author | hoelzl |
Fri, 14 Jan 2011 14:21:48 +0100 | |
changeset 41544 | c3b977fee8a3 |
parent 41097 | a1abfa4e2b44 |
child 41661 | baf1964bc468 |
permissions | -rw-r--r-- |
38656 | 1 |
theory Radon_Nikodym |
2 |
imports Lebesgue_Integration |
|
3 |
begin |
|
4 |
||
40859 | 5 |
lemma less_\<omega>_Ex_of_nat: "x < \<omega> \<longleftrightarrow> (\<exists>n. x < of_nat n)" |
6 |
proof safe |
|
7 |
assume "x < \<omega>" |
|
8 |
then obtain r where "0 \<le> r" "x = Real r" by (cases x) auto |
|
9 |
moreover obtain n where "r < of_nat n" using ex_less_of_nat by auto |
|
10 |
ultimately show "\<exists>n. x < of_nat n" by (auto simp: real_eq_of_nat) |
|
11 |
qed auto |
|
12 |
||
38656 | 13 |
lemma (in sigma_finite_measure) Ex_finite_integrable_function: |
14 |
shows "\<exists>h\<in>borel_measurable M. positive_integral h \<noteq> \<omega> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<omega>)" |
|
15 |
proof - |
|
16 |
obtain A :: "nat \<Rightarrow> 'a set" where |
|
17 |
range: "range A \<subseteq> sets M" and |
|
18 |
space: "(\<Union>i. A i) = space M" and |
|
19 |
measure: "\<And>i. \<mu> (A i) \<noteq> \<omega>" and |
|
20 |
disjoint: "disjoint_family A" |
|
21 |
using disjoint_sigma_finite by auto |
|
22 |
let "?B i" = "2^Suc i * \<mu> (A i)" |
|
23 |
have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)" |
|
24 |
proof |
|
25 |
fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)" |
|
26 |
proof cases |
|
27 |
assume "\<mu> (A i) = 0" |
|
28 |
then show ?thesis by (auto intro!: exI[of _ 1]) |
|
29 |
next |
|
30 |
assume not_0: "\<mu> (A i) \<noteq> 0" |
|
31 |
then have "?B i \<noteq> \<omega>" using measure[of i] by auto |
|
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it is known as the extended reals, not the infinite reals
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40871
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32 |
then have "inverse (?B i) \<noteq> 0" unfolding pextreal_inverse_eq_0 by simp |
38656 | 33 |
then show ?thesis using measure[of i] not_0 |
34 |
by (auto intro!: exI[of _ "inverse (?B i) / 2"] |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
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35 |
simp: pextreal_noteq_omega_Ex zero_le_mult_iff zero_less_mult_iff mult_le_0_iff power_le_zero_eq) |
38656 | 36 |
qed |
37 |
qed |
|
38 |
from choice[OF this] obtain n where n: "\<And>i. 0 < n i" |
|
39 |
"\<And>i. n i < inverse (2^Suc i * \<mu> (A i))" by auto |
|
40 |
let "?h x" = "\<Sum>\<^isub>\<infinity> i. n i * indicator (A i) x" |
|
41 |
show ?thesis |
|
42 |
proof (safe intro!: bexI[of _ ?h] del: notI) |
|
39092 | 43 |
have "\<And>i. A i \<in> sets M" |
44 |
using range by fastsimp+ |
|
45 |
then have "positive_integral ?h = (\<Sum>\<^isub>\<infinity> i. n i * \<mu> (A i))" |
|
46 |
by (simp add: positive_integral_psuminf positive_integral_cmult_indicator) |
|
38656 | 47 |
also have "\<dots> \<le> (\<Sum>\<^isub>\<infinity> i. Real ((1 / 2)^Suc i))" |
48 |
proof (rule psuminf_le) |
|
49 |
fix N show "n N * \<mu> (A N) \<le> Real ((1 / 2) ^ Suc N)" |
|
50 |
using measure[of N] n[of N] |
|
39092 | 51 |
by (cases "n N") |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
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parents:
40871
diff
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|
52 |
(auto simp: pextreal_noteq_omega_Ex field_simps zero_le_mult_iff |
39092 | 53 |
mult_le_0_iff mult_less_0_iff power_less_zero_eq |
54 |
power_le_zero_eq inverse_eq_divide less_divide_eq |
|
55 |
power_divide split: split_if_asm) |
|
38656 | 56 |
qed |
57 |
also have "\<dots> = Real 1" |
|
58 |
by (rule suminf_imp_psuminf, rule power_half_series, auto) |
|
59 |
finally show "positive_integral ?h \<noteq> \<omega>" by auto |
|
60 |
next |
|
61 |
fix x assume "x \<in> space M" |
|
62 |
then obtain i where "x \<in> A i" using space[symmetric] by auto |
|
63 |
from psuminf_cmult_indicator[OF disjoint, OF this] |
|
64 |
have "?h x = n i" by simp |
|
65 |
then show "0 < ?h x" and "?h x < \<omega>" using n[of i] by auto |
|
66 |
next |
|
67 |
show "?h \<in> borel_measurable M" using range |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
68 |
by (auto intro!: borel_measurable_psuminf borel_measurable_pextreal_times) |
38656 | 69 |
qed |
70 |
qed |
|
71 |
||
40871 | 72 |
subsection "Absolutely continuous" |
73 |
||
38656 | 74 |
definition (in measure_space) |
41023
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it is known as the extended reals, not the infinite reals
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|
75 |
"absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: pextreal))" |
38656 | 76 |
|
40859 | 77 |
lemma (in sigma_finite_measure) absolutely_continuous_AE: |
78 |
assumes "measure_space M \<nu>" "absolutely_continuous \<nu>" "AE x. P x" |
|
79 |
shows "measure_space.almost_everywhere M \<nu> P" |
|
80 |
proof - |
|
81 |
interpret \<nu>: measure_space M \<nu> by fact |
|
82 |
from `AE x. P x` obtain N where N: "N \<in> null_sets" and "{x\<in>space M. \<not> P x} \<subseteq> N" |
|
83 |
unfolding almost_everywhere_def by auto |
|
84 |
show "\<nu>.almost_everywhere P" |
|
85 |
proof (rule \<nu>.AE_I') |
|
86 |
show "{x\<in>space M. \<not> P x} \<subseteq> N" by fact |
|
87 |
from `absolutely_continuous \<nu>` show "N \<in> \<nu>.null_sets" |
|
88 |
using N unfolding absolutely_continuous_def by auto |
|
89 |
qed |
|
90 |
qed |
|
91 |
||
39097 | 92 |
lemma (in finite_measure_space) absolutely_continuousI: |
93 |
assumes "finite_measure_space M \<nu>" |
|
94 |
assumes v: "\<And>x. \<lbrakk> x \<in> space M ; \<mu> {x} = 0 \<rbrakk> \<Longrightarrow> \<nu> {x} = 0" |
|
95 |
shows "absolutely_continuous \<nu>" |
|
96 |
proof (unfold absolutely_continuous_def sets_eq_Pow, safe) |
|
97 |
fix N assume "\<mu> N = 0" "N \<subseteq> space M" |
|
98 |
interpret v: finite_measure_space M \<nu> by fact |
|
99 |
have "\<nu> N = \<nu> (\<Union>x\<in>N. {x})" by simp |
|
100 |
also have "\<dots> = (\<Sum>x\<in>N. \<nu> {x})" |
|
101 |
proof (rule v.measure_finitely_additive''[symmetric]) |
|
102 |
show "finite N" using `N \<subseteq> space M` finite_space by (auto intro: finite_subset) |
|
103 |
show "disjoint_family_on (\<lambda>i. {i}) N" unfolding disjoint_family_on_def by auto |
|
104 |
