M\<close>], subst nn_integral_suminf) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
712 |
finally have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M (Q0 \<inter> A)" . |
47694 | 713 |
moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)" |
61808 | 714 |
using Q Q_sets \<open>A \<in> sets M\<close> |
47694 | 715 |
by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq) |
716 |
moreover have "\<infinity> * emeasure M (Q0 \<inter> A) = N (Q0 \<inter> A)" |
|
717 |
proof - |
|
61808 | 718 |
have "Q0 \<inter> A \<in> sets M" using Q0(1) \<open>A \<in> sets M\<close> by blast |
47694 | 719 |
from in_Q0[OF this] show ?thesis by auto |
720 |
qed |
|
721 |
moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M" |
|
61808 | 722 |
using Q_sets \<open>A \<in> sets M\<close> Q0(1) by auto |
47694 | 723 |
moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}" |
61808 | 724 |
using \<open>A \<in> sets M\<close> sets.sets_into_space Q0 by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
725 |
ultimately have "N A = (\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M)" |
47694 | 726 |
using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "Q0 \<inter> A"] by (simp add: sets_eq) |
61808 | 727 |
with \<open>A \<in> sets M\<close> borel Q Q0(1) show "emeasure (density M ?f) A = N A" |
50003 | 728 |
by (auto simp: subset_eq emeasure_density) |
47694 | 729 |
qed (simp add: sets_eq) |
38656 | 730 |
qed |
731 |
qed |
|
732 |
||
733 |
lemma (in sigma_finite_measure) Radon_Nikodym: |
|
47694 | 734 |
assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M" |
735 |
shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> density M f = N" |
|
38656 | 736 |
proof - |
737 |
from Ex_finite_integrable_function |
|
56996 | 738 |
obtain h where finite: "integral\<^sup>N M h \<noteq> \<infinity>" and |
38656 | 739 |
borel: "h \<in> borel_measurable M" and |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
740 |
nn: "\<And>x. 0 \<le> h x" and |
38656 | 741 |
pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
742 |
"\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
743 |
let ?T = "\<lambda>A. (\<integral>\<^sup>+x. h x * indicator A x \<partial>M)" |
47694 | 744 |
let ?MT = "density M h" |
745 |
from borel finite nn interpret T: finite_measure ?MT |
|
56996 | 746 |
by (auto intro!: finite_measureI cong: nn_integral_cong simp: emeasure_density) |
47694 | 747 |
have "absolutely_continuous ?MT N" "sets N = sets ?MT" |
748 |
proof (unfold absolutely_continuous_def, safe) |
|
749 |
fix A assume "A \<in> null_sets ?MT" |
|
750 |
with borel have "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> h x \<le> 0" |
|
751 |
by (auto simp add: null_sets_density_iff) |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
752 |
with pos sets.sets_into_space have "AE x in M. x \<notin> A" |
61810 | 753 |
by (elim eventually_mono) (auto simp: not_le[symmetric]) |
47694 | 754 |
then have "A \<in> null_sets M" |
61808 | 755 |
using \<open>A \<in> sets M\<close> by (simp add: AE_iff_null_sets) |
47694 | 756 |
with ac show "A \<in> null_sets N" |
757 |
by (auto simp: absolutely_continuous_def) |
|
758 |
qed (auto simp add: sets_eq) |
|
759 |
from T.Radon_Nikodym_finite_measure_infinite[OF this] |
|
760 |
obtain f where f_borel: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "density ?MT f = N" by auto |
|
761 |
with nn borel show ?thesis |
|
762 |
by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq) |
|
38656 | 763 |
qed |
764 |
||
61808 | 765 |
subsection \<open>Uniqueness of densities\<close> |
40859 | 766 |
|
47694 | 767 |
lemma finite_density_unique: |
40859 | 768 |
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
47694 | 769 |
assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" |
56996 | 770 |
and fin: "integral\<^sup>N M f \<noteq> \<infinity>" |
49785 | 771 |
shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)" |
40859 | 772 |
proof (intro iffI ballI) |
47694 | 773 |
fix A assume eq: "AE x in M. f x = g x" |
49785 | 774 |
with borel show "density M f = density M g" |
775 |
by (auto intro: density_cong) |
|
40859 | 776 |
next |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
777 |
let ?P = "\<lambda>f A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M" |
49785 | 778 |
assume "density M f = density M g" |
779 |
with borel have eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" |
|
780 |
by (simp add: emeasure_density[symmetric]) |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
781 |
from this[THEN bspec, OF sets.top] fin |
56996 | 782 |
have g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" by (simp cong: nn_integral_cong) |
40859 | 783 |
{ fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
47694 | 784 |
and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" |
56996 | 785 |
and g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" |
40859 | 786 |
let ?N = "{x\<in>space M. g x < f x}" |
787 |
have N: "?N \<in> sets M" using borel by simp |
|
56996 | 788 |
have "?P g ?N \<le> integral\<^sup>N M g" using pos |
789 |
by (intro nn_integral_mono_AE) (auto split: split_indicator) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
790 |
then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
791 |
have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^sup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)" |
