author | hoelzl |
Mon, 08 Aug 2016 14:13:14 +0200 | |
changeset 63627 | 6ddb43c6b711 |
parent 63626 | src/HOL/Multivariate_Analysis/Radon_Nikodym.thy@44ce6b524ff3 |
child 64283 | 979cdfdf7a79 |
permissions | -rw-r--r-- |
63627 | 1 |
(* Title: HOL/Analysis/Radon_Nikodym.thy |
42067 | 2 |
Author: Johannes Hölzl, TU München |
3 |
*) |
|
4 |
||
61808 | 5 |
section \<open>Radon-Nikod{\'y}m derivative\<close> |
42067 | 6 |
|
38656 | 7 |
theory Radon_Nikodym |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
8 |
imports Bochner_Integration |
38656 | 9 |
begin |
10 |
||
63329 | 11 |
definition diff_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure" |
12 |
where |
|
13 |
"diff_measure M N = measure_of (space M) (sets M) (\<lambda>A. emeasure M A - emeasure N A)" |
|
47694 | 14 |
|
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
|
15 |
lemma |
47694 | 16 |
shows space_diff_measure[simp]: "space (diff_measure M N) = space M" |
17 |
and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M" |
|
18 |
by (auto simp: diff_measure_def) |
|
19 |
||
20 |
lemma emeasure_diff_measure: |
|
21 |
assumes fin: "finite_measure M" "finite_measure N" and sets_eq: "sets M = sets N" |
|
22 |
assumes pos: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure N A \<le> emeasure M A" and A: "A \<in> sets M" |
|
23 |
shows "emeasure (diff_measure M N) A = emeasure M A - emeasure N A" (is "_ = ?\<mu> A") |
|
24 |
unfolding diff_measure_def |
|
25 |
proof (rule emeasure_measure_of_sigma) |
|
26 |
show "sigma_algebra (space M) (sets M)" .. |
|
27 |
show "positive (sets M) ?\<mu>" |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
28 |
using pos by (simp add: positive_def) |
47694 | 29 |
show "countably_additive (sets M) ?\<mu>" |
30 |
proof (rule countably_additiveI) |
|
31 |
fix A :: "nat \<Rightarrow> _" assume A: "range A \<subseteq> sets M" and "disjoint_family A" |
|
32 |
then have suminf: |
|
33 |
"(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)" |
|
34 |
"(\<Sum>i. emeasure N (A i)) = emeasure N (\<Union>i. A i)" |
|
35 |
by (simp_all add: suminf_emeasure sets_eq) |
|
36 |
with A have "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) = |
|
37 |
(\<Sum>i. emeasure M (A i)) - (\<Sum>i. emeasure N (A i))" |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
38 |
using fin pos[of "A _"] |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
39 |
by (intro ennreal_suminf_minus) |
63329 | 40 |
(auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure) |
47694 | 41 |
then show "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) = |
42 |
emeasure M (\<Union>i. A i) - emeasure N (\<Union>i. A i) " |
|
43 |
by (simp add: suminf) |
|
44 |
qed |
|
45 |
qed fact |
|
46 |
||
38656 | 47 |
lemma (in sigma_finite_measure) Ex_finite_integrable_function: |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
48 |
"\<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>)" |
38656 | 49 |
proof - |
50 |
obtain A :: "nat \<Rightarrow> 'a set" where |
|
50003 | 51 |
range[measurable]: "range A \<subseteq> sets M" and |
38656 | 52 |
space: "(\<Union>i. A i) = space M" and |
47694 | 53 |
measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and |
38656 | 54 |
disjoint: "disjoint_family A" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62083
diff
changeset
|
55 |
using sigma_finite_disjoint by blast |
47694 | 56 |
let ?B = "\<lambda>i. 2^Suc i * emeasure M (A i)" |
63329 | 57 |
have [measurable]: "\<And>i. A i \<in> sets M" |
58 |
using range by fastforce+ |
|
38656 | 59 |
have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)" |
60 |
proof |
|
47694 | 61 |
fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
62 |
using measure[of i] |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
63 |
by (auto intro!: dense simp: ennreal_inverse_positive ennreal_mult_eq_top_iff power_eq_top_ennreal) |
38656 | 64 |
qed |
65 |
from choice[OF this] obtain n where n: "\<And>i. 0 < n i" |
|
47694 | 66 |
"\<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:
41832
diff
changeset
|
67 |
{ fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this |
46731 | 68 |
let ?h = "\<lambda>x. \<Sum>i. n i * indicator (A i) x" |
38656 | 69 |
show ?thesis |
70 |
proof (safe intro!: bexI[of _ ?h] del: notI) |
|
63329 | 71 |
have "integral\<^sup>N M ?h = (\<Sum>i. n i * emeasure M (A i))" using pos |
56996 | 72 |
by (simp add: nn_integral_suminf nn_integral_cmult_indicator) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
73 |
also have "\<dots> \<le> (\<Sum>i. ennreal ((1/2)^Suc i))" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
74 |
proof (intro suminf_le allI) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
75 |
fix N |
47694 | 76 |
have "n N * emeasure M (A N) \<le> inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
77 |
using n[of N] by (intro mult_right_mono) auto |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
78 |
also have "\<dots> = (1/2)^Suc N * (inverse (emeasure M (A N)) * emeasure M (A N))" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
79 |
using measure[of N] |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
80 |
by (simp add: ennreal_inverse_power divide_ennreal_def ennreal_inverse_mult |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
81 |
power_eq_top_ennreal less_top[symmetric] mult_ac |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
82 |
del: power_Suc) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
83 |
also have "\<dots> \<le> inverse (ennreal 2) ^ Suc N" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
84 |
using measure[of N] |
63329 | 85 |
by (cases "emeasure M (A N)"; cases "emeasure M (A N) = 0") |
86 |
(auto simp: inverse_ennreal ennreal_mult[symmetric] divide_ennreal_def simp del: power_Suc) |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
87 |
also have "\<dots> = ennreal (inverse 2 ^ Suc N)" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
88 |
by (subst ennreal_power[symmetric], simp) (simp add: inverse_ennreal) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
89 |
finally show "n N * emeasure M (A N) \<le> ennreal ((1/2)^Suc N)" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
90 |
by simp |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
91 |
qed auto |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
92 |
also have "\<dots> < top" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
93 |
unfolding less_top[symmetric] |
63329 | 94 |
by (rule ennreal_suminf_neq_top) |
95 |
(auto simp: summable_geometric summable_Suc_iff simp del: power_Suc) |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
96 |
finally show "integral\<^sup>N M ?h \<noteq> \<infinity>" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
97 |
by (auto simp: top_unique) |
38656 | 98 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
99 |
{ 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
|
100 |
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
|
101 |
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
|
102 |
by (auto intro!: suminf_cmult_indicator intro: less_imp_le) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
103 |
then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by (auto simp: less_top[symmetric]) } |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
104 |
note pos = this |
50003 | 105 |
qed measurable |
38656 | 106 |
qed |
107 |
||
40871 | 108 |
subsection "Absolutely continuous" |
109 |
||
47694 | 110 |
definition absolutely_continuous :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where |
111 |
"absolutely_continuous M N \<longleftrightarrow> null_sets M \<subseteq> null_sets N" |
|
112 |
||
113 |
lemma absolutely_continuousI_count_space: "absolutely_continuous (count_space A) M" |
|
114 |
unfolding absolutely_continuous_def by (auto simp: null_sets_count_space) |
|
38656 | 115 |
|
47694 | 116 |
lemma absolutely_continuousI_density: |
117 |
"f \<in> borel_measurable M \<Longrightarrow> absolutely_continuous M (density M f)" |
|
118 |
by (force simp add: absolutely_continuous_def null_sets_density_iff dest: AE_not_in) |
|
119 |
||
120 |
lemma absolutely_continuousI_point_measure_finite: |
|
121 |
"(\<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)" |
|
122 |
unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff) |
|
123 |
||
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
124 |
lemma absolutely_continuousD: |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
125 |
"absolutely_continuous M N \<Longrightarrow> A \<in> sets M \<Longrightarrow> emeasure M A = 0 \<Longrightarrow> emeasure N A = 0" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
126 |
by (auto simp: absolutely_continuous_def null_sets_def) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
127 |
|
47694 | 128 |
lemma absolutely_continuous_AE: |
129 |
assumes sets_eq: "sets M' = sets M" |
|
130 |
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
|
131 |
shows "AE x in M'. P x" |
40859 | 132 |
proof - |
61808 | 133 |
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 | 134 |
unfolding eventually_ae_filter by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
135 |
show "AE x in M'. P x" |
47694 | 136 |
proof (rule AE_I') |
137 |
show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp |
|
61808 | 138 |
from \<open>absolutely_continuous M M'\<close> show "N \<in> null_sets M'" |
47694 | 139 |
using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto |
40859 | 140 |
qed |
141 |
qed |
|
142 |
||
40871 | 143 |
subsection "Existence of the Radon-Nikodym derivative" |
144 |
||
38656 | 145 |