fix x assume "x \<in> N" thus "{x} \<in> sets M" using `N \<subseteq> space M` sets_eq_Pow by auto |
|
105 |
qed |
|
106 |
also have "\<dots> = 0" |
|
107 |
proof (safe intro!: setsum_0') |
|
108 |
fix x assume "x \<in> N" |
|
109 |
hence "\<mu> {x} \<le> \<mu> N" using sets_eq_Pow `N \<subseteq> space M` by (auto intro!: measure_mono) |
|
110 |
hence "\<mu> {x} = 0" using `\<mu> N = 0` by simp |
|
111 |
thus "\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto |
|
112 |
qed |
|
113 |
finally show "\<nu> N = 0" . |
|
114 |
qed |
|
115 |
||
40871 | 116 |
lemma (in measure_space) density_is_absolutely_continuous: |
41544 | 117 |
assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)" |
40871 | 118 |
shows "absolutely_continuous \<nu>" |
119 |
using assms unfolding absolutely_continuous_def |
|
120 |
by (simp add: positive_integral_null_set) |
|
121 |
||
122 |
subsection "Existence of the Radon-Nikodym derivative" |
|
123 |
||
38656 | 124 |
lemma (in finite_measure) Radon_Nikodym_aux_epsilon: |
125 |
fixes e :: real assumes "0 < e" |
|
126 |
assumes "finite_measure M s" |
|
127 |
shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (s (space M)) \<le> |
|
128 |
real (\<mu> A) - real (s A) \<and> |
|
129 |
(\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < real (\<mu> B) - real (s B))" |
|
130 |
proof - |
|
131 |
let "?d A" = "real (\<mu> A) - real (s A)" |
|
132 |
interpret M': finite_measure M s by fact |
|
133 |
let "?A A" = "if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B) |
|
134 |
then {} |
|
135 |
else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)" |
|
136 |
def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}" |
|
137 |
have A_simps[simp]: |
|
138 |
"A 0 = {}" |
|
139 |
"\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all |
|
140 |
{ fix A assume "A \<in> sets M" |
|
141 |
have "?A A \<in> sets M" |
|
142 |
by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) } |
|
143 |
note A'_in_sets = this |
|
144 |
{ fix n have "A n \<in> sets M" |
|
145 |
proof (induct n) |
|
146 |
case (Suc n) thus "A (Suc n) \<in> sets M" |
|
147 |
using A'_in_sets[of "A n"] by (auto split: split_if_asm) |
|
148 |
qed (simp add: A_def) } |
|
149 |
note A_in_sets = this |
|
150 |
hence "range A \<subseteq> sets M" by auto |
|
151 |
{ fix n B |
|
152 |
assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> -e" |
|
153 |
hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B)" by (auto simp: not_less) |
|
154 |
have "?d (A (Suc n)) \<le> ?d (A n) - e" unfolding A_simps if_not_P[OF False] |
|
155 |
proof (rule someI2_ex[OF Ex]) |
|
156 |
fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e" |
|
157 |
hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto |
|
158 |
hence "?d (A n \<union> B) = ?d (A n) + ?d B" |
|
159 |
using `A n \<in> sets M` real_finite_measure_Union M'.real_finite_measure_Union by simp |
|
160 |
also have "\<dots> \<le> ?d (A n) - e" using dB by simp |
|
161 |
finally show "?d (A n \<union> B) \<le> ?d (A n) - e" . |
|
162 |
qed } |
|
163 |
note dA_epsilon = this |
|
164 |
{ fix n have "?d (A (Suc n)) \<le> ?d (A n)" |
|
165 |
proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e") |
|
166 |
case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp |
|
167 |
next |
|
168 |
case False |
|
169 |
hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le) |
|
170 |
thus ?thesis by simp |
|
171 |
qed } |
|
172 |
note dA_mono = this |
|
173 |
show ?thesis |
|
174 |
proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B") |
|
175 |
case True then obtain n where B: "\<And>B. \<lbrakk> B \<in> sets M; B \<subseteq> space M - A n\<rbrakk> \<Longrightarrow> -e < ?d B" by blast |
|
176 |
show ?thesis |
|
177 |
proof (safe intro!: bexI[of _ "space M - A n"]) |
|
178 |
fix B assume "B \<in> sets M" "B \<subseteq> space M - A n" |
|
179 |
from B[OF this] show "-e < ?d B" . |
|
180 |
next |
|
181 |
show "space M - A n \<in> sets M" by (rule compl_sets) fact |
|
182 |
next |
|
183 |
show "?d (space M) \<le> ?d (space M - A n)" |
|
184 |
proof (induct n) |
|
185 |
fix n assume "?d (space M) \<le> ?d (space M - A n)" |
|
186 |
also have "\<dots> \<le> ?d (space M - A (Suc n))" |
|
187 |
using A_in_sets sets_into_space dA_mono[of n] |
|
188 |
real_finite_measure_Diff[of "space M"] |
|
189 |
real_finite_measure_Diff[of "space M"] |
|
190 |
M'.real_finite_measure_Diff[of "space M"] |
|
191 |
M'.real_finite_measure_Diff[of "space M"] |
|
192 |
by (simp del: A_simps) |
|
193 |
finally show "?d (space M) \<le> ?d (space M - A (Suc n))" . |
|
194 |
qed simp |
|
195 |
qed |
|
196 |
next |
|
197 |
case False hence B: "\<And>n. \<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e" |
|
198 |
by (auto simp add: not_less) |
|
199 |
{ fix n have "?d (A n) \<le> - real n * e" |
|
200 |
proof (induct n) |
|
201 |
case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps) |
|
202 |
qed simp } note dA_less = this |
|
203 |
have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq |
|
204 |
proof (rule incseq_SucI) |
|
205 |
fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto |
|
206 |
qed |
|
207 |
from real_finite_continuity_from_below[of A] `range A \<subseteq> sets M` |
|
208 |
M'.real_finite_continuity_from_below[of A] |
|
209 |
have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)" |
|
210 |
by (auto intro!: LIMSEQ_diff) |
|
211 |
obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto |
|
212 |
moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less] |
|
213 |
have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps) |
|
214 |
ultimately show ?thesis by auto |
|
215 |
qed |
|
216 |
qed |
|
217 |
||
218 |
lemma (in finite_measure) Radon_Nikodym_aux: |
|
219 |
assumes "finite_measure M s" |
|
220 |
shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (s (space M)) \<le> |
|
221 |
real (\<mu> A) - real (s A) \<and> |
|
222 |
(\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> real (\<mu> B) - real (s B))" |
|
223 |
proof - |
|
224 |
let "?d A" = "real (\<mu> A) - real (s A)" |
|
225 |
let "?P A B n" = "A \<in> sets M \<and> A \<subseteq> B \<and> ?d B \<le> ?d A \<and> (\<forall>C\<in>sets M. C \<subseteq> A \<longrightarrow> - 1 / real (Suc n) < ?d C)" |
|
226 |
interpret M': finite_measure M s by fact |