56996 | 792 |
by (auto intro!: nn_integral_cong simp: indicator_def) |
40859 | 793 |
also have "\<dots> = ?P f ?N - ?P g ?N" |
56996 | 794 |
proof (rule nn_integral_diff) |
40859 | 795 |
show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M" |
796 |
using borel N by auto |
|
47694 | 797 |
show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x" |
798 |
"AE x in M. 0 \<le> g x * indicator ?N x" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
799 |
using pos by (auto split: split_indicator) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
800 |
qed fact |
40859 | 801 |
also have "\<dots> = 0" |
56996 | 802 |
unfolding eq[THEN bspec, OF N] using nn_integral_nonneg[of M] Pg_fin by auto |
47694 | 803 |
finally have "AE x in M. f x \<le> g x" |
56996 | 804 |
using pos borel nn_integral_PInf_AE[OF borel(2) g_fin] |
805 |
by (subst (asm) nn_integral_0_iff_AE) |
|
43920 | 806 |
(auto split: split_indicator simp: not_less ereal_minus_le_iff) } |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
807 |
from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq |
47694 | 808 |
show "AE x in M. f x = g x" by auto |
40859 | 809 |
qed |
810 |
||
811 |
lemma (in finite_measure) density_unique_finite_measure: |
|
812 |
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M" |
|
47694 | 813 |
assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> f' x" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
814 |
assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. f' x * indicator A x \<partial>M)" |
40859 | 815 |
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A") |
47694 | 816 |
shows "AE x in M. f x = f' x" |
40859 | 817 |
proof - |
47694 | 818 |
let ?D = "\<lambda>f. density M f" |
819 |
let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A" |
|
46731 | 820 |
let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x" |
47694 | 821 |
|
822 |
have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets 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:
61169
diff
changeset
|
823 |
using borel by (auto intro!: absolutely_continuousI_density) |
47694 | 824 |
from split_space_into_finite_sets_and_rest[OF this] |
40859 | 825 |
obtain Q0 and Q :: "nat \<Rightarrow> 'a set" |
826 |
where Q: "disjoint_family Q" "range Q \<subseteq> sets M" |
|
827 |
and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)" |
|
47694 | 828 |
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> ?D f A = 0 \<or> 0 < emeasure M A \<and> ?D f A = \<infinity>" |
829 |
and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force |
|
830 |
with borel pos have in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> emeasure M A = 0 \<and> ?N A = 0 \<or> 0 < emeasure M A \<and> ?N A = \<infinity>" |
|
831 |
and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq) |
|
832 |
||
40859 | 833 |
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto |
47694 | 834 |
let ?D = "{x\<in>space M. f x \<noteq> f' x}" |
835 |
have "?D \<in> sets M" using borel by auto |
|
43920 | 836 |
have *: "\<And>i x A. \<And>y::ereal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x" |
40859 | 837 |
unfolding indicator_def by auto |
47694 | 838 |
have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos |
40859 | 839 |
by (intro finite_density_unique[THEN iffD1] allI) |
50003 | 840 |
(auto intro!: f measure_eqI simp: emeasure_density * subset_eq) |
47694 | 841 |
moreover have "AE x in M. ?f Q0 x = ?f' Q0 x" |
40859 | 842 |
proof (rule AE_I') |
43920 | 843 |
{ fix f :: "'a \<Rightarrow> ereal" assume borel: "f \<in> borel_measurable M" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
844 |
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)" |
46731 | 845 |
let ?A = "\<lambda>i. Q0 \<inter> {x \<in> space M. f x < (i::nat)}" |
47694 | 846 |
have "(\<Union>i. ?A i) \<in> null_sets M" |
40859 | 847 |
proof (rule null_sets_UN) |
43923 | 848 |
fix i ::nat have "?A i \<in> sets M" |
40859 | 849 |
using borel Q0(1) by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
850 |
have "?N (?A i) \<le> (\<integral>\<^sup>+x. (i::ereal) * indicator (?A i) x \<partial>M)" |
61808 | 851 |
unfolding eq[OF \<open>?A i \<in> sets M\<close>] |
56996 | 852 |
by (auto intro!: nn_integral_mono simp: indicator_def) |
47694 | 853 |
also have "\<dots> = i * emeasure M (?A i)" |
61808 | 854 |
using \<open>?A i \<in> sets M\<close> by (auto intro!: nn_integral_cmult_indicator) |
47694 | 855 |
also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by simp |
856 |
finally have "?N (?A i) \<noteq> \<infinity>" by simp |
|
61808 | 857 |
then show "?A i \<in> null_sets M" using in_Q0[OF \<open>?A i \<in> sets M\<close>] \<open>?A i \<in> sets M\<close> by auto |
40859 | 858 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
859 |
also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}" |
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:
61169
diff
changeset
|
860 |
by (auto simp: less_PInf_Ex_of_nat) |
47694 | 861 |
finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp } |
40859 | 862 |
from this[OF borel(1) refl] this[OF borel(2) f] |
47694 | 863 |
have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all |
864 |
then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets M" by (rule null_sets.Un) |
|
40859 | 865 |