lemma (in finite_measure) Radon_Nikodym_finite_measure: |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
146 |
assumes "finite_measure N" and sets_eq[simp]: "sets N = sets M" |
47694 | 147 |
assumes "absolutely_continuous M N" |
63329 | 148 |
shows "\<exists>f \<in> borel_measurable M. density M f = N" |
38656 | 149 |
proof - |
47694 | 150 |
interpret N: finite_measure N by fact |
63329 | 151 |
define G where "G = {g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A}" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
152 |
have [measurable_dest]: "f \<in> G \<Longrightarrow> f \<in> borel_measurable M" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
153 |
and G_D: "\<And>A. f \<in> G \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) \<le> N A" for f |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
154 |
by (auto simp: G_def) |
50003 | 155 |
note this[measurable_dest] |
38656 | 156 |
have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto |
157 |
hence "G \<noteq> {}" by auto |
|
63329 | 158 |
{ fix f g assume f[measurable]: "f \<in> G" and g[measurable]: "g \<in> G" |
38656 | 159 |
have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def |
160 |
proof safe |
|
161 |
let ?A = "{x \<in> space M. f x \<le> g x}" |
|
162 |
have "?A \<in> sets M" using f g unfolding G_def by auto |
|
63329 | 163 |
fix A assume [measurable]: "A \<in> sets M" |
38656 | 164 |
have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A" |
61808 | 165 |
using sets.sets_into_space[OF \<open>A \<in> sets M\<close>] by auto |
38656 | 166 |
have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x = |
167 |
g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x" |
|
168 |
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
|
169 |
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
|
170 |
(\<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
|
171 |
(\<integral>\<^sup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)" |
56996 | 172 |
by (auto cong: nn_integral_cong intro!: nn_integral_add) |
47694 | 173 |
also have "\<dots> \<le> N (?A \<inter> A) + N ((space M - ?A) \<inter> A)" |
63329 | 174 |
using f g unfolding G_def by (auto intro!: add_mono) |
47694 | 175 |
also have "\<dots> = N A" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
176 |
using union by (subst plus_emeasure) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
177 |
finally show "(\<integral>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> N A" . |
63329 | 178 |
qed auto } |
38656 | 179 |
note max_in_G = this |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
180 |
{ fix f assume "incseq f" and f: "\<And>i. f i \<in> G" |
50003 | 181 |
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
|
182 |
have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def |
38656 | 183 |
proof safe |
50003 | 184 |
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
|
185 |
next |
38656 | 186 |
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
|
187 |
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
|
188 |
(\<integral>\<^sup>+x. (SUP i. f i x * indicator A x) \<partial>M)" |
56996 | 189 |
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
|
190 |
also have "\<dots> = (SUP i. (\<integral>\<^sup>+x. f i x * indicator A x \<partial>M))" |
61808 | 191 |
using \<open>incseq f\<close> f \<open>A \<in> sets M\<close> |
56996 | 192 |
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
|
193 |
(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
|
194 |
finally show "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
195 |
using f \<open>A \<in> sets M\<close> by (auto intro!: SUP_least simp: G_D) |
38656 | 196 |
qed } |
197 |
note SUP_in_G = this |
|
56996 | 198 |
let ?y = "SUP g : G. integral\<^sup>N M g" |
47694 | 199 |
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
|
200 |
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
|
201 |
fix g assume "\<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A" |
56996 | 202 |
from this[THEN bspec, OF sets.top] show "integral\<^sup>N M g \<le> N (space M)" |
203 |
by (simp cong: nn_integral_cong) |
|
38656 | 204 |
qed |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
205 |
from ennreal_SUP_countable_SUP [OF \<open>G \<noteq> {}\<close>, of "integral\<^sup>N M"] guess ys .. note ys = this |
56996 | 206 |
then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n" |
38656 | 207 |
proof safe |
56996 | 208 |
fix n assume "range ys \<subseteq> integral\<^sup>N M ` G" |
209 |
hence "ys n \<in> integral\<^sup>N M ` G" by auto |
|
210 |
thus "\<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n" by auto |
|
38656 | 211 |
qed |
56996 | 212 |
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 |
213 |
hence y_eq: "?y = (SUP i. integral\<^sup>N M (gs i))" using ys by auto |
|
46731 | 214 |
let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})" |
63040 | 215 |
define f where [abs_def]: "f x = (SUP i. ?g i x)" for x |
46731 | 216 |
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
|
217 |
have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto |
38656 | 218 |
{ fix i have "?g i \<in> G" |
219 |
proof (induct i) |
|
220 |
case 0 thus ?case by simp fact |
|
221 |
next |
|
222 |
case (Suc i) |
|
61808 | 223 |
with Suc gs_not_empty \<open>gs (Suc i) \<in> G\<close> show ?case |
38656 | 224 |
by (auto simp add: atMost_Suc intro!: max_in_G) |
225 |
qed } |
|
226 |
note g_in_G = this |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
227 |
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
|
228 |
by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc) |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
229 |
|
50003 | 230 |
from SUP_in_G[OF this g_in_G] have [measurable]: "f \<in> G" unfolding f_def . |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
231 |
then have [measurable]: "f \<in> borel_measurable M" unfolding G_def by auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
232 |
|
56996 | 233 |
have "integral\<^sup>N M f = (SUP i. integral\<^sup>N M (?g i))" unfolding f_def |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
234 |
using g_in_G \<open>incseq ?g\<close> by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def) |
38656 | 235 |
also have "\<dots> = ?y" |
236 |
proof (rule antisym) |
|
56996 | 237 |
show "(SUP i. integral\<^sup>N M (?g i)) \<le> ?y" |
56166 | 238 |
using g_in_G by (auto intro: SUP_mono) |
56996 | 239 |
show "?y \<le> (SUP i. integral\<^sup>N M (?g i))" unfolding y_eq |
240 |
by (auto intro!: SUP_mono nn_integral_mono Max_ge) |
|
38656 | 241 |
qed |
56996 | 242 |
finally have int_f_eq_y: "integral\<^sup>N M f = ?y" . |
47694 | 243 |
|
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
244 |
have upper_bound: "\<forall>A\<in>sets M. N A \<le> density M f A" |
38656 | 245 |
proof (rule ccontr) |
246 |
assume "\<not> ?thesis" |
|
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
247 |
then obtain A where A[measurable]: "A \<in> sets M" and f_less_N: "density M f A < N A" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
248 |
by (auto simp: not_le) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
249 |
then have pos_A: "0 < M A" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
250 |
using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, OF A] |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
251 |
by (auto simp: zero_less_iff_neq_zero) |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
252 |
|
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
253 |
define b where "b = (N A - density M f A) / M A / 2" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
254 |
with f_less_N pos_A have "0 < b" "b \<noteq> top" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
255 |
by (auto intro!: diff_gr0_ennreal simp: zero_less_iff_neq_zero diff_eq_0_iff_ennreal ennreal_divide_eq_top_iff) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
256 |
|
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
257 |
let ?f = "\<lambda>x. f x + b" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
258 |
have "nn_integral M f \<noteq> top" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
259 |
using `f \<in> G`[THEN G_D, of "space M"] by (auto simp: top_unique cong: nn_integral_cong) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
260 |
with \<open>b \<noteq> top\<close> interpret Mf: finite_measure "density M ?f" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
261 |
by (intro finite_measureI) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
262 |
(auto simp: field_simps mult_indicator_subset ennreal_mult_eq_top_iff |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
263 |
emeasure_density nn_integral_cmult_indicator nn_integral_add |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
264 |
cong: nn_integral_cong) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
265 |
|
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
266 |
from unsigned_Hahn_decomposition[of "density M ?f" N A] |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
267 |
obtain Y where [measurable]: "Y \<in> sets M" and [simp]: "Y \<subseteq> A" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
268 |
and Y1: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> density M ?f C \<le> N C" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
269 |
and Y2: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> A \<Longrightarrow> C \<inter> Y = {} \<Longrightarrow> N C \<le> density M ?f C" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
270 |
by auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
271 |
|
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