|
39092 | 227 |
let "?r S" = "restricted_space S" |
38656 | 228 |
{ fix S n |
229 |
assume S: "S \<in> sets M" |
|
230 |
hence **: "\<And>X. X \<in> op \<inter> S ` sets M \<longleftrightarrow> X \<in> sets M \<and> X \<subseteq> S" by auto |
|
231 |
from M'.restricted_finite_measure[of S] restricted_finite_measure[of S] S |
|
232 |
have "finite_measure (?r S) \<mu>" "0 < 1 / real (Suc n)" |
|
233 |
"finite_measure (?r S) s" by auto |
|
234 |
from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. |
|
235 |
hence "?P X S n" |
|
236 |
proof (simp add: **, safe) |
|
237 |
fix C assume C: "C \<in> sets M" "C \<subseteq> X" "X \<subseteq> S" and |
|
238 |
*: "\<forall>B\<in>sets M. S \<inter> B \<subseteq> X \<longrightarrow> - (1 / real (Suc n)) < ?d (S \<inter> B)" |
|
239 |
hence "C \<subseteq> S" "C \<subseteq> X" "S \<inter> C = C" by auto |
|
240 |
with *[THEN bspec, OF `C \<in> sets M`] |
|
241 |
show "- (1 / real (Suc n)) < ?d C" by auto |
|
242 |
qed |
|
243 |
hence "\<exists>A. ?P A S n" by auto } |
|
244 |
note Ex_P = this |
|
245 |
def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)" |
|
246 |
have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def) |
|
247 |
have A_0[simp]: "A 0 = space M" unfolding A_def by simp |
|
248 |
{ fix i have "A i \<in> sets M" unfolding A_def |
|
249 |
proof (induct i) |
|
250 |
case (Suc i) |
|
251 |
from Ex_P[OF this, of i] show ?case unfolding nat_rec_Suc |
|
252 |
by (rule someI2_ex) simp |
|
253 |
qed simp } |
|
254 |
note A_in_sets = this |
|
255 |
{ fix n have "?P (A (Suc n)) (A n) n" |
|
256 |
using Ex_P[OF A_in_sets] unfolding A_Suc |
|
257 |
by (rule someI2_ex) simp } |
|
258 |
note P_A = this |
|
259 |
have "range A \<subseteq> sets M" using A_in_sets by auto |
|
260 |
have A_mono: "\<And>i. A (Suc i) \<subseteq> A i" using P_A by simp |
|
261 |
have mono_dA: "mono (\<lambda>i. ?d (A i))" using P_A by (simp add: mono_iff_le_Suc) |
|
262 |
have epsilon: "\<And>C i. \<lbrakk> C \<in> sets M; C \<subseteq> A (Suc i) \<rbrakk> \<Longrightarrow> - 1 / real (Suc i) < ?d C" |
|
263 |
using P_A by auto |
|
264 |
show ?thesis |
|
265 |
proof (safe intro!: bexI[of _ "\<Inter>i. A i"]) |
|
266 |
show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto |
|
267 |
from `range A \<subseteq> sets M` A_mono |
|
268 |
real_finite_continuity_from_above[of A] |
|
269 |
M'.real_finite_continuity_from_above[of A] |
|
270 |
have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (auto intro!: LIMSEQ_diff) |
|
271 |
thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _] |
|
272 |
by (rule_tac LIMSEQ_le_const) (auto intro!: exI) |
|
273 |
next |
|
274 |
fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)" |
|
275 |
show "0 \<le> ?d B" |
|
276 |
proof (rule ccontr) |
|
277 |
assume "\<not> 0 \<le> ?d B" |
|
278 |
hence "0 < - ?d B" by auto |
|
279 |
from ex_inverse_of_nat_Suc_less[OF this] |
|
280 |
obtain n where *: "?d B < - 1 / real (Suc n)" |
|
281 |
by (auto simp: real_eq_of_nat inverse_eq_divide field_simps) |
|
282 |
have "B \<subseteq> A (Suc n)" using B by (auto simp del: nat_rec_Suc) |
|
283 |
from epsilon[OF B(1) this] * |
|
284 |
show False by auto |
|
285 |
qed |
|
286 |
qed |
|
287 |
qed |
|
288 |
||
289 |
lemma (in finite_measure) Radon_Nikodym_finite_measure: |
|
290 |
assumes "finite_measure M \<nu>" |
|
291 |
assumes "absolutely_continuous \<nu>" |
|
41544 | 292 |
shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)" |
38656 | 293 |
proof - |
294 |
interpret M': finite_measure M \<nu> using assms(1) . |
|
41544 | 295 |
def G \<equiv> "{g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x) \<le> \<nu> A}" |
38656 | 296 |
have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto |
297 |
hence "G \<noteq> {}" by auto |
|
298 |
{ fix f g assume f: "f \<in> G" and g: "g \<in> G" |
|
299 |
have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def |
|
300 |
proof safe |
|
301 |
show "?max \<in> borel_measurable M" using f g unfolding G_def by auto |
|
302 |
let ?A = "{x \<in> space M. f x \<le> g x}" |
|
303 |
have "?A \<in> sets M" using f g unfolding G_def by auto |
|
304 |
fix A assume "A \<in> sets M" |
|
305 |
hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto |
|
306 |
have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A" |
|
307 |
using sets_into_space[OF `A \<in> sets M`] by auto |
|
308 |
have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x = |
|
309 |
g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x" |
|
310 |
by (auto simp: indicator_def max_def) |
|
41544 | 311 |
hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x) = |
312 |
(\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x) + |
|
313 |
(\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x)" |
|
38656 | 314 |
using f g sets unfolding G_def |
315 |
by (auto cong: positive_integral_cong intro!: positive_integral_add borel_measurable_indicator) |
|
316 |
also have "\<dots> \<le> \<nu> (?A \<inter> A) + \<nu> ((space M - ?A) \<inter> A)" |
|
317 |
using f g sets unfolding G_def by (auto intro!: add_mono) |
|
318 |
also have "\<dots> = \<nu> A" |
|
319 |
using M'.measure_additive[OF sets] union by auto |
|
41544 | 320 |
finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x) \<le> \<nu> A" . |
38656 | 321 |
qed } |
322 |
note max_in_G = this |
|
323 |
{ fix f g assume "f \<up> g" and f: "\<And>i. f i \<in> G" |
|
324 |
have "g \<in> G" unfolding G_def |
|
325 |
proof safe |
|
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset
|
326 |
from `f \<up> g` have [simp]: "g = (\<lambda>x. SUP i. f i x)" |
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset
|
327 |
unfolding isoton_def fun_eq_iff SUPR_apply by simp |
38656 | 328 |
have f_borel: "\<And>i. f i \<in> borel_measurable M" using f unfolding G_def by simp |
41097
a1abfa4e2b44
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents:
41095
diff
changeset
|
329 |
thus "g \<in> borel_measurable M" by auto |
38656 | 330 |
fix A assume "A \<in> sets M" |
331 |
hence "\<And>i. (\<lambda>x. f i x * indicator A x) \<in> borel_measurable M" |
|
332 |
using f_borel by (auto intro!: borel_measurable_indicator) |
|
333 |
from positive_integral_isoton[OF isoton_indicator[OF `f \<up> g`] this] |
|
41544 | 334 |
have SUP: "(\<integral>\<^isup>+x. g x * indicator A x) = |