show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq> |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
866 |
(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def) |
40859 | 867 |
qed |
47694 | 868 |
moreover have "AE x in M. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow> |
40859 | 869 |
?f (space M) x = ?f' (space M) x" |
870 |
by (auto simp: indicator_def Q0) |
|
47694 | 871 |
ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x" |
872 |
unfolding AE_all_countable[symmetric] |
|
873 |
by eventually_elim (auto intro!: AE_I2 split: split_if_asm simp: indicator_def) |
|
874 |
then show "AE x in M. f x = f' x" by auto |
|
40859 | 875 |
qed |
876 |
||
877 |
lemma (in sigma_finite_measure) density_unique: |
|
47694 | 878 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
879 |
assumes f': "f' \<in> borel_measurable M" "AE x in M. 0 \<le> f' x" |
|
880 |
assumes density_eq: "density M f = density M f'" |
|
881 |
shows "AE x in M. f x = f' x" |
|
40859 | 882 |
proof - |
883 |
obtain h where h_borel: "h \<in> borel_measurable M" |
|
56996 | 884 |
and fin: "integral\<^sup>N M h \<noteq> \<infinity>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<infinity>" "\<And>x. 0 \<le> h x" |
40859 | 885 |
using Ex_finite_integrable_function by auto |
47694 | 886 |
then have h_nn: "AE x in M. 0 \<le> h x" by auto |
887 |
let ?H = "density M h" |
|
888 |
interpret h: finite_measure ?H |
|
889 |
using fin h_borel pos |
|
56996 | 890 |
by (intro finite_measureI) (simp cong: nn_integral_cong emeasure_density add: fin) |
47694 | 891 |
let ?fM = "density M f" |
892 |
let ?f'M = "density M f'" |
|
40859 | 893 |
{ fix A assume "A \<in> sets M" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
894 |
then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
895 |
using pos(1) sets.sets_into_space by (force simp: indicator_def) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
896 |
then have "(\<integral>\<^sup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M" |
61808 | 897 |
using h_borel \<open>A \<in> sets M\<close> h_nn by (subst nn_integral_0_iff) auto } |
40859 | 898 |
note h_null_sets = this |
899 |
{ fix A assume "A \<in> sets M" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
900 |
have "(\<integral>\<^sup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?fM)" |
61808 | 901 |
using \<open>A \<in> sets M\<close> h_borel h_nn f f' |
56996 | 902 |
by (intro nn_integral_density[symmetric]) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
903 |
also have "\<dots> = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?f'M)" |
47694 | 904 |
by (simp_all add: density_eq) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
905 |
also have "\<dots> = (\<integral>\<^sup>+x. f' x * (h x * indicator A x) \<partial>M)" |
61808 | 906 |
using \<open>A \<in> sets M\<close> h_borel h_nn f f' |
56996 | 907 |
by (intro nn_integral_density) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
908 |
finally have "(\<integral>\<^sup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * (f' x * indicator A x) \<partial>M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
909 |
by (simp add: ac_simps) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
910 |
then have "(\<integral>\<^sup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^sup>+x. (f' x * indicator A x) \<partial>?H)" |
61808 | 911 |
using \<open>A \<in> sets M\<close> h_borel h_nn f f' |
56996 | 912 |
by (subst (asm) (1 2) nn_integral_density[symmetric]) auto } |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
913 |
then have "AE x in ?H. f x = f' x" using h_borel h_nn f f' |
47694 | 914 |
by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M]) |
915 |
(auto simp add: AE_density) |
|
916 |
then show "AE x in M. f x = f' x" |
|
917 |
unfolding eventually_ae_filter using h_borel pos |
|
918 |
by (auto simp add: h_null_sets null_sets_density_iff not_less[symmetric] |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
919 |
AE_iff_null_sets[symmetric]) blast |
40859 | 920 |
qed |
921 |
||
47694 | 922 |
lemma (in sigma_finite_measure) density_unique_iff: |
923 |
assumes f: "f \<in> borel_measurable M" and "AE x in M. 0 \<le> f x" |
|
924 |
assumes f': "f' \<in> borel_measurable M" and "AE x in M. 0 \<le> f' x" |
|
925 |
shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)" |
|
926 |
using density_unique[OF assms] density_cong[OF f f'] by auto |
|
927 |
||
49785 | 928 |
lemma sigma_finite_density_unique: |
929 |
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
|
930 |
assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" |
|
931 |
and fin: "sigma_finite_measure (density M f)" |
|
932 |
shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)" |
|
933 |
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:
61169
diff
changeset
|
934 |
assume "AE x in M. f x = g x" with borel show "density M f = density M g" |
49785 | 935 |
by (auto intro: density_cong) |
936 |
next |
|
937 |
assume eq: "density M f = density M g" |
|
61605 | 938 |
interpret f: sigma_finite_measure "density M f" by fact |
49785 | 939 |
from f.sigma_finite_incseq guess A . note cover = this |
940 |
||
941 |
have "AE x in M. \<forall>i. x \<in> A i \<longrightarrow> f x = g x" |
|
942 |
unfolding AE_all_countable |
|
943 |
proof |
|
944 |
fix i |
|
945 |
have "density (density M f) (indicator (A i)) = density (density M g) (indicator (A i))" |