272 |
let ?f' = "\<lambda>x. f x + b * indicator Y x" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
273 |
have "M Y \<noteq> 0" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
274 |
proof |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
275 |
assume "M Y = 0" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
276 |
then have "N Y = 0" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
277 |
using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, of Y] by auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
278 |
then have "N A = N (A - Y)" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
279 |
by (subst emeasure_Diff) auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
280 |
also have "\<dots> \<le> density M ?f (A - Y)" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
281 |
by (rule Y2) auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
282 |
also have "\<dots> \<le> density M ?f A - density M ?f Y" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
283 |
by (subst emeasure_Diff) auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
284 |
also have "\<dots> \<le> density M ?f A - 0" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
285 |
by (intro ennreal_minus_mono) auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
286 |
also have "density M ?f A = b * M A + density M f A" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
287 |
by (simp add: emeasure_density field_simps mult_indicator_subset nn_integral_add nn_integral_cmult_indicator) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
288 |
also have "\<dots> < N A" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
289 |
using f_less_N pos_A |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
290 |
by (cases "density M f A"; cases "M A"; cases "N A") |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
291 |
(auto simp: b_def ennreal_less_iff ennreal_minus divide_ennreal ennreal_numeral[symmetric] |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
292 |
ennreal_plus[symmetric] ennreal_mult[symmetric] field_simps |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
293 |
simp del: ennreal_numeral ennreal_plus) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
294 |
finally show False |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
295 |
by simp |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
296 |
qed |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
297 |
then have "nn_integral M f < nn_integral M ?f'" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
298 |
using \<open>0 < b\<close> \<open>nn_integral M f \<noteq> top\<close> |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
299 |
by (simp add: nn_integral_add nn_integral_cmult_indicator ennreal_zero_less_mult_iff zero_less_iff_neq_zero) |
38656 | 300 |
moreover |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
301 |
have "?f' \<in> G" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
302 |
unfolding G_def |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
303 |
proof safe |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
304 |
fix X assume [measurable]: "X \<in> sets M" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
305 |
have "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) = density M f (X - Y) + density M ?f (X \<inter> Y)" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
306 |
by (auto simp add: emeasure_density nn_integral_add[symmetric] split: split_indicator intro!: nn_integral_cong) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
307 |
also have "\<dots> \<le> N (X - Y) + N (X \<inter> Y)" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
308 |
using G_D[OF \<open>f \<in> G\<close>] by (intro add_mono Y1) (auto simp: emeasure_density) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
309 |
also have "\<dots> = N X" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
310 |
by (subst plus_emeasure) (auto intro!: arg_cong2[where f=emeasure]) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
311 |
finally show "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) \<le> N X" . |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
312 |
qed simp |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
313 |
then have "nn_integral M ?f' \<le> ?y" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
314 |
by (rule SUP_upper) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
315 |
ultimately show False |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
316 |
by (simp add: int_f_eq_y) |
38656 | 317 |
qed |
318 |
show ?thesis |
|
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
319 |
proof (intro bexI[of _ f] measure_eqI conjI antisym) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
320 |
fix A assume "A \<in> sets (density M f)" then show "emeasure (density M f) A \<le> emeasure N A" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
321 |
by (auto simp: emeasure_density intro!: G_D[OF \<open>f \<in> G\<close>]) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
322 |
next |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
323 |
fix A assume A: "A \<in> sets (density M f)" then show "emeasure N A \<le> emeasure (density M f) A" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
324 |
using upper_bound by auto |
47694 | 325 |
qed auto |
38656 | 326 |
qed |
327 |
||
40859 | 328 |
lemma (in finite_measure) split_space_into_finite_sets_and_rest: |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
329 |
assumes ac: "absolutely_continuous M N" and sets_eq[simp]: "sets N = sets M" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
330 |
shows "\<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> (\<forall>i. N (B i) \<noteq> \<infinity>) \<and> |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
331 |
(\<forall>A\<in>sets M. A \<inter> (\<Union>i. B i) = {} \<longrightarrow> (emeasure M A = 0 \<and> N A = 0) \<or> (emeasure M A > 0 \<and> N A = \<infinity>))" |
38656 | 332 |
proof - |
47694 | 333 |
let ?Q = "{Q\<in>sets M. N Q \<noteq> \<infinity>}" |
334 |
let ?a = "SUP Q:?Q. emeasure M Q" |
|
335 |
have "{} \<in> ?Q" by auto |
|
38656 | 336 |
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
|
337 |
have "?a \<le> emeasure M (space M)" using sets.sets_into_space |
47694 | 338 |
by (auto intro!: SUP_least emeasure_mono) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
339 |
then have "?a \<noteq> \<infinity>" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
340 |
using finite_emeasure_space |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
341 |
by (auto simp: less_top[symmetric] top_unique simp del: SUP_eq_top_iff Sup_eq_top_iff) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
342 |
from ennreal_SUP_countable_SUP [OF Q_not_empty, of "emeasure M"] |
47694 | 343 |
obtain Q'' where "range Q'' \<subseteq> emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)" |
38656 | 344 |
by auto |
47694 | 345 |
then have "\<forall>i. \<exists>Q'. Q'' i = emeasure M Q' \<and> Q' \<in> ?Q" by auto |
346 |
from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = emeasure M (Q' i)" "\<And>i. Q' i \<in> ?Q" |
|
38656 | 347 |
by auto |
47694 | 348 |
then have a_Lim: "?a = (SUP i::nat. emeasure M (Q' i))" using a by simp |
46731 | 349 |
let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i" |
47694 | 350 |
have Union: "(SUP i. emeasure M (?O i)) = emeasure M (\<Union>i. ?O i)" |
351 |
proof (rule SUP_emeasure_incseq[of ?O]) |
|
352 |
show "range ?O \<subseteq> sets M" using Q' by auto |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44568
diff
changeset
|
353 |
show "incseq ?O" by (fastforce intro!: incseq_SucI) |
38656 | 354 |
qed |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
355 |
have Q'_sets[measurable]: "\<And>i. Q' i \<in> sets M" using Q' by auto |
47694 | 356 |
have O_sets: "\<And>i. ?O i \<in> sets M" using Q' by auto |
38656 | 357 |
then have O_in_G: "\<And>i. ?O i \<in> ?Q" |
358 |
proof (safe del: notI) |
|
47694 | 359 |
fix i have "Q' ` {..i} \<subseteq> sets M" using Q' by auto |
360 |
then have "N (?O i) \<le> (\<Sum>i\<le>i. N (Q' i))" |
|
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
361 |
by (simp add: emeasure_subadditive_finite) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
362 |
also have "\<dots> < \<infinity>" using Q' by (simp add: less_top) |
47694 | 363 |
finally show "N (?O i) \<noteq> \<infinity>" by simp |
38656 | 364 |
qed auto |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44568
diff
changeset
|
365 |
have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce |
47694 | 366 |
have a_eq: "?a = emeasure M (\<Union>i. ?O i)" unfolding Union[symmetric] |
38656 | 367 |
proof (rule antisym) |
47694 | 368 |
show "?a \<le> (SUP i. emeasure M (?O i))" unfolding a_Lim |
369 |
using Q' by (auto intro!: SUP_mono emeasure_mono) |
|
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62083
diff
changeset
|
370 |
show "(SUP i. emeasure M (?O i)) \<le> ?a" |
38656 | 371 |
proof (safe intro!: Sup_mono, unfold bex_simps) |
372 |
fix i |
|
52141
eff000cab70f
weaker precendence of syntax for big intersection and union on sets
haftmann
parents:
51329
diff
changeset
|
373 |
have *: "(\<Union>(Q' ` {..i})) = ?O i" by auto |
47694 | 374 |
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
|
375 |
emeasure M (\<Union>(Q' ` {..i})) \<le> emeasure M x" |
38656 | 376 |
using O_in_G[of i] by (auto intro!: exI[of _ "?O i"]) |
377 |
qed |
|
378 |
qed |
|
46731 | 379 |
let ?O_0 = "(\<Union>i. ?O i)" |
38656 | 380 |
have "?O_0 \<in> sets M" using Q' by auto |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