335 |
(SUP i. (\<integral>\<^isup>+x. f i x * indicator A x))" |
|
38656 | 336 |
unfolding isoton_def by simp |
41544 | 337 |
show "(\<integral>\<^isup>+x. g x * indicator A x) \<le> \<nu> A" unfolding SUP |
38656 | 338 |
using f `A \<in> sets M` unfolding G_def by (auto intro!: SUP_leI) |
339 |
qed } |
|
340 |
note SUP_in_G = this |
|
341 |
let ?y = "SUP g : G. positive_integral g" |
|
342 |
have "?y \<le> \<nu> (space M)" unfolding G_def |
|
343 |
proof (safe intro!: SUP_leI) |
|
41544 | 344 |
fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x) \<le> \<nu> A" |
38656 | 345 |
from this[THEN bspec, OF top] show "positive_integral g \<le> \<nu> (space M)" |
346 |
by (simp cong: positive_integral_cong) |
|
347 |
qed |
|
348 |
hence "?y \<noteq> \<omega>" using M'.finite_measure_of_space by auto |
|
349 |
from SUPR_countable_SUPR[OF this `G \<noteq> {}`] guess ys .. note ys = this |
|
350 |
hence "\<forall>n. \<exists>g. g\<in>G \<and> positive_integral g = ys n" |
|
351 |
proof safe |
|
352 |
fix n assume "range ys \<subseteq> positive_integral ` G" |
|
353 |
hence "ys n \<in> positive_integral ` G" by auto |
|
354 |
thus "\<exists>g. g\<in>G \<and> positive_integral g = ys n" by auto |
|
355 |
qed |
|
356 |
from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. positive_integral (gs n) = ys n" by auto |
|
357 |
hence y_eq: "?y = (SUP i. positive_integral (gs i))" using ys by auto |
|
358 |
let "?g i x" = "Max ((\<lambda>n. gs n x) ` {..i})" |
|
359 |
def f \<equiv> "SUP i. ?g i" |
|
360 |
have gs_not_empty: "\<And>i. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto |
|
361 |
{ fix i have "?g i \<in> G" |
|
362 |
proof (induct i) |
|
363 |
case 0 thus ?case by simp fact |
|
364 |
next |
|
365 |
case (Suc i) |
|
366 |
with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case |
|
367 |
by (auto simp add: atMost_Suc intro!: max_in_G) |
|
368 |
qed } |
|
369 |
note g_in_G = this |
|
370 |
have "\<And>x. \<forall>i. ?g i x \<le> ?g (Suc i) x" |
|
371 |
using gs_not_empty by (simp add: atMost_Suc) |
|
372 |
hence isoton_g: "?g \<up> f" by (simp add: isoton_def le_fun_def f_def) |
|
373 |
from SUP_in_G[OF this g_in_G] have "f \<in> G" . |
|
374 |
hence [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto |
|
375 |
have "(\<lambda>i. positive_integral (?g i)) \<up> positive_integral f" |
|
376 |
using isoton_g g_in_G by (auto intro!: positive_integral_isoton simp: G_def f_def) |
|
377 |
hence "positive_integral f = (SUP i. positive_integral (?g i))" |
|
378 |
unfolding isoton_def by simp |
|
379 |
also have "\<dots> = ?y" |
|
380 |
proof (rule antisym) |
|
381 |
show "(SUP i. positive_integral (?g i)) \<le> ?y" |
|
382 |
using g_in_G by (auto intro!: exI Sup_mono simp: SUPR_def) |
|
383 |
show "?y \<le> (SUP i. positive_integral (?g i))" unfolding y_eq |
|
384 |
by (auto intro!: SUP_mono positive_integral_mono Max_ge) |
|
385 |
qed |
|
386 |
finally have int_f_eq_y: "positive_integral f = ?y" . |
|
41544 | 387 |
let "?t A" = "\<nu> A - (\<integral>\<^isup>+x. f x * indicator A x)" |
38656 | 388 |
have "finite_measure M ?t" |
389 |
proof |
|
390 |
show "?t {} = 0" by simp |
|
391 |
show "?t (space M) \<noteq> \<omega>" using M'.finite_measure by simp |
|
392 |
show "countably_additive M ?t" unfolding countably_additive_def |
|
393 |
proof safe |
|
394 |
fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets M" "disjoint_family A" |
|
41544 | 395 |
have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x)) |
396 |
= (\<integral>\<^isup>+x. (\<Sum>\<^isub>\<infinity>n. f x * indicator (A n) x))" |
|
38656 | 397 |
using `range A \<subseteq> sets M` |
398 |
by (rule_tac positive_integral_psuminf[symmetric]) (auto intro!: borel_measurable_indicator) |
|
41544 | 399 |
also have "\<dots> = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x)" |
38656 | 400 |
apply (rule positive_integral_cong) |
401 |
apply (subst psuminf_cmult_right) |
|
402 |
unfolding psuminf_indicator[OF `disjoint_family A`] .. |
|
41544 | 403 |
finally have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x)) |
404 |
= (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x)" . |
|
38656 | 405 |
moreover have "(\<Sum>\<^isub>\<infinity>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)" |
406 |
using M'.measure_countably_additive A by (simp add: comp_def) |
|
41544 | 407 |
moreover have "\<And>i. (\<integral>\<^isup>+x. f x * indicator (A i) x) \<le> \<nu> (A i)" |
38656 | 408 |
using A `f \<in> G` unfolding G_def by auto |
409 |
moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<omega>" using M'.finite_measure A by (simp add: countable_UN) |
|
410 |
moreover { |
|
41544 | 411 |
have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x) \<le> \<nu> (\<Union>i. A i)" |
38656 | 412 |
using A `f \<in> G` unfolding G_def by (auto simp: countable_UN) |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
413 |
also have "\<nu> (\<Union>i. A i) < \<omega>" using v_fin by (simp add: pextreal_less_\<omega>) |
41544 | 414 |
finally have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x) \<noteq> \<omega>" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
415 |
by (simp add: pextreal_less_\<omega>) } |
38656 | 416 |
ultimately |
417 |
show "(\<Sum>\<^isub>\<infinity> n. ?t (A n)) = ?t (\<Union>i. A i)" |
|
418 |
apply (subst psuminf_minus) by simp_all |
|
419 |
qed |
|
420 |
qed |
|
421 |
then interpret M: finite_measure M ?t . |
|
422 |
have ac: "absolutely_continuous ?t" using `absolutely_continuous \<nu>` unfolding absolutely_continuous_def by auto |
|
423 |
have upper_bound: "\<forall>A\<in>sets M. ?t A \<le> 0" |
|
424 |
proof (rule ccontr) |
|
425 |
assume "\<not> ?thesis" |
|
426 |
then obtain A where A: "A \<in> sets M" and pos: "0 < ?t A" |
|
427 |
by (auto simp: not_le) |
|
428 |
note pos |
|
429 |
also have "?t A \<le> ?t (space M)" |
|
430 |
using M.measure_mono[of A "space M"] A sets_into_space by simp |
|
431 |
finally have pos_t: "0 < ?t (space M)" by simp |
|
432 |
moreover |
|
433 |
hence pos_M: "0 < \<mu> (space M)" |
|
434 |
using ac top unfolding absolutely_continuous_def by auto |
|
435 |
moreover |
|
41544 | 436 |
have "(\<integral>\<^isup>+x. f x * indicator (space M) x) \<le> \<nu> (space M)" |
38656 | 437 |