|
946 |
unfolding eq .. |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
947 |
moreover have "(\<integral>\<^sup>+x. f x * indicator (A i) x \<partial>M) \<noteq> \<infinity>" |
49785 | 948 |
using cover(1) cover(3)[of i] borel by (auto simp: emeasure_density subset_eq) |
949 |
ultimately have "AE x in M. f x * indicator (A i) x = g x * indicator (A i) x" |
|
950 |
using borel pos cover(1) pos |
|
951 |
by (intro finite_density_unique[THEN iffD1]) |
|
952 |
(auto simp: density_density_eq subset_eq) |
|
953 |
then show "AE x in M. x \<in> A i \<longrightarrow> f x = g x" |
|
954 |
by auto |
|
955 |
qed |
|
956 |
with AE_space show "AE x in M. f x = g x" |
|
957 |
apply eventually_elim |
|
958 |
using cover(2)[symmetric] |
|
959 |
apply auto |
|
960 |
done |
|
961 |
qed |
|
962 |
||
49778
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
963 |
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite': |
47694 | 964 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
965 |
shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)" |
|
966 |
(is "sigma_finite_measure ?N \<longleftrightarrow> _") |
|
40859 | 967 |
proof |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
968 |
assume "sigma_finite_measure ?N" |
47694 | 969 |
then interpret N: sigma_finite_measure ?N . |
970 |
from N.Ex_finite_integrable_function obtain h where |
|
56996 | 971 |
h: "h \<in> borel_measurable M" "integral\<^sup>N ?N h \<noteq> \<infinity>" and |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
972 |
h_nn: "\<And>x. 0 \<le> h x" and |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
973 |
fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" by auto |
47694 | 974 |
have "AE x in M. f x * h x \<noteq> \<infinity>" |
40859 | 975 |
proof (rule AE_I') |
56996 | 976 |
have "integral\<^sup>N ?N h = (\<integral>\<^sup>+x. f x * h x \<partial>M)" using f h h_nn |
977 |
by (auto intro!: nn_integral_density) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
978 |
then have "(\<integral>\<^sup>+x. f x * h x \<partial>M) \<noteq> \<infinity>" |
40859 | 979 |
using h(2) by simp |
47694 | 980 |
then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets M" |
56996 | 981 |
using f h(1) by (auto intro!: nn_integral_PInf borel_measurable_vimage) |
40859 | 982 |
qed auto |
47694 | 983 |
then show "AE x in M. f x \<noteq> \<infinity>" |
41705 | 984 |
using fin by (auto elim!: AE_Ball_mp) |
40859 | 985 |
next |
47694 | 986 |
assume AE: "AE x in M. f x \<noteq> \<infinity>" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
987 |
from sigma_finite guess Q . note Q = this |
43923 | 988 |
def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ereal(of_nat (Suc n))}) \<inter> space M" |
40859 | 989 |
{ fix i j have "A i \<inter> Q j \<in> sets M" |
990 |
unfolding A_def using f Q |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
991 |
apply (rule_tac sets.Int) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
992 |
by (cases i) (auto intro: measurable_sets[OF f(1)]) } |
40859 | 993 |
note A_in_sets = this |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
994 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
995 |
show "sigma_finite_measure ?N" |
61169 | 996 |
proof (standard, intro exI conjI ballI) |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
997 |
show "countable (range (\<lambda>(i, j). A i \<inter> Q j))" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
998 |
by auto |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
999 |
show "range (\<lambda>(i, j). A i \<inter> Q j) \<subseteq> sets (density M f)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1000 |
using A_in_sets by auto |
40859 | 1001 |
next |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1002 |
have "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = (\<Union>i j. A i \<inter> Q j)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1003 |
by auto |
40859 | 1004 |
also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto |
1005 |
also have "(\<Union>i. A i) = space M" |
|
1006 |
proof safe |
|
1007 |
fix x assume x: "x \<in> space M" |
|
1008 |
show "x \<in> (\<Union>i. A i)" |
|
1009 |
proof (cases "f x") |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
1010 |
case PInf with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0]) |
40859 | 1011 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
1012 |
case (real r) |
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:
61169
diff
changeset
|
1013 |
with less_PInf_Ex_of_nat[of "f x"] obtain n :: nat where "f x < n" by auto |
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:
61169
diff
changeset
|
1014 |
then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
1015 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
1016 |
case MInf with x show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
1017 |
unfolding A_def by (auto intro!: exI[of _ "Suc 0"]) |
40859 | 1018 |
qed |
1019 |
qed (auto simp: A_def) |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1020 |
finally show "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = space ?N" by simp |
40859 | 1021 |
next |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1022 |
fix X assume "X \<in> range (\<lambda>(i, j). A i \<inter> Q j)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1023 |