381 |
have "disjointed Q' i \<in> sets M" for i |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
382 |
using sets.range_disjointed_sets[of Q' M] using Q'_sets by (auto simp: subset_eq) |
38656 | 383 |
note Q_sets = this |
40859 | 384 |
show ?thesis |
385 |
proof (intro bexI exI conjI ballI impI allI) |
|
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
386 |
show "disjoint_family (disjointed Q')" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
387 |
by (rule disjoint_family_disjointed) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
388 |
show "range (disjointed Q') \<subseteq> sets M" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
389 |
using Q'_sets by (intro sets.range_disjointed_sets) auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
390 |
{ fix A assume A: "A \<in> sets M" "A \<inter> (\<Union>i. disjointed Q' i) = {}" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
391 |
then have A1: "A \<inter> (\<Union>i. Q' i) = {}" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
392 |
unfolding UN_disjointed_eq by auto |
47694 | 393 |
show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>" |
40859 | 394 |
proof (rule disjCI, simp) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
395 |
assume *: "emeasure M A = 0 \<or> N A \<noteq> top" |
47694 | 396 |
show "emeasure M A = 0 \<and> N A = 0" |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
397 |
proof (cases "emeasure M A = 0") |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
398 |
case True |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
399 |
with ac A have "N A = 0" |
40859 | 400 |
unfolding absolutely_continuous_def by auto |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
401 |
with True show ?thesis by simp |
40859 | 402 |
next |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
403 |
case False |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
404 |
with * have "N A \<noteq> \<infinity>" by auto |
47694 | 405 |
with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
406 |
using Q' A1 by (auto intro!: plus_emeasure simp: set_eq_iff) |
47694 | 407 |
also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))" |
408 |
proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified]) |
|
40859 | 409 |
show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M" |
61808 | 410 |
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
|
411 |
qed (fastforce intro!: incseq_SucI) |
40859 | 412 |
also have "\<dots> \<le> ?a" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44918
diff
changeset
|
413 |
proof (safe intro!: SUP_least) |
40859 | 414 |
fix i have "?O i \<union> A \<in> ?Q" |
415 |
proof (safe del: notI) |
|
416 |
show "?O i \<union> A \<in> sets M" using O_sets A by auto |
|
47694 | 417 |
from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
418 |
using emeasure_subadditive[of "?O i" N A] A O_sets by auto |
47694 | 419 |
with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
420 |
using \<open>N A \<noteq> \<infinity>\<close> by (auto simp: top_unique) |
40859 | 421 |
qed |
47694 | 422 |
then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper) |
40859 | 423 |
qed |
47694 | 424 |
finally have "emeasure M A = 0" |
425 |
unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure) |
|
61808 | 426 |
with \<open>emeasure M A \<noteq> 0\<close> show ?thesis by auto |
40859 | 427 |
qed |
428 |
qed } |
|
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
429 |
{ fix i |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
430 |
have "N (disjointed Q' i) \<le> N (Q' i)" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
431 |
by (auto intro!: emeasure_mono simp: disjointed_def) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
432 |
then show "N (disjointed Q' i) \<noteq> \<infinity>" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
433 |
using Q'(2)[of i] by (auto simp: top_unique) } |
40859 | 434 |
qed |
435 |
qed |
|
436 |
||
437 |
lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite: |
|
47694 | 438 |
assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M" |
63329 | 439 |
shows "\<exists>f\<in>borel_measurable M. density M f = N" |
40859 | 440 |
proof - |
441 |
from split_space_into_finite_sets_and_rest[OF assms] |
|
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
442 |
obtain Q :: "nat \<Rightarrow> 'a set" |
40859 | 443 |
where Q: "disjoint_family Q" "range Q \<subseteq> sets M" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
444 |
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>" |
47694 | 445 |
and Q_fin: "\<And>i. N (Q i) \<noteq> \<infinity>" by force |
40859 | 446 |
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto |
47694 | 447 |
let ?N = "\<lambda>i. density N (indicator (Q i))" and ?M = "\<lambda>i. density M (indicator (Q i))" |
63329 | 448 |
have "\<forall>i. \<exists>f\<in>borel_measurable (?M i). density (?M i) f = ?N i" |
47694 | 449 |
proof (intro allI finite_measure.Radon_Nikodym_finite_measure) |
38656 | 450 |
fix i |
47694 | 451 |
from Q show "finite_measure (?M i)" |
56996 | 452 |
by (auto intro!: finite_measureI cong: nn_integral_cong |
47694 | 453 |
simp add: emeasure_density subset_eq sets_eq) |
454 |
from Q have "emeasure (?N i) (space N) = emeasure N (Q i)" |
|
56996 | 455 |
by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: nn_integral_cong) |
47694 | 456 |
with Q_fin show "finite_measure (?N i)" |
457 |
by (auto intro!: finite_measureI) |
|
458 |
show "sets (?N i) = sets (?M i)" by (simp add: sets_eq) |
|
50003 | 459 |
have [measurable]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq) |
47694 | 460 |
show "absolutely_continuous (?M i) (?N i)" |
61808 | 461 |
using \<open>absolutely_continuous M N\<close> \<open>Q i \<in> sets M\<close> |
47694 | 462 |
by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq |
463 |
intro!: absolutely_continuous_AE[OF sets_eq]) |
|
38656 | 464 |
qed |
47694 | 465 |
from choice[OF this[unfolded Bex_def]] |
466 |
obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x" |
|
467 |
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
|
468 |
by force |
47694 | 469 |
{ 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
|
470 |
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 | 471 |
by (auto simp add: emeasure_density nn_integral_density subset_eq |
472 |
intro!: nn_integral_cong split: split_indicator) |
|
47694 | 473 |
also have "\<dots> = emeasure N (Q i \<inter> A)" |
474 |
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
|
475 |
finally have "emeasure N (Q i \<inter> A) = (\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)" .. } |
47694 | 476 |
note integral_eq = this |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
477 |
let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator (space M - (\<Union>i. Q i)) x" |
38656 | 478 |
show ?thesis |
479 |
proof (safe intro!: bexI[of _ ?f]) |
|
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
480 |
show "?f \<in> borel_measurable M" using borel Q_sets |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
481 |
by (auto intro!: measurable_If) |
47694 | 482 |
show "density M ?f = N" |
483 |
proof (rule measure_eqI) |
|
484 |
fix A assume "A \<in> sets (density M ?f)" |
|
485 |
then have "A \<in> sets M" by simp |
|
486 |
have Qi: "\<And>i. Q i \<in> sets M" using Q by auto |
|
487 |
have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M" |
|
488 |
"\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x" |
|
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
489 |
using borel Qi \<open>A \<in> sets M\<close> by auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
490 |
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 ((space M - (\<Union>i. Q i)) \<inter> A) x \<partial>M)" |
56996 | 491 |
using borel by (intro nn_integral_cong) (auto simp: indicator_def) |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
492 |
also have "\<dots> = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
493 |
using borel Qi \<open>A \<in> sets M\<close> |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
494 |
by (subst nn_integral_add) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
495 |
(auto simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le) |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
496 |
also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)" |
61808 | 497 |
by (subst integral_eq[OF \<open>A \<in> sets M\<close>], subst nn_integral_suminf) auto |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
498 |
finally have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)" . |
47694 | 499 |
moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)" |
61808 | 500 |
using Q Q_sets \<open>A \<in> sets M\<close> |
47694 | 501 |
by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq) |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
502 |
moreover |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
503 |
have "(space M - (\<Union>x. Q x)) \<inter> A \<inter> (\<Union>x. Q x) = {}" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
504 |
by auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
505 |
then have "\<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A) = N ((space M - (\<Union>i. Q i)) \<inter> A)" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
506 |
using in_Q0[of "(space M - (\<Union>i. Q i)) \<inter> A"] \<open>A \<in> sets M\<close> Q by (auto simp: ennreal_top_mult) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
507 |
moreover have "(space M - (\<Union>i. Q i)) \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