using `f \<in> G` unfolding G_def by auto |
41544 | 438 |
hence "(\<integral>\<^isup>+x. f x * indicator (space M) x) \<noteq> \<omega>" |
38656 | 439 |
using M'.finite_measure_of_space by auto |
440 |
moreover |
|
441 |
def b \<equiv> "?t (space M) / \<mu> (space M) / 2" |
|
442 |
ultimately have b: "b \<noteq> 0" "b \<noteq> \<omega>" |
|
443 |
using M'.finite_measure_of_space |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
444 |
by (auto simp: pextreal_inverse_eq_0 finite_measure_of_space) |
38656 | 445 |
have "finite_measure M (\<lambda>A. b * \<mu> A)" (is "finite_measure M ?b") |
446 |
proof |
|
447 |
show "?b {} = 0" by simp |
|
448 |
show "?b (space M) \<noteq> \<omega>" using finite_measure_of_space b by auto |
|
449 |
show "countably_additive M ?b" |
|
450 |
unfolding countably_additive_def psuminf_cmult_right |
|
451 |
using measure_countably_additive by auto |
|
452 |
qed |
|
453 |
from M.Radon_Nikodym_aux[OF this] |
|
454 |
obtain A0 where "A0 \<in> sets M" and |
|
455 |
space_less_A0: "real (?t (space M)) - real (b * \<mu> (space M)) \<le> real (?t A0) - real (b * \<mu> A0)" and |
|
456 |
*: "\<And>B. \<lbrakk> B \<in> sets M ; B \<subseteq> A0 \<rbrakk> \<Longrightarrow> 0 \<le> real (?t B) - real (b * \<mu> B)" by auto |
|
457 |
{ fix B assume "B \<in> sets M" "B \<subseteq> A0" |
|
458 |
with *[OF this] have "b * \<mu> B \<le> ?t B" |
|
459 |
using M'.finite_measure b finite_measure |
|
460 |
by (cases "b * \<mu> B", cases "?t B") (auto simp: field_simps) } |
|
461 |
note bM_le_t = this |
|
462 |
let "?f0 x" = "f x + b * indicator A0 x" |
|
463 |
{ fix A assume A: "A \<in> sets M" |
|
464 |
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto |
|
41544 | 465 |
have "(\<integral>\<^isup>+x. ?f0 x * indicator A x) = |
466 |
(\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x)" |
|
38656 | 467 |
by (auto intro!: positive_integral_cong simp: field_simps indicator_inter_arith) |
41544 | 468 |
hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x) = |
469 |
(\<integral>\<^isup>+x. f x * indicator A x) + b * \<mu> (A \<inter> A0)" |
|
38656 | 470 |
using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A |
471 |
by (simp add: borel_measurable_indicator positive_integral_add positive_integral_cmult_indicator) } |
|
472 |
note f0_eq = this |
|
473 |
{ fix A assume A: "A \<in> sets M" |
|
474 |
hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto |
|
41544 | 475 |
have f_le_v: "(\<integral>\<^isup>+x. f x * indicator A x) \<le> \<nu> A" |
38656 | 476 |
using `f \<in> G` A unfolding G_def by auto |
477 |
note f0_eq[OF A] |
|
41544 | 478 |
also have "(\<integral>\<^isup>+x. f x * indicator A x) + b * \<mu> (A \<inter> A0) \<le> |
479 |
(\<integral>\<^isup>+x. f x * indicator A x) + ?t (A \<inter> A0)" |
|
38656 | 480 |
using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M` |
481 |
by (auto intro!: add_left_mono) |
|
41544 | 482 |
also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x) + ?t A" |
38656 | 483 |
using M.measure_mono[simplified, OF _ `A \<inter> A0 \<in> sets M` `A \<in> sets M`] |
484 |
by (auto intro!: add_left_mono) |
|
485 |
also have "\<dots> \<le> \<nu> A" |
|
486 |
using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`] |
|
41544 | 487 |
by (cases "(\<integral>\<^isup>+x. f x * indicator A x)", cases "\<nu> A", auto) |
488 |
finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x) \<le> \<nu> A" . } |
|
38656 | 489 |
hence "?f0 \<in> G" using `A0 \<in> sets M` unfolding G_def |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
490 |
by (auto intro!: borel_measurable_indicator borel_measurable_pextreal_add borel_measurable_pextreal_times) |
38656 | 491 |
have real: "?t (space M) \<noteq> \<omega>" "?t A0 \<noteq> \<omega>" |
492 |
"b * \<mu> (space M) \<noteq> \<omega>" "b * \<mu> A0 \<noteq> \<omega>" |
|
493 |
using `A0 \<in> sets M` b |
|
494 |
finite_measure[of A0] M.finite_measure[of A0] |
|
495 |
finite_measure_of_space M.finite_measure_of_space |
|
496 |
by auto |
|
497 |
have int_f_finite: "positive_integral f \<noteq> \<omega>" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
498 |
using M'.finite_measure_of_space pos_t unfolding pextreal_zero_less_diff_iff |
38656 | 499 |
by (auto cong: positive_integral_cong) |
500 |
have "?t (space M) > b * \<mu> (space M)" unfolding b_def |
|
501 |
apply (simp add: field_simps) |
|
502 |
apply (subst mult_assoc[symmetric]) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
503 |
apply (subst pextreal_mult_inverse) |
38656 | 504 |
using finite_measure_of_space M'.finite_measure_of_space pos_t pos_M |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
505 |
using pextreal_mult_strict_right_mono[of "Real (1/2)" 1 "?t (space M)"] |
38656 | 506 |
by simp_all |
507 |
hence "0 < ?t (space M) - b * \<mu> (space M)" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
508 |
by (simp add: pextreal_zero_less_diff_iff) |
38656 | 509 |
also have "\<dots> \<le> ?t A0 - b * \<mu> A0" |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
510 |
using space_less_A0 pos_M pos_t b real[unfolded pextreal_noteq_omega_Ex] by auto |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
511 |
finally have "b * \<mu> A0 < ?t A0" unfolding pextreal_zero_less_diff_iff . |
38656 | 512 |
hence "0 < ?t A0" by auto |
513 |
hence "0 < \<mu> A0" using ac unfolding absolutely_continuous_def |
|
514 |
using `A0 \<in> sets M` by auto |
|
515 |
hence "0 < b * \<mu> A0" using b by auto |
|
516 |
from int_f_finite this |
|
517 |
have "?y + 0 < positive_integral f + b * \<mu> A0" unfolding int_f_eq_y |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
518 |
by (rule pextreal_less_add) |
38656 | 519 |
also have "\<dots> = positive_integral ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space |
520 |
by (simp cong: positive_integral_cong) |
|
521 |
finally have "?y < positive_integral ?f0" by simp |
|
522 |
moreover from `?f0 \<in> G` have "positive_integral ?f0 \<le> ?y" by (auto intro!: le_SUPI) |
|
523 |
ultimately show False by auto |
|
524 |
qed |
|
525 |
show ?thesis |
|
526 |
proof (safe intro!: bexI[of _ f]) |
|
527 |
fix A assume "A\<in>sets M" |
|
41544 | 528 |
show "\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)" |
38656 | 529 |
proof (rule antisym) |
41544 | 530 |