then obtain i j where [simp]:"X = A i \<inter> Q j" by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1024 |
have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>" |
40859 | 1025 |
proof (cases i) |
1026 |
case 0 |
|
47694 | 1027 |
have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0" |
61808 | 1028 |
using AE by (auto simp: A_def \<open>i = 0\<close>) |
56996 | 1029 |
from nn_integral_cong_AE[OF this] show ?thesis by simp |
40859 | 1030 |
next |
1031 |
case (Suc n) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1032 |
then have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le> |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1033 |
(\<integral>\<^sup>+x. (Suc n :: ereal) * indicator (Q j) x \<partial>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:
61169
diff
changeset
|
1034 |
by (auto intro!: nn_integral_mono simp: indicator_def A_def) |
47694 | 1035 |
also have "\<dots> = Suc n * emeasure M (Q j)" |
56996 | 1036 |
using Q by (auto intro!: nn_integral_cmult_indicator) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
1037 |
also have "\<dots> < \<infinity>" |
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:
61169
diff
changeset
|
1038 |
using Q by auto |
40859 | 1039 |
finally show ?thesis by simp |
1040 |
qed |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1041 |
then show "emeasure ?N X \<noteq> \<infinity>" |
47694 | 1042 |
using A_in_sets Q f by (auto simp: emeasure_density) |
40859 | 1043 |
qed |
1044 |
qed |
|
1045 |
||
49778
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
1046 |
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite: |
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
1047 |
"f \<in> borel_measurable M \<Longrightarrow> sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)" |
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
1048 |
apply (subst density_max_0) |
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
1049 |
apply (subst sigma_finite_iff_density_finite') |
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
1050 |
apply (auto simp: max_def intro!: measurable_If) |
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
1051 |
done |
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
1052 |
|
61808 | 1053 |
subsection \<open>Radon-Nikodym derivative\<close> |
38656 | 1054 |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1055 |
definition RN_deriv :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a \<Rightarrow> ereal" where |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1056 |
"RN_deriv M N = |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1057 |
(if \<exists>f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1058 |
then SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1059 |
else (\<lambda>_. 0))" |
38656 | 1060 |
|
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:
61169
diff
changeset
|
1061 |
lemma RN_derivI: |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1062 |
assumes "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "density M f = N" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1063 |
shows "density M (RN_deriv M N) = N" |
40859 | 1064 |
proof - |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1065 |
have "\<exists>f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1066 |
using assms by auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1067 |
moreover then have "density M (SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N) = N" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1068 |
by (rule someI2_ex) auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1069 |
ultimately show ?thesis |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1070 |
by (auto simp: RN_deriv_def) |
40859 | 1071 |
qed |
1072 |
||
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:
61169
diff
changeset
|
1073 |
lemma |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1074 |
shows borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M" (is ?m) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1075 |
and RN_deriv_nonneg: "0 \<le> RN_deriv M N x" (is ?nn) |
38656 | 1076 |
proof - |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1077 |
{ assume ex: "\<exists>f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1078 |
have 1: "(SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N) \<in> borel_measurable M" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1079 |
using ex by (rule someI2_ex) auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1080 |
moreover |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1081 |
have 2: "0 \<le> (SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N) x" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1082 |
using ex by (rule someI2_ex) auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1083 |
note 1 2 } |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1084 |
from this show ?m ?nn |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1085 |
by (auto simp: RN_deriv_def) |
38656 | 1086 |
qed |
1087 |
||
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1088 |
lemma density_RN_deriv_density: |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1089 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1090 |
shows "density M (RN_deriv M (density M f)) = density M f" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1091 |