508 |
using Q_sets \<open>A \<in> sets M\<close> by auto |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
509 |
moreover have "((\<Union>i. Q i) \<inter> A) \<union> ((space M - (\<Union>i. Q i)) \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> ((space M - (\<Union>i. Q i)) \<inter> A) = {}" |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
510 |
using \<open>A \<in> sets M\<close> sets.sets_into_space by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
511 |
ultimately have "N A = (\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M)" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
512 |
using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "(space M - (\<Union>i. Q i)) \<inter> A"] by (simp add: sets_eq) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
513 |
with \<open>A \<in> sets M\<close> borel Q show "emeasure (density M ?f) A = N A" |
50003 | 514 |
by (auto simp: subset_eq emeasure_density) |
47694 | 515 |
qed (simp add: sets_eq) |
38656 | 516 |
qed |
517 |
qed |
|
518 |
||
519 |
lemma (in sigma_finite_measure) Radon_Nikodym: |
|
47694 | 520 |
assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M" |
63329 | 521 |
shows "\<exists>f \<in> borel_measurable M. density M f = N" |
38656 | 522 |
proof - |
523 |
from Ex_finite_integrable_function |
|
56996 | 524 |
obtain h where finite: "integral\<^sup>N M h \<noteq> \<infinity>" and |
38656 | 525 |
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
|
526 |
nn: "\<And>x. 0 \<le> h x" and |
38656 | 527 |
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
|
528 |
"\<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
|
529 |
let ?T = "\<lambda>A. (\<integral>\<^sup>+x. h x * indicator A x \<partial>M)" |
47694 | 530 |
let ?MT = "density M h" |
531 |
from borel finite nn interpret T: finite_measure ?MT |
|
56996 | 532 |
by (auto intro!: finite_measureI cong: nn_integral_cong simp: emeasure_density) |
47694 | 533 |
have "absolutely_continuous ?MT N" "sets N = sets ?MT" |
534 |
proof (unfold absolutely_continuous_def, safe) |
|
535 |
fix A assume "A \<in> null_sets ?MT" |
|
536 |
with borel have "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> h x \<le> 0" |
|
537 |
by (auto simp add: null_sets_density_iff) |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
538 |
with pos sets.sets_into_space have "AE x in M. x \<notin> A" |
61810 | 539 |
by (elim eventually_mono) (auto simp: not_le[symmetric]) |
47694 | 540 |
then have "A \<in> null_sets M" |
61808 | 541 |
using \<open>A \<in> sets M\<close> by (simp add: AE_iff_null_sets) |
47694 | 542 |
with ac show "A \<in> null_sets N" |
543 |
by (auto simp: absolutely_continuous_def) |
|
544 |
qed (auto simp add: sets_eq) |
|
545 |
from T.Radon_Nikodym_finite_measure_infinite[OF this] |
|
63329 | 546 |
obtain f where f_borel: "f \<in> borel_measurable M" "density ?MT f = N" by auto |
47694 | 547 |
with nn borel show ?thesis |
548 |
by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq) |
|
38656 | 549 |
qed |
550 |
||
61808 | 551 |
subsection \<open>Uniqueness of densities\<close> |
40859 | 552 |
|
47694 | 553 |
lemma finite_density_unique: |
40859 | 554 |
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
47694 | 555 |
assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" |
56996 | 556 |
and fin: "integral\<^sup>N M f \<noteq> \<infinity>" |
49785 | 557 |
shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)" |
40859 | 558 |
proof (intro iffI ballI) |
47694 | 559 |
fix A assume eq: "AE x in M. f x = g x" |
49785 | 560 |
with borel show "density M f = density M g" |
561 |
by (auto intro: density_cong) |
|
40859 | 562 |
next |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
563 |
let ?P = "\<lambda>f A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M" |
49785 | 564 |
assume "density M f = density M g" |
565 |
with borel have eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" |
|
566 |
by (simp add: emeasure_density[symmetric]) |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
567 |
from this[THEN bspec, OF sets.top] fin |
56996 | 568 |
have g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" by (simp cong: nn_integral_cong) |
40859 | 569 |
{ fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
47694 | 570 |
and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x" |
56996 | 571 |
and g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" |
40859 | 572 |
let ?N = "{x\<in>space M. g x < f x}" |
573 |
have N: "?N \<in> sets M" using borel by simp |
|
56996 | 574 |
have "?P g ?N \<le> integral\<^sup>N M g" using pos |
575 |
by (intro nn_integral_mono_AE) (auto split: split_indicator) |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
576 |
then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by (auto simp: top_unique) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
577 |
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 | 578 |
by (auto intro!: nn_integral_cong simp: indicator_def) |
40859 | 579 |
also have "\<dots> = ?P f ?N - ?P g ?N" |
56996 | 580 |
proof (rule nn_integral_diff) |
40859 | 581 |
show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M" |
582 |
using borel N by auto |
|
47694 | 583 |
show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
584 |
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
|
585 |
qed fact |
40859 | 586 |
also have "\<dots> = 0" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
587 |
unfolding eq[THEN bspec, OF N] using Pg_fin by auto |
47694 | 588 |
finally have "AE x in M. f x \<le> g x" |
56996 | 589 |
using pos borel nn_integral_PInf_AE[OF borel(2) g_fin] |
590 |
by (subst (asm) nn_integral_0_iff_AE) |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
591 |
(auto split: split_indicator simp: not_less ennreal_minus_eq_0) } |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
592 |
from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq |
47694 | 593 |
show "AE x in M. f x = g x" by auto |
40859 | 594 |
qed |
595 |
||
596 |
lemma (in finite_measure) density_unique_finite_measure: |
|
597 |
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M" |
|
47694 | 598 |
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
|
599 |
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 | 600 |
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A") |
47694 | 601 |
shows "AE x in M. f x = f' x" |
40859 | 602 |
proof - |
47694 | 603 |
let ?D = "\<lambda>f. density M f" |
604 |
let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A" |
|
46731 | 605 |
let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x" |
47694 | 606 |
|
607 |
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
|
608 |
using borel by (auto intro!: absolutely_continuousI_density) |
47694 | 609 |
from split_space_into_finite_sets_and_rest[OF this] |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
610 |
obtain Q :: "nat \<Rightarrow> 'a set" |
40859 | 611 |
where Q: "disjoint_family Q" "range Q \<subseteq> sets M" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
612 |
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> ?D f A = 0 \<or> 0 < emeasure M A \<and> ?D f A = \<infinity>" |
47694 | 613 |
and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
614 |
with borel pos have in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> ?N A = 0 \<or> 0 < emeasure M A \<and> ?N A = \<infinity>" |
47694 | 615 |
and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq) |
616 |
||
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
617 |
from Q have Q_sets[measurable]: "\<And>i. Q i \<in> sets M" by auto |
47694 | 618 |
let ?D = "{x\<in>space M. f x \<noteq> f' x}" |
619 |
have "?D \<in> sets M" using borel by auto |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
620 |
have *: "\<And>i x A. \<And>y::ennreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x" |
40859 | 621 |
unfolding indicator_def by auto |
47694 | 622 |
have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos |
40859 | 623 |
by (intro finite_density_unique[THEN iffD1] allI) |
50003 | 624 |
(auto intro!: f measure_eqI simp: emeasure_density * subset_eq) |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
625 |
moreover have "AE x in M. ?f (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x" |
40859 | 626 |
proof (rule AE_I') |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
627 |
{ fix f :: "'a \<Rightarrow> ennreal" 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
|
628 |
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
629 |
let ?A = "\<lambda>i. (space M - (\<Union>i. Q i)) \<inter> {x \<in> space M. f x < (i::nat)}" |
47694 | 630 |
have "(\<Union>i. ?A i) \<in> null_sets M" |
40859 | 631 |
proof (rule null_sets_UN) |
43923 | 632 |
fix i ::nat have "?A i \<in> sets M" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
633 |
using borel by auto |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
634 |
have "?N (?A i) \<le> (\<integral>\<^sup>+x. (i::ennreal) * indicator (?A i) x \<partial>M)" |
61808 | 635 |
unfolding eq[OF \<open>?A i \<in> sets M\<close>] |
56996 | 636 |
by (auto intro!: nn_integral_mono simp: indicator_def) |
47694 | 637 |
also have "\<dots> = i * emeasure M (?A i)" |
61808 | 638 |
using \<open>?A i \<in> sets M\<close> by (auto intro!: nn_integral_cmult_indicator) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
639 |
also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by (auto simp: ennreal_mult_less_top of_nat_less_top) |
47694 | 640 |
finally have "?N (?A i) \<noteq> \<infinity>" by simp |
61808 | 641 |
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 | 642 |