show "(\<integral>\<^isup>+x. f x * indicator A x) \<le> \<nu> A" |
38656 | 531 |
using `f \<in> G` `A \<in> sets M` unfolding G_def by auto |
41544 | 532 |
show "\<nu> A \<le> (\<integral>\<^isup>+x. f x * indicator A x)" |
38656 | 533 |
using upper_bound[THEN bspec, OF `A \<in> sets M`] |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
534 |
by (simp add: pextreal_zero_le_diff) |
38656 | 535 |
qed |
536 |
qed simp |
|
537 |
qed |
|
538 |
||
40859 | 539 |
lemma (in finite_measure) split_space_into_finite_sets_and_rest: |
38656 | 540 |
assumes "measure_space M \<nu>" |
40859 | 541 |
assumes ac: "absolutely_continuous \<nu>" |
542 |
shows "\<exists>\<Omega>0\<in>sets M. \<exists>\<Omega>::nat\<Rightarrow>'a set. disjoint_family \<Omega> \<and> range \<Omega> \<subseteq> sets M \<and> \<Omega>0 = space M - (\<Union>i. \<Omega> i) \<and> |
|
543 |
(\<forall>A\<in>sets M. A \<subseteq> \<Omega>0 \<longrightarrow> (\<mu> A = 0 \<and> \<nu> A = 0) \<or> (\<mu> A > 0 \<and> \<nu> A = \<omega>)) \<and> |
|
544 |
(\<forall>i. \<nu> (\<Omega> i) \<noteq> \<omega>)" |
|
38656 | 545 |
proof - |
546 |
interpret v: measure_space M \<nu> by fact |
|
547 |
let ?Q = "{Q\<in>sets M. \<nu> Q \<noteq> \<omega>}" |
|
548 |
let ?a = "SUP Q:?Q. \<mu> Q" |
|
549 |
have "{} \<in> ?Q" using v.empty_measure by auto |
|
550 |
then have Q_not_empty: "?Q \<noteq> {}" by blast |
|
551 |
have "?a \<le> \<mu> (space M)" using sets_into_space |
|
552 |
by (auto intro!: SUP_leI measure_mono top) |
|
553 |
then have "?a \<noteq> \<omega>" using finite_measure_of_space |
|
554 |
by auto |
|
555 |
from SUPR_countable_SUPR[OF this Q_not_empty] |
|
556 |
obtain Q'' where "range Q'' \<subseteq> \<mu> ` ?Q" and a: "?a = (SUP i::nat. Q'' i)" |
|
557 |
by auto |
|
558 |
then have "\<forall>i. \<exists>Q'. Q'' i = \<mu> Q' \<and> Q' \<in> ?Q" by auto |
|
559 |
from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = \<mu> (Q' i)" "\<And>i. Q' i \<in> ?Q" |
|
560 |
by auto |
|
561 |
then have a_Lim: "?a = (SUP i::nat. \<mu> (Q' i))" using a by simp |
|
562 |
let "?O n" = "\<Union>i\<le>n. Q' i" |
|
563 |
have Union: "(SUP i. \<mu> (?O i)) = \<mu> (\<Union>i. ?O i)" |
|
564 |
proof (rule continuity_from_below[of ?O]) |
|
565 |
show "range ?O \<subseteq> sets M" using Q' by (auto intro!: finite_UN) |
|
566 |
show "\<And>i. ?O i \<subseteq> ?O (Suc i)" by fastsimp |
|
567 |
qed |
|
568 |
have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto |
|
569 |
have O_sets: "\<And>i. ?O i \<in> sets M" |
|
570 |
using Q' by (auto intro!: finite_UN Un) |
|
571 |
then have O_in_G: "\<And>i. ?O i \<in> ?Q" |
|
572 |
proof (safe del: notI) |
|
573 |
fix i have "Q' ` {..i} \<subseteq> sets M" |
|
574 |
using Q' by (auto intro: finite_UN) |
|
575 |
with v.measure_finitely_subadditive[of "{.. i}" Q'] |
|
576 |
have "\<nu> (?O i) \<le> (\<Sum>i\<le>i. \<nu> (Q' i))" by auto |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
577 |
also have "\<dots> < \<omega>" unfolding setsum_\<omega> pextreal_less_\<omega> using Q' by auto |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
578 |
finally show "\<nu> (?O i) \<noteq> \<omega>" unfolding pextreal_less_\<omega> by auto |
38656 | 579 |
qed auto |
580 |
have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastsimp |
|
581 |
have a_eq: "?a = \<mu> (\<Union>i. ?O i)" unfolding Union[symmetric] |
|
582 |
proof (rule antisym) |
|
583 |
show "?a \<le> (SUP i. \<mu> (?O i))" unfolding a_Lim |
|
584 |
using Q' by (auto intro!: SUP_mono measure_mono finite_UN) |
|
585 |
show "(SUP i. \<mu> (?O i)) \<le> ?a" unfolding SUPR_def |
|
586 |
proof (safe intro!: Sup_mono, unfold bex_simps) |
|
587 |
fix i |
|
588 |
have *: "(\<Union>Q' ` {..i}) = ?O i" by auto |
|
589 |
then show "\<exists>x. (x \<in> sets M \<and> \<nu> x \<noteq> \<omega>) \<and> |
|
590 |
\<mu> (\<Union>Q' ` {..i}) \<le> \<mu> x" |
|
591 |
using O_in_G[of i] by (auto intro!: exI[of _ "?O i"]) |
|
592 |
qed |
|
593 |
qed |
|
594 |
let "?O_0" = "(\<Union>i. ?O i)" |
|
595 |
have "?O_0 \<in> sets M" using Q' by auto |
|
40859 | 596 |
def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n" |
38656 | 597 |
{ fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) } |
598 |
note Q_sets = this |
|
40859 | 599 |
show ?thesis |
600 |
proof (intro bexI exI conjI ballI impI allI) |
|
601 |
show "disjoint_family Q" |
|
602 |
by (fastsimp simp: disjoint_family_on_def Q_def |
|
603 |
split: nat.split_asm) |
|
604 |
show "range Q \<subseteq> sets M" |
|
605 |
using Q_sets by auto |
|
606 |
{ fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0" |
|
607 |
show "\<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<omega>" |
|
608 |
proof (rule disjCI, simp) |
|
609 |
assume *: "0 < \<mu> A \<longrightarrow> \<nu> A \<noteq> \<omega>" |
|
610 |
show "\<mu> A = 0 \<and> \<nu> A = 0" |
|
611 |
proof cases |
|
612 |
assume "\<mu> A = 0" moreover with ac A have "\<nu> A = 0" |
|
613 |
unfolding absolutely_continuous_def by auto |
|
614 |
ultimately show ?thesis by simp |
|
615 |
next |
|
616 |
assume "\<mu> A \<noteq> 0" with * have "\<nu> A \<noteq> \<omega>" by auto |
|
617 |
with A have "\<mu> ?O_0 + \<mu> A = \<mu> (?O_0 \<union> A)" |
|
618 |
using Q' by (auto intro!: measure_additive countable_UN) |
|
619 |
also have "\<dots> = (SUP i. \<mu> (?O i \<union> A))" |
|
620 |
proof (rule continuity_from_below[of "\<lambda>i. ?O i \<union> A", symmetric, simplified]) |
|
621 |
show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M" |
|
622 |
using `\<nu> A \<noteq> \<omega>` O_sets A by auto |
|
623 |
qed fastsimp |
|
624 |
also have "\<dots> \<le> ?a" |
|
625 |
proof (safe intro!: SUPR_bound) |
|
626 |
fix i have "?O i \<union> A \<in> ?Q" |
|
627 |
proof (safe del: notI) |
|
628 |
show "?O i \<union> A \<in> sets M" using O_sets A by auto |
|
629 |
from O_in_G[of i] |
|
630 |
moreover have "\<nu> (?O i \<union> A) \<le> \<nu> (?O i) + \<nu> A" |
|
631 |
using v.measure_subadditive[of "?O i" A] A O_sets by auto |
|
632 |
ultimately show "\<nu> (?O i \<union> A) \<noteq> \<omega>" |
|
633 |
using `\<nu> A \<noteq> \<omega>` by auto |
|
634 |
qed |
|
635 |
then show "\<mu> (?O i \<union> A) \<le> ?a" by (rule le_SUPI) |
|
636 |
qed |
|
637 |
finally have "\<mu> A = 0" unfolding a_eq using finite_measure[OF `?O_0 \<in> sets M`] |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