proof (rule RN_derivI) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1092 |
show "(\<lambda>x. max 0 (f x)) \<in> borel_measurable M" "\<And>x. 0 \<le> max 0 (f x)" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1093 |
using f by auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1094 |
show "density M (\<lambda>x. max 0 (f x)) = density M f" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1095 |
using f by (intro density_cong) (auto simp: max_def) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1096 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1097 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1098 |
lemma (in sigma_finite_measure) density_RN_deriv: |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1099 |
"absolutely_continuous M N \<Longrightarrow> sets N = sets M \<Longrightarrow> density M (RN_deriv M N) = N" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1100 |
by (metis RN_derivI Radon_Nikodym) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1101 |
|
56996 | 1102 |
lemma (in sigma_finite_measure) RN_deriv_nn_integral: |
47694 | 1103 |
assumes N: "absolutely_continuous M N" "sets N = sets M" |
40859 | 1104 |
and f: "f \<in> borel_measurable M" |
56996 | 1105 |
shows "integral\<^sup>N N f = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)" |
40859 | 1106 |
proof - |
56996 | 1107 |
have "integral\<^sup>N N f = integral\<^sup>N (density M (RN_deriv M N)) f" |
47694 | 1108 |
using N by (simp add: density_RN_deriv) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1109 |
also have "\<dots> = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)" |
56996 | 1110 |
using f by (simp add: nn_integral_density RN_deriv_nonneg) |
47694 | 1111 |
finally show ?thesis by simp |
40859 | 1112 |
qed |
1113 |
||
47694 | 1114 |
lemma null_setsD_AE: "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N" |
1115 |
using AE_iff_null_sets[of N M] by auto |
|
1116 |
||
1117 |
lemma (in sigma_finite_measure) RN_deriv_unique: |
|
1118 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
|
1119 |
and eq: "density M f = N" |
|
1120 |
shows "AE x in M. f x = RN_deriv M N x" |
|
49785 | 1121 |
unfolding eq[symmetric] |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1122 |
by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1123 |
RN_deriv_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric]) |
49785 | 1124 |
|
1125 |
lemma RN_deriv_unique_sigma_finite: |
|
1126 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
|
1127 |
and eq: "density M f = N" and fin: "sigma_finite_measure N" |
|
1128 |
shows "AE x in M. f x = RN_deriv M N x" |
|
1129 |
using fin unfolding eq[symmetric] |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1130 |
by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1131 |
RN_deriv_nonneg[THEN AE_I2] density_RN_deriv_density[symmetric]) |
47694 | 1132 |
|
1133 |
lemma (in sigma_finite_measure) RN_deriv_distr: |
|
1134 |
fixes T :: "'a \<Rightarrow> 'b" |
|
1135 |
assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M" |
|
1136 |
and inv: "\<forall>x\<in>space M. T' (T x) = x" |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1137 |
and ac[simp]: "absolutely_continuous (distr M M' T) (distr N M' T)" |
47694 | 1138 |
and N: "sets N = sets M" |
1139 |
shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x" |
|
41832 | 1140 |
proof (rule RN_deriv_unique) |
47694 | 1141 |
have [simp]: "sets N = sets M" by fact |
1142 |
note sets_eq_imp_space_eq[OF N, simp] |
|
1143 |
have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def) |
|
1144 |
{ fix A assume "A \<in> sets M" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
1145 |
with inv T T' sets.sets_into_space[OF this] |
47694 | 1146 |
have "T -` T' -` A \<inter> T -` space M' \<inter> space M = A" |
1147 |
by (auto simp: measurable_def) } |
|
1148 |
note eq = this[simp] |
|
1149 |
{ fix A assume "A \<in> sets M" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
1150 |
with inv T T' sets.sets_into_space[OF this] |
47694 | 1151 |
have "(T' \<circ> T) -` A \<inter> space M = A" |
1152 |
by (auto simp: measurable_def) } |
|
1153 |
note eq2 = this[simp] |
|
1154 |
let ?M' = "distr M M' T" and ?N' = "distr N M' T" |
|
1155 |
interpret M': sigma_finite_measure ?M' |
|
41832 | 1156 |
proof |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1157 |
from sigma_finite_countable guess F .. note F = this |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1158 |
show "\<exists>A. countable A \<and> A \<subseteq> sets (distr M M' T) \<and> \<Union>A = space (distr M M' T) \<and> (\<forall>a\<in>A. emeasure (distr M M' T) a \<noteq> \<infinity>)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1159 |
proof (intro exI conjI ballI) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1160 |
show *: "(\<lambda>A. T' -` A \<inter> space ?M') ` F \<subseteq> sets ?M'" |
47694 | 1161 |
using F T' by (auto simp: measurable_def) |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1162 |
show "\<Union>((\<lambda>A. T' -` A \<inter> space ?M')`F) = space ?M'" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1163 |
using F T'[THEN measurable_space] by (auto simp: set_eq_iff) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1164 |
next |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1165 |
fix X assume "X \<in> (\<lambda>A. T' -` A \<inter> space ?M')`F" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1166 |