qed |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
643 |
also have "(\<Union>i. ?A i) = (space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
644 |
by (auto simp: ennreal_Ex_less_of_nat less_top[symmetric]) |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
645 |
finally have "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp } |
40859 | 646 |
from this[OF borel(1) refl] this[OF borel(2) f] |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
647 |
have "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
648 |
then show "((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets M" by (rule null_sets.Un) |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
649 |
show "{x \<in> space M. ?f (space M - (\<Union>i. Q i)) x \<noteq> ?f' (space M - (\<Union>i. Q i)) x} \<subseteq> |
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
650 |
((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def) |
40859 | 651 |
qed |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
652 |
moreover have "AE x in M. (?f (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow> |
40859 | 653 |
?f (space M) x = ?f' (space M) x" |
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
654 |
by (auto simp: indicator_def) |
47694 | 655 |
ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x" |
656 |
unfolding AE_all_countable[symmetric] |
|
63330
8d591640c3bd
Probability: introduce Hahn decomposition; use it to clean up Radon_Nikodym
hoelzl
parents:
63329
diff
changeset
|
657 |
by eventually_elim (auto split: if_split_asm simp: indicator_def) |
47694 | 658 |
then show "AE x in M. f x = f' x" by auto |
40859 | 659 |
qed |
660 |
||
661 |
lemma (in sigma_finite_measure) density_unique: |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
662 |
assumes f: "f \<in> borel_measurable M" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
663 |
assumes f': "f' \<in> borel_measurable M" |
47694 | 664 |
assumes density_eq: "density M f = density M f'" |
665 |
shows "AE x in M. f x = f' x" |
|
40859 | 666 |
proof - |
667 |
obtain h where h_borel: "h \<in> borel_measurable M" |
|
56996 | 668 |
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 | 669 |
using Ex_finite_integrable_function by auto |
47694 | 670 |
then have h_nn: "AE x in M. 0 \<le> h x" by auto |
671 |
let ?H = "density M h" |
|
672 |
interpret h: finite_measure ?H |
|
673 |
using fin h_borel pos |
|
56996 | 674 |
by (intro finite_measureI) (simp cong: nn_integral_cong emeasure_density add: fin) |
47694 | 675 |
let ?fM = "density M f" |
676 |
let ?f'M = "density M f'" |
|
40859 | 677 |
{ 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
|
678 |
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
|
679 |
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
|
680 |
then have "(\<integral>\<^sup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M" |
61808 | 681 |
using h_borel \<open>A \<in> sets M\<close> h_nn by (subst nn_integral_0_iff) auto } |
40859 | 682 |
note h_null_sets = this |
683 |
{ 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
|
684 |
have "(\<integral>\<^sup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?fM)" |
61808 | 685 |
using \<open>A \<in> sets M\<close> h_borel h_nn f f' |
56996 | 686 |
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
|
687 |
also have "\<dots> = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?f'M)" |
47694 | 688 |
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
|
689 |
also have "\<dots> = (\<integral>\<^sup>+x. f' x * (h x * indicator A x) \<partial>M)" |
61808 | 690 |
using \<open>A \<in> sets M\<close> h_borel h_nn f f' |
56996 | 691 |
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
|
692 |
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
|
693 |
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
|
694 |
then have "(\<integral>\<^sup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^sup>+x. (f' x * indicator A x) \<partial>?H)" |
61808 | 695 |
using \<open>A \<in> sets M\<close> h_borel h_nn f f' |
56996 | 696 |
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
|
697 |
then have "AE x in ?H. f x = f' x" using h_borel h_nn f f' |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
698 |
by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M]) auto |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
699 |
with AE_space[of M] pos show "AE x in M. f x = f' x" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
700 |
unfolding AE_density[OF h_borel] by auto |
40859 | 701 |
qed |
702 |
||
47694 | 703 |
lemma (in sigma_finite_measure) density_unique_iff: |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
704 |
assumes f: "f \<in> borel_measurable M" and f': "f' \<in> borel_measurable M" |
47694 | 705 |
shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)" |
706 |
using density_unique[OF assms] density_cong[OF f f'] by auto |
|
707 |
||
49785 | 708 |
lemma sigma_finite_density_unique: |
709 |
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
|
710 |
and fin: "sigma_finite_measure (density M f)" |
|
711 |
shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)" |
|
712 |
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
|
713 |
assume "AE x in M. f x = g x" with borel show "density M f = density M g" |
49785 | 714 |
by (auto intro: density_cong) |
715 |
next |
|
716 |
assume eq: "density M f = density M g" |
|
61605 | 717 |
interpret f: sigma_finite_measure "density M f" by fact |
49785 | 718 |
from f.sigma_finite_incseq guess A . note cover = this |
719 |
||
720 |
have "AE x in M. \<forall>i. x \<in> A i \<longrightarrow> f x = g x" |
|
721 |
unfolding AE_all_countable |
|
722 |
proof |
|
723 |
fix i |
|
724 |
have "density (density M f) (indicator (A i)) = density (density M g) (indicator (A i))" |
|
725 |
unfolding eq .. |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
726 |
moreover have "(\<integral>\<^sup>+x. f x * indicator (A i) x \<partial>M) \<noteq> \<infinity>" |
49785 | 727 |
using cover(1) cover(3)[of i] borel by (auto simp: emeasure_density subset_eq) |
728 |
ultimately have "AE x in M. f x * indicator (A i) x = g x * indicator (A i) x" |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
729 |
using borel cover(1) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
730 |
by (intro finite_density_unique[THEN iffD1]) (auto simp: density_density_eq subset_eq) |
49785 | 731 |
then show "AE x in M. x \<in> A i \<longrightarrow> f x = g x" |
732 |
by auto |
|
733 |
qed |
|
734 |
with AE_space show "AE x in M. f x = g x" |
|
735 |
apply eventually_elim |
|
736 |
using cover(2)[symmetric] |
|
737 |
apply auto |
|
738 |
done |
|
739 |
qed |
|
740 |
||
49778
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
741 |
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite': |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
742 |
assumes f: "f \<in> borel_measurable M" |
47694 | 743 |
shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)" |
744 |
(is "sigma_finite_measure ?N \<longleftrightarrow> _") |
|
40859 | 745 |
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
|
746 |
assume "sigma_finite_measure ?N" |
47694 | 747 |
then interpret N: sigma_finite_measure ?N . |
748 |
from N.Ex_finite_integrable_function obtain h where |
|
56996 | 749 |
h: "h \<in> borel_measurable M" "integral\<^sup>N ?N h \<noteq> \<infinity>" and |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
750 |
fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
751 |
by auto |
47694 | 752 |
have "AE x in M. f x * h x \<noteq> \<infinity>" |
40859 | 753 |
proof (rule AE_I') |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
754 |
have "integral\<^sup>N ?N h = (\<integral>\<^sup>+x. f x * h x \<partial>M)" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
755 |
using f h 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
|
756 |
then have "(\<integral>\<^sup>+x. f x * h x \<partial>M) \<noteq> \<infinity>" |
40859 | 757 |
using h(2) by simp |
47694 | 758 |
then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets M" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
759 |
using f h(1) by (auto intro!: nn_integral_PInf[unfolded infinity_ennreal_def] borel_measurable_vimage) |
40859 | 760 |
qed auto |
47694 | 761 |
then show "AE x in M. f x \<noteq> \<infinity>" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
762 |
using fin by (auto elim!: AE_Ball_mp simp: less_top ennreal_mult_less_top) |
40859 | 763 |
next |
47694 | 764 |
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
|
765 |
from sigma_finite guess Q . note Q = this |
63040 | 766 |
define A where "A i = |
767 |
f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ennreal(of_nat (Suc n))}) \<inter> space M" for i |
|
40859 | 768 |
{ fix i j have "A i \<inter> Q j \<in> sets M" |
769 |
unfolding A_def using f Q |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
770 |
apply (rule_tac sets.Int) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
771 |
by (cases i) (auto intro: measurable_sets[OF f(1)]) } |
40859 | 772 |
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
|
773 |
|
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
|
774 |
show "sigma_finite_measure ?N" |
61169 | 775 |
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
|
776 |
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
|
777 |
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
|
778 |
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
|
779 |
using A_in_sets by auto |
40859 | 780 |
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
|
781 |
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