638 |
by (cases "\<mu> A") (auto simp: pextreal_noteq_omega_Ex) |
40859 | 639 |
with `\<mu> A \<noteq> 0` show ?thesis by auto |
640 |
qed |
|
641 |
qed } |
|
642 |
{ fix i show "\<nu> (Q i) \<noteq> \<omega>" |
|
643 |
proof (cases i) |
|
644 |
case 0 then show ?thesis |
|
645 |
unfolding Q_def using Q'[of 0] by simp |
|
646 |
next |
|
647 |
case (Suc n) |
|
648 |
then show ?thesis unfolding Q_def |
|
649 |
using `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q` O_mono |
|
650 |
using v.measure_Diff[of "?O n" "?O (Suc n)"] by auto |
|
651 |
qed } |
|
652 |
show "space M - ?O_0 \<in> sets M" using Q'_sets by auto |
|
653 |
{ fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)" |
|
654 |
proof (induct j) |
|
655 |
case 0 then show ?case by (simp add: Q_def) |
|
656 |
next |
|
657 |
case (Suc j) |
|
658 |
have eq: "\<And>j. (\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q' i)" by fastsimp |
|
659 |
have "{..j} \<union> {..Suc j} = {..Suc j}" by auto |
|
660 |
then have "(\<Union>i\<le>Suc j. Q' i) = (\<Union>i\<le>j. Q' i) \<union> Q (Suc j)" |
|
661 |
by (simp add: UN_Un[symmetric] Q_def del: UN_Un) |
|
662 |
then show ?case using Suc by (auto simp add: eq atMost_Suc) |
|
663 |
qed } |
|
664 |
then have "(\<Union>j. (\<Union>i\<le>j. ?O i)) = (\<Union>j. (\<Union>i\<le>j. Q i))" by simp |
|
665 |
then show "space M - ?O_0 = space M - (\<Union>i. Q i)" by fastsimp |
|
666 |
qed |
|
667 |
qed |
|
668 |
||
669 |
lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite: |
|
670 |
assumes "measure_space M \<nu>" |
|
671 |
assumes "absolutely_continuous \<nu>" |
|
41544 | 672 |
shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)" |
40859 | 673 |
proof - |
674 |
interpret v: measure_space M \<nu> by fact |
|
675 |
from split_space_into_finite_sets_and_rest[OF assms] |
|
676 |
obtain Q0 and Q :: "nat \<Rightarrow> 'a set" |
|
677 |
where Q: "disjoint_family Q" "range Q \<subseteq> sets M" |
|
678 |
and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)" |
|
679 |
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<omega>" |
|
680 |
and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<omega>" by force |
|
681 |
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto |
|
38656 | 682 |
have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M. |
41544 | 683 |
\<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x))" |
38656 | 684 |
proof |
685 |
fix i |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
686 |
have indicator_eq: "\<And>f x A. (f x :: pextreal) * indicator (Q i \<inter> A) x * indicator (Q i) x |
38656 | 687 |
= (f x * indicator (Q i) x) * indicator A x" |
688 |
unfolding indicator_def by auto |
|
39092 | 689 |
have fm: "finite_measure (restricted_space (Q i)) \<mu>" |
38656 | 690 |
(is "finite_measure ?R \<mu>") by (rule restricted_finite_measure[OF Q_sets[of i]]) |
691 |
then interpret R: finite_measure ?R . |
|
692 |
have fmv: "finite_measure ?R \<nu>" |
|
693 |
unfolding finite_measure_def finite_measure_axioms_def |
|
694 |
proof |
|
695 |
show "measure_space ?R \<nu>" |
|
696 |
using v.restricted_measure_space Q_sets[of i] by auto |
|
697 |
show "\<nu> (space ?R) \<noteq> \<omega>" |
|
40859 | 698 |
using Q_fin by simp |
38656 | 699 |
qed |
700 |
have "R.absolutely_continuous \<nu>" |
|
701 |
using `absolutely_continuous \<nu>` `Q i \<in> sets M` |
|
702 |
by (auto simp: R.absolutely_continuous_def absolutely_continuous_def) |
|
703 |
from finite_measure.Radon_Nikodym_finite_measure[OF fm fmv this] |
|
704 |
obtain f where f: "(\<lambda>x. f x * indicator (Q i) x) \<in> borel_measurable M" |
|
41544 | 705 |
and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. (f x * indicator (Q i) x) * indicator A x)" |
38656 | 706 |
unfolding Bex_def borel_measurable_restricted[OF `Q i \<in> sets M`] |
707 |
positive_integral_restricted[OF `Q i \<in> sets M`] by (auto simp: indicator_eq) |
|
708 |
then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M. |
|
41544 | 709 |
\<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x))" |
38656 | 710 |
by (fastsimp intro!: exI[of _ "\<lambda>x. f x * indicator (Q i) x"] positive_integral_cong |
711 |
simp: indicator_def) |
|
712 |
qed |
|
713 |
from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M" |
|
714 |
and f: "\<And>A i. A \<in> sets M \<Longrightarrow> |
|
41544 | 715 |
\<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x)" |
38656 | 716 |
by auto |
717 |
let "?f x" = |
|
40859 | 718 |
"(\<Sum>\<^isub>\<infinity> i. f i x * indicator (Q i) x) + \<omega> * indicator Q0 x" |
38656 | 719 |
show ?thesis |
720 |
proof (safe intro!: bexI[of _ ?f]) |
|
721 |
show "?f \<in> borel_measurable M" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
722 |
by (safe intro!: borel_measurable_psuminf borel_measurable_pextreal_times |
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
723 |
borel_measurable_pextreal_add borel_measurable_indicator |
40859 | 724 |
borel_measurable_const borel Q_sets Q0 Diff countable_UN) |
38656 | 725 |
fix A assume "A \<in> sets M" |
40859 | 726 |
have *: |
38656 | 727 |
"\<And>x i. indicator A x * (f i x * indicator (Q i) x) = |
728 |
f i x * indicator (Q i \<inter> A) x" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
729 |
"\<And>x i. (indicator A x * indicator Q0 x :: pextreal) = |
40859 | 730 |
indicator (Q0 \<inter> A) x" by (auto simp: indicator_def) |
41544 | 731 |
have "(\<integral>\<^isup>+x. ?f x * indicator A x) = |
40859 | 732 |
(\<Sum>\<^isub>\<infinity> i. \<nu> (Q i \<inter> A)) + \<omega> * \<mu> (Q0 \<inter> A)" |
38656 | 733 |
unfolding f[OF `A \<in> sets M`] |
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
734 |
apply (simp del: pextreal_times(2) add: field_simps *) |
38656 | 735 |
apply (subst positive_integral_add) |
40859 | 736 |
apply (fastsimp intro: Q0 `A \<in> sets M`) |
737 |
apply (fastsimp intro: Q_sets `A \<in> sets M` borel_measurable_psuminf borel) |
|
738 |
apply (subst positive_integral_cmult_indicator) |
|
739 |
apply (fastsimp intro: Q0 `A \<in> sets M`) |
|
38656 | 740 |
unfolding psuminf_cmult_right[symmetric] |
741 |
apply (subst positive_integral_psuminf) |
|
40859 | 742 |
apply (fastsimp intro: `A \<in> sets M` Q_sets borel) |