then obtain A where [simp]: "X = T' -` A \<inter> space ?M'" and "A \<in> F" by auto |
61808 | 1167 |
have "X \<in> sets M'" using F T' \<open>A\<in>F\<close> by auto |
41832 | 1168 |
moreover |
61808 | 1169 |
have Fi: "A \<in> sets M" using F \<open>A\<in>F\<close> by auto |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1170 |
ultimately show "emeasure ?M' X \<noteq> \<infinity>" |
61808 | 1171 |
using F T T' \<open>A\<in>F\<close> by (simp add: emeasure_distr) |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
1172 |
qed (insert F, auto) |
41832 | 1173 |
qed |
47694 | 1174 |
have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M" |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1175 |
using T ac by measurable |
47694 | 1176 |
then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M" |
41832 | 1177 |
by (simp add: comp_def) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1178 |
show "AE x in M. 0 \<le> RN_deriv ?M' ?N' (T x)" by (auto intro: RN_deriv_nonneg) |
47694 | 1179 |
|
1180 |
have "N = distr N M (T' \<circ> T)" |
|
1181 |
by (subst measure_of_of_measure[of N, symmetric]) |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
1182 |
(auto simp add: distr_def sets.sigma_sets_eq intro!: measure_of_eq sets.space_closed) |
47694 | 1183 |
also have "\<dots> = distr (distr N M' T) M T'" |
1184 |
using T T' by (simp add: distr_distr) |
|
1185 |
also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'" |
|
1186 |
using ac by (simp add: M'.density_RN_deriv) |
|
1187 |
also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)" |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1188 |
by (simp add: distr_density_distr[OF T T', OF inv]) |
47694 | 1189 |
finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N" |
1190 |
by (simp add: comp_def) |
|
41832 | 1191 |
qed |
1192 |
||
40859 | 1193 |
lemma (in sigma_finite_measure) RN_deriv_finite: |
47694 | 1194 |
assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M" |
1195 |
shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>" |
|
40859 | 1196 |
proof - |
47694 | 1197 |
interpret N: sigma_finite_measure N by fact |
1198 |
from N show ?thesis |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1199 |
using sigma_finite_iff_density_finite[OF borel_measurable_RN_deriv, of N] |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1200 |
density_RN_deriv[OF ac] |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1201 |
by (simp add: RN_deriv_nonneg) |
40859 | 1202 |
qed |
1203 |
||
1204 |
lemma (in sigma_finite_measure) |
|
47694 | 1205 |
assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M" |
40859 | 1206 |
and f: "f \<in> borel_measurable M" |
47694 | 1207 |
shows RN_deriv_integrable: "integrable N f \<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:
61169
diff
changeset
|
1208 |
integrable M (\<lambda>x. real_of_ereal (RN_deriv M N x) * f x)" (is ?integrable) |
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:
61169
diff
changeset
|
1209 |
and RN_deriv_integral: "integral\<^sup>L N f = (\<integral>x. real_of_ereal (RN_deriv M N x) * f x \<partial>M)" (is ?integral) |
40859 | 1210 |
proof - |
47694 | 1211 |
note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp] |
1212 |
interpret N: sigma_finite_measure N by fact |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1213 |
|
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:
61169
diff
changeset
|
1214 |
have eq: "density M (RN_deriv M N) = density M (\<lambda>x. real_of_ereal (RN_deriv M N x))" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1215 |
proof (rule density_cong) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1216 |
from RN_deriv_finite[OF assms(1,2,3)] |
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:
61169
diff
changeset
|
1217 |
show "AE x in M. RN_deriv M N x = ereal (real_of_ereal (RN_deriv M N x))" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1218 |
by eventually_elim (insert RN_deriv_nonneg[of M N], auto simp: ereal_real) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1219 |
qed (insert ac, auto) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1220 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1221 |
show ?integrable |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1222 |
apply (subst density_RN_deriv[OF ac, symmetric]) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1223 |
unfolding eq |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1224 |
apply (intro integrable_real_density f AE_I2 real_of_ereal_pos RN_deriv_nonneg) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1225 |
apply (insert ac, auto) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1226 |
done |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1227 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1228 |
show ?integral |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1229 |
apply (subst density_RN_deriv[OF ac, symmetric]) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1230 |
unfolding eq |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1231 |
apply (intro integral_real_density f AE_I2 real_of_ereal_pos RN_deriv_nonneg) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1232 |
apply (insert ac, auto) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1233 |
done |
40859 | 1234 |
qed |
1235 |
||