|
782 |
by auto |
40859 | 783 |
also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto |
784 |
also have "(\<Union>i. A i) = space M" |
|
785 |
proof safe |
|
786 |
fix x assume x: "x \<in> space M" |
|
787 |
show "x \<in> (\<Union>i. A i)" |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
788 |
proof (cases "f x" rule: ennreal_cases) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
789 |
case top with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0]) |
40859 | 790 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41832
diff
changeset
|
791 |
case (real r) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
792 |
with ennreal_Ex_less_of_nat[of "f x"] obtain n :: nat where "f x < n" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
793 |
by auto |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
794 |
also have "n < (Suc n :: ennreal)" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
795 |
by simp |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
796 |
finally show ?thesis |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
797 |
using x real by (auto simp: A_def ennreal_of_nat_eq_real_of_nat intro!: exI[of _ "Suc n"]) |
40859 | 798 |
qed |
799 |
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
|
800 |
finally show "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = space ?N" by simp |
40859 | 801 |
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
|
802 |
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
|
803 |
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
|
804 |
have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>" |
40859 | 805 |
proof (cases i) |
806 |
case 0 |
|
47694 | 807 |
have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0" |
61808 | 808 |
using AE by (auto simp: A_def \<open>i = 0\<close>) |
56996 | 809 |
from nn_integral_cong_AE[OF this] show ?thesis by simp |
40859 | 810 |
next |
811 |
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
|
812 |
then have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le> |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
813 |
(\<integral>\<^sup>+x. (Suc n :: ennreal) * indicator (Q j) x \<partial>M)" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
814 |
by (auto intro!: nn_integral_mono simp: indicator_def A_def ennreal_of_nat_eq_real_of_nat) |
47694 | 815 |
also have "\<dots> = Suc n * emeasure M (Q j)" |
56996 | 816 |
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
|
817 |
also have "\<dots> < \<infinity>" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
818 |
using Q by (auto simp: ennreal_mult_less_top less_top of_nat_less_top) |
40859 | 819 |
finally show ?thesis by simp |
820 |
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
|
821 |
then show "emeasure ?N X \<noteq> \<infinity>" |
47694 | 822 |
using A_in_sets Q f by (auto simp: emeasure_density) |
40859 | 823 |
qed |
824 |
qed |
|
825 |
||
49778
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
826 |
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
|
827 |
"f \<in> borel_measurable M \<Longrightarrow> sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
828 |
by (subst sigma_finite_iff_density_finite') |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
829 |
(auto simp: max_def intro!: measurable_If) |
49778
bbbc0f492780
sigma_finite_iff_density_finite does not require a positive density function
hoelzl
parents:
47694
diff
changeset
|
830 |
|
61808 | 831 |
subsection \<open>Radon-Nikodym derivative\<close> |
38656 | 832 |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
833 |
definition RN_deriv :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a \<Rightarrow> ennreal" where |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
834 |
"RN_deriv M N = |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
835 |
(if \<exists>f. f \<in> borel_measurable M \<and> density M f = N |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
836 |
then SOME f. f \<in> borel_measurable M \<and> density M f = N |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
837 |
else (\<lambda>_. 0))" |
38656 | 838 |
|
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
|
839 |
lemma RN_derivI: |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
840 |
assumes "f \<in> borel_measurable M" "density M f = N" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
841 |
shows "density M (RN_deriv M N) = N" |
40859 | 842 |
proof - |
63540 | 843 |
have *: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
844 |
using assms by auto |
63540 | 845 |
then have "density M (SOME f. f \<in> borel_measurable M \<and> density M f = N) = N" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
846 |
by (rule someI2_ex) auto |
63540 | 847 |
with * 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
|
848 |
by (auto simp: RN_deriv_def) |
40859 | 849 |
qed |
850 |
||
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
851 |
lemma borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M" |
38656 | 852 |
proof - |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
853 |
{ assume ex: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
854 |
have 1: "(SOME f. f \<in> borel_measurable M \<and> density M f = N) \<in> borel_measurable M" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
855 |
using ex by (rule someI2_ex) auto } |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
856 |
from this 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
|
857 |
by (auto simp: RN_deriv_def) |
38656 | 858 |
qed |
859 |
||
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
860 |
lemma density_RN_deriv_density: |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
861 |
assumes f: "f \<in> borel_measurable M" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
862 |
shows "density M (RN_deriv M (density M f)) = density M f" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
863 |
by (rule RN_derivI[OF f]) simp |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
864 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
865 |
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
|
866 |
"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
|
867 |
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
|
868 |
|
56996 | 869 |
lemma (in sigma_finite_measure) RN_deriv_nn_integral: |
47694 | 870 |
assumes N: "absolutely_continuous M N" "sets N = sets M" |
40859 | 871 |
and f: "f \<in> borel_measurable M" |
56996 | 872 |
shows "integral\<^sup>N N f = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)" |
40859 | 873 |
proof - |
56996 | 874 |
have "integral\<^sup>N N f = integral\<^sup>N (density M (RN_deriv M N)) f" |
47694 | 875 |
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
|
876 |
also have "\<dots> = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
877 |
using f by (simp add: nn_integral_density) |
47694 | 878 |
finally show ?thesis by simp |
40859 | 879 |
qed |
880 |
||
47694 | 881 |
lemma null_setsD_AE: "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N" |
882 |
using AE_iff_null_sets[of N M] by auto |
|
883 |
||
884 |
lemma (in sigma_finite_measure) RN_deriv_unique: |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
885 |
assumes f: "f \<in> borel_measurable M" |
47694 | 886 |
and eq: "density M f = N" |
887 |
shows "AE x in M. f x = RN_deriv M N x" |
|
49785 | 888 |
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
|
889 |
by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
890 |
density_RN_deriv_density[symmetric]) |
49785 | 891 |
|
892 |
lemma RN_deriv_unique_sigma_finite: |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
893 |
assumes f: "f \<in> borel_measurable M" |
49785 | 894 |
and eq: "density M f = N" and fin: "sigma_finite_measure N" |
895 |
shows "AE x in M. f x = RN_deriv M N x" |
|
896 |
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
|
897 |
by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
898 |
density_RN_deriv_density[symmetric]) |
47694 | 899 |
|
900 |
lemma (in sigma_finite_measure) RN_deriv_distr: |
|
901 |
fixes T :: "'a \<Rightarrow> 'b" |
|
902 |
assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M" |
|
903 |
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
|
904 |
and ac[simp]: "absolutely_continuous (distr M M' T) (distr N M' T)" |
47694 | 905 |
and N: "sets N = sets M" |
906 |
shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x" |
|
41832 | 907 |
proof (rule RN_deriv_unique) |
47694 | 908 |
have [simp]: "sets N = sets M" by fact |
909 |
note sets_eq_imp_space_eq[OF N, simp] |
|
910 |
have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def) |
|
911 |
{ fix A assume "A \<in> sets M" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
912 |
with inv T T' sets.sets_into_space[OF this] |
47694 | 913 |
have "T -` T' -` A \<inter> T -` space M' \<inter> space M = A" |
914 |
by (auto simp: measurable_def) } |
|
915 |
note eq = this[simp] |
|
916 |
{ fix A assume "A \<in> sets M" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50021
diff
changeset
|
917 |
with inv T T' sets.sets_into_space[OF this] |
47694 | 918 |
have "(T' \<circ> T) -` A \<inter> space M = A" |
919 |
by (auto simp: measurable_def) } |
|
920 |
note eq2 = this[simp] |
|
921 |
let ?M' = "distr M M' T" and ?N' = "distr N M' T" |
|
922 |
interpret M': sigma_finite_measure ?M' |
|
41832 | 923 |
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
|
924 |
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
|
925 |
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
|
926 |
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
|
927 |