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1236 |
lemma (in sigma_finite_measure) real_RN_deriv: |
47694 | 1237 |
assumes "finite_measure N" |
1238 |
assumes ac: "absolutely_continuous M N" "sets N = sets M" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1239 |
obtains D where "D \<in> borel_measurable M" |
47694 | 1240 |
and "AE x in M. RN_deriv M N x = ereal (D x)" |
1241 |
and "AE x in N. 0 < D x" |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1242 |
and "\<And>x. 0 \<le> D x" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1243 |
proof |
47694 | 1244 |
interpret N: finite_measure N by fact |
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:
61169
diff
changeset
|
1245 |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1246 |
note RN = borel_measurable_RN_deriv density_RN_deriv[OF ac] RN_deriv_nonneg[of M N] |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1247 |
|
47694 | 1248 |
let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1249 |
|
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:
61169
diff
changeset
|
1250 |
show "(\<lambda>x. real_of_ereal (RN_deriv M N x)) \<in> borel_measurable M" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1251 |
using RN by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1252 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1253 |
have "N (?RN \<infinity>) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)" |
47694 | 1254 |
using RN(1,3) by (subst RN(2)[symmetric]) (auto simp: emeasure_density) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1255 |
also have "\<dots> = (\<integral>\<^sup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)" |
56996 | 1256 |
by (intro nn_integral_cong) (auto simp: indicator_def) |
47694 | 1257 |
also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)" |
56996 | 1258 |
using RN by (intro nn_integral_cmult_indicator) auto |
47694 | 1259 |
finally have eq: "N (?RN \<infinity>) = \<infinity> * emeasure M (?RN \<infinity>)" . |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1260 |
moreover |
47694 | 1261 |
have "emeasure M (?RN \<infinity>) = 0" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1262 |
proof (rule ccontr) |
47694 | 1263 |
assume "emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>} \<noteq> 0" |
1264 |
moreover from RN have "0 \<le> emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" by auto |
|
1265 |
ultimately have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" by auto |
|
1266 |
with eq have "N (?RN \<infinity>) = \<infinity>" by simp |
|
1267 |
with N.emeasure_finite[of "?RN \<infinity>"] RN show False by auto |
|
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1268 |
qed |
47694 | 1269 |
ultimately have "AE x in M. RN_deriv M N x < \<infinity>" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1270 |
using RN by (intro AE_iff_measurable[THEN iffD2]) auto |
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:
61169
diff
changeset
|
1271 |
then show "AE x in M. RN_deriv M N x = ereal (real_of_ereal (RN_deriv M N x))" |
43920 | 1272 |
using RN(3) by (auto simp: ereal_real) |
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:
61169
diff
changeset
|
1273 |
then have eq: "AE x in N. RN_deriv M N x = ereal (real_of_ereal (RN_deriv M N x))" |
47694 | 1274 |
using ac absolutely_continuous_AE by auto |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1275 |
|
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:
61169
diff
changeset
|
1276 |
show "\<And>x. 0 \<le> real_of_ereal (RN_deriv M N x)" |
43920 | 1277 |
using RN by (auto intro: real_of_ereal_pos) |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1278 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1279 |
have "N (?RN 0) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)" |
47694 | 1280 |
using RN(1,3) by (subst RN(2)[symmetric]) (auto simp: emeasure_density) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1281 |
also have "\<dots> = (\<integral>\<^sup>+ x. 0 \<partial>M)" |
56996 | 1282 |
by (intro nn_integral_cong) (auto simp: indicator_def) |
47694 | 1283 |
finally have "AE x in N. RN_deriv M N x \<noteq> 0" |
1284 |
using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq) |
|
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:
61169
diff
changeset
|
1285 |
with RN(3) eq show "AE x in N. 0 < real_of_ereal (RN_deriv M N x)" |
43920 | 1286 |
by (auto simp: zero_less_real_of_ereal le_less) |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1287 |
qed |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1288 |
|
38656 | 1289 |
lemma (in sigma_finite_measure) RN_deriv_singleton: |
47694 | 1290 |
assumes ac: "absolutely_continuous M N" "sets N = sets M" |
1291 |
and x: "{x} \<in> sets M" |
|
1292 |
shows "N {x} = RN_deriv M N x * emeasure M {x}" |
|
38656 | 1293 |
proof - |
61808 | 1294 |
from \<open>{x} \<in> sets M\<close> |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1295 |
have "density M (RN_deriv M N) {x} = (\<integral>\<^sup>+w. RN_deriv M N x * indicator {x} w \<partial>M)" |
56996 | 1296 |
by (auto simp: indicator_def emeasure_density intro!: nn_integral_cong) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
1297 |
with x density_RN_deriv[OF ac] RN_deriv_nonneg[of M N] show ?thesis |
62083 | 1298 |
by (auto simp: max_def) |
38656 | 1299 |
qed |
1300 |
||
1301 |
end |