show *: "(\<lambda>A. T' -` A \<inter> space ?M') ` F \<subseteq> sets ?M'" |
47694 | 928 |
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
|
929 |
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
|
930 |
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
|
931 |
next |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56996
diff
changeset
|
932 |
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
|
933 |
then obtain A where [simp]: "X = T' -` A \<inter> space ?M'" and "A \<in> F" by auto |
61808 | 934 |
have "X \<in> sets M'" using F T' \<open>A\<in>F\<close> by auto |
41832 | 935 |
moreover |
61808 | 936 |
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
|
937 |
ultimately show "emeasure ?M' X \<noteq> \<infinity>" |
61808 | 938 |
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
|
939 |
qed (insert F, auto) |
41832 | 940 |
qed |
47694 | 941 |
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
|
942 |
using T ac by measurable |
47694 | 943 |
then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M" |
41832 | 944 |
by (simp add: comp_def) |
47694 | 945 |
|
946 |
have "N = distr N M (T' \<circ> T)" |
|
947 |
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
|
948 |
(auto simp add: distr_def sets.sigma_sets_eq intro!: measure_of_eq sets.space_closed) |
47694 | 949 |
also have "\<dots> = distr (distr N M' T) M T'" |
950 |
using T T' by (simp add: distr_distr) |
|
951 |
also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'" |
|
952 |
using ac by (simp add: M'.density_RN_deriv) |
|
953 |
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
|
954 |
by (simp add: distr_density_distr[OF T T', OF inv]) |
47694 | 955 |
finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N" |
956 |
by (simp add: comp_def) |
|
41832 | 957 |
qed |
958 |
||
40859 | 959 |
lemma (in sigma_finite_measure) RN_deriv_finite: |
47694 | 960 |
assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M" |
961 |
shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>" |
|
40859 | 962 |
proof - |
47694 | 963 |
interpret N: sigma_finite_measure N by fact |
964 |
from N show ?thesis |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
965 |
using sigma_finite_iff_density_finite[OF borel_measurable_RN_deriv, of N] density_RN_deriv[OF ac] |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
966 |
by simp |
40859 | 967 |
qed |
968 |
||
969 |
lemma (in sigma_finite_measure) |
|
47694 | 970 |
assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M" |
40859 | 971 |
and f: "f \<in> borel_measurable M" |
47694 | 972 |
shows RN_deriv_integrable: "integrable N f \<longleftrightarrow> |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
973 |
integrable M (\<lambda>x. enn2real (RN_deriv M N x) * f x)" (is ?integrable) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
974 |
and RN_deriv_integral: "integral\<^sup>L N f = (\<integral>x. enn2real (RN_deriv M N x) * f x \<partial>M)" (is ?integral) |
40859 | 975 |
proof - |
47694 | 976 |
note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp] |
977 |
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
|
978 |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
979 |
have eq: "density M (RN_deriv M N) = density M (\<lambda>x. enn2real (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
|
980 |
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
|
981 |
from RN_deriv_finite[OF assms(1,2,3)] |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
982 |
show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
983 |
by eventually_elim (auto simp: less_top) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
984 |
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
|
985 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
986 |
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
|
987 |
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
|
988 |
unfolding eq |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
989 |
apply (intro integrable_real_density f AE_I2 enn2real_nonneg) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
990 |
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
|
991 |
done |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
992 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
993 |
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
|
994 |
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
|
995 |
unfolding eq |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
996 |
apply (intro integral_real_density f AE_I2 enn2real_nonneg) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56537
diff
changeset
|
997 |
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
|
998 |
done |
40859 | 999 |
qed |
1000 |
||
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1001 |
lemma (in sigma_finite_measure) real_RN_deriv: |
47694 | 1002 |
assumes "finite_measure N" |
1003 |
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
|
1004 |
obtains D where "D \<in> borel_measurable M" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1005 |
and "AE x in M. RN_deriv M N x = ennreal (D x)" |
47694 | 1006 |
and "AE x in N. 0 < D x" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1007 |
and "\<And>x. 0 \<le> D x" |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1008 |
proof |
47694 | 1009 |
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
|
1010 |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1011 |
note RN = borel_measurable_RN_deriv density_RN_deriv[OF ac] |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1012 |
|
47694 | 1013 |
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
|
1014 |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1015 |
show "(\<lambda>x. enn2real (RN_deriv M N x)) \<in> borel_measurable M" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1016 |
using RN by auto |
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1017 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1018 |
have "N (?RN \<infinity>) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1019 |
using RN(1) 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
|
1020 |
also have "\<dots> = (\<integral>\<^sup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)" |
56996 | 1021 |
by (intro nn_integral_cong) (auto simp: indicator_def) |
47694 | 1022 |
also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)" |
56996 | 1023 |
using RN by (intro nn_integral_cmult_indicator) auto |
47694 | 1024 |
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
|
1025 |
moreover |
47694 | 1026 |
have "emeasure M (?RN \<infinity>) = 0" |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1027 |
proof (rule ccontr) |
47694 | 1028 |
assume "emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>} \<noteq> 0" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1029 |
then have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1030 |
by (auto simp: zero_less_iff_neq_zero) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1031 |
with eq have "N (?RN \<infinity>) = \<infinity>" by (simp add: ennreal_mult_eq_top_iff) |
47694 | 1032 |
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
|
1033 |
qed |
47694 | 1034 |
ultimately have "AE x in M. RN_deriv M N x < \<infinity>" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1035 |
using RN by (intro AE_iff_measurable[THEN iffD2]) (auto simp: less_top[symmetric]) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1036 |
then show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1037 |
by auto |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1038 |
then have eq: "AE x in N. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))" |
47694 | 1039 |
using ac absolutely_continuous_AE by auto |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1040 |
|
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1041 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
52141
diff
changeset
|
1042 |
have "N (?RN 0) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)" |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1043 |
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
|
1044 |
also have "\<dots> = (\<integral>\<^sup>+ x. 0 \<partial>M)" |
56996 | 1045 |
by (intro nn_integral_cong) (auto simp: indicator_def) |
47694 | 1046 |
finally have "AE x in N. RN_deriv M N x \<noteq> 0" |
1047 |
using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq) |
|
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1048 |
with eq show "AE x in N. 0 < enn2real (RN_deriv M N x)" |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1049 |
by (auto simp: enn2real_positive_iff less_top[symmetric] zero_less_iff_neq_zero) |
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1050 |
qed (rule enn2real_nonneg) |
43340
60e181c4eae4
lemma: independence is equal to mutual information = 0
hoelzl
parents:
42866
diff
changeset
|
1051 |
|
38656 | 1052 |
lemma (in sigma_finite_measure) RN_deriv_singleton: |
47694 | 1053 |
assumes ac: "absolutely_continuous M N" "sets N = sets M" |
1054 |
and x: "{x} \<in> sets M" |
|
1055 |
shows "N {x} = RN_deriv M N x * emeasure M {x}" |
|
38656 | 1056 |
proof - |
61808 | 1057 |
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
|
1058 |
have "density M (RN_deriv M N) {x} = (\<integral>\<^sup>+w. RN_deriv M N x * indicator {x} w \<partial>M)" |
56996 | 1059 |
by (auto simp: indicator_def emeasure_density intro!: nn_integral_cong) |
62975
1d066f6ab25d
Probability: move emeasure and nn_integral from ereal to ennreal
hoelzl
parents:
62623
diff
changeset
|
1060 |
with x density_RN_deriv[OF ac] show ?thesis |
62083 | 1061 |
by (auto simp: max_def) |
38656 | 1062 |
qed |
1063 |
||
1064 |
end |