743 |
apply (simp add: *) |
|
744 |
done |
|
38656 | 745 |
moreover have "(\<Sum>\<^isub>\<infinity>i. \<nu> (Q i \<inter> A)) = \<nu> ((\<Union>i. Q i) \<inter> A)" |
40859 | 746 |
using Q Q_sets `A \<in> sets M` |
747 |
by (intro v.measure_countably_additive[of "\<lambda>i. Q i \<inter> A", unfolded comp_def, simplified]) |
|
748 |
(auto simp: disjoint_family_on_def) |
|
749 |
moreover have "\<omega> * \<mu> (Q0 \<inter> A) = \<nu> (Q0 \<inter> A)" |
|
750 |
proof - |
|
751 |
have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast |
|
752 |
from in_Q0[OF this] show ?thesis by auto |
|
38656 | 753 |
qed |
40859 | 754 |
moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M" |
755 |
using Q_sets `A \<in> sets M` Q0(1) by (auto intro!: countable_UN) |
|
756 |
moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}" |
|
757 |
using `A \<in> sets M` sets_into_space Q0 by auto |
|
41544 | 758 |
ultimately show "\<nu> A = (\<integral>\<^isup>+x. ?f x * indicator A x)" |
40859 | 759 |
using v.measure_additive[simplified, of "(\<Union>i. Q i) \<inter> A" "Q0 \<inter> A"] |
760 |
by simp |
|
38656 | 761 |
qed |
762 |
qed |
|
763 |
||
764 |
lemma (in sigma_finite_measure) Radon_Nikodym: |
|
765 |
assumes "measure_space M \<nu>" |
|
766 |
assumes "absolutely_continuous \<nu>" |
|
41544 | 767 |
shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)" |
38656 | 768 |
proof - |
769 |
from Ex_finite_integrable_function |
|
770 |
obtain h where finite: "positive_integral h \<noteq> \<omega>" and |
|
771 |
borel: "h \<in> borel_measurable M" and |
|
772 |
pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and |
|
773 |
"\<And>x. x \<in> space M \<Longrightarrow> h x < \<omega>" by auto |
|
41544 | 774 |
let "?T A" = "(\<integral>\<^isup>+x. h x * indicator A x)" |
38656 | 775 |
from measure_space_density[OF borel] finite |
776 |
interpret T: finite_measure M ?T |
|
777 |
unfolding finite_measure_def finite_measure_axioms_def |
|
778 |
by (simp cong: positive_integral_cong) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
779 |
have "\<And>N. N \<in> sets M \<Longrightarrow> {x \<in> space M. h x \<noteq> 0 \<and> indicator N x \<noteq> (0::pextreal)} = N" |
38656 | 780 |
using sets_into_space pos by (force simp: indicator_def) |
781 |
then have "T.absolutely_continuous \<nu>" using assms(2) borel |
|
782 |
unfolding T.absolutely_continuous_def absolutely_continuous_def |
|
783 |
by (fastsimp simp: borel_measurable_indicator positive_integral_0_iff) |
|
784 |
from T.Radon_Nikodym_finite_measure_infinite[simplified, OF assms(1) this] |
|
785 |
obtain f where f_borel: "f \<in> borel_measurable M" and |
|
786 |
fT: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = T.positive_integral (\<lambda>x. f x * indicator A x)" by auto |
|
787 |
show ?thesis |
|
788 |
proof (safe intro!: bexI[of _ "\<lambda>x. h x * f x"]) |
|
789 |
show "(\<lambda>x. h x * f x) \<in> borel_measurable M" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
790 |
using borel f_borel by (auto intro: borel_measurable_pextreal_times) |
38656 | 791 |
fix A assume "A \<in> sets M" |
792 |
then have "(\<lambda>x. f x * indicator A x) \<in> borel_measurable M" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
793 |
using f_borel by (auto intro: borel_measurable_pextreal_times borel_measurable_indicator) |
38656 | 794 |
from positive_integral_translated_density[OF borel this] |
41544 | 795 |
show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x)" |
38656 | 796 |
unfolding fT[OF `A \<in> sets M`] by (simp add: field_simps) |
797 |
qed |
|
798 |
qed |
|
799 |
||
40859 | 800 |
section "Uniqueness of densities" |
801 |
||
802 |
lemma (in measure_space) finite_density_unique: |
|
803 |
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
|
804 |
and fin: "positive_integral f < \<omega>" |
|
41544 | 805 |
shows "(\<forall>A\<in>sets M. (\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. g x * indicator A x)) |
40859 | 806 |
\<longleftrightarrow> (AE x. f x = g x)" |
807 |
(is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _") |
|
808 |
proof (intro iffI ballI) |
|
809 |
fix A assume eq: "AE x. f x = g x" |
|
810 |
show "?P f A = ?P g A" |
|
811 |
by (rule positive_integral_cong_AE[OF AE_mp[OF eq]]) simp |
|
812 |
next |
|
813 |
assume eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" |
|
814 |
from this[THEN bspec, OF top] fin |
|
815 |
have g_fin: "positive_integral g < \<omega>" by (simp cong: positive_integral_cong) |
|
816 |
{ fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
|
817 |
and g_fin: "positive_integral g < \<omega>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A" |
|
818 |
let ?N = "{x\<in>space M. g x < f x}" |
|
819 |
have N: "?N \<in> sets M" using borel by simp |
|
41544 | 820 |
have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x)" |
40859 | 821 |
by (auto intro!: positive_integral_cong simp: indicator_def) |
822 |
also have "\<dots> = ?P f ?N - ?P g ?N" |
|
823 |
proof (rule positive_integral_diff) |
|
824 |
show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M" |
|
825 |
using borel N by auto |
|
826 |
have "?P g ?N \<le> positive_integral g" |
|
827 |
by (auto intro!: positive_integral_mono simp: indicator_def) |
|
828 |
then show "?P g ?N \<noteq> \<omega>" using g_fin by auto |
|
829 |
fix x assume "x \<in> space M" |
|
830 |
show "g x * indicator ?N x \<le> f x * indicator ?N x" |
|
831 |
by (auto simp: indicator_def) |
|
832 |
qed |
|
833 |
also have "\<dots> = 0" |
|
834 |
using eq[THEN bspec, OF N] by simp |
|
835 |
finally have "\<mu> {x\<in>space M. (f x - g x) * indicator ?N x \<noteq> 0} = 0" |
|
836 |
using borel N by (subst (asm) positive_integral_0_iff) auto |
|
837 |
moreover have "{x\<in>space M. (f x - g x) * indicator ?N x \<noteq> 0} = ?N" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
838 |
by (auto simp: pextreal_zero_le_diff) |
40859 | 839 |
ultimately have "?N \<in> null_sets" using N by simp } |
840 |
from this[OF borel g_fin eq] this[OF borel(2,1) fin] |
|
841 |
have "{x\<in>space M. g x < f x} \<union> {x\<in>space M. f x < g x} \<in> null_sets" |
|
842 |
using eq by (intro null_sets_Un) auto |
|
843 |
also have "{x\<in>space M. g x < f x} \<union> {x\<in>space M. f x < g x} = {x\<in>space M. f x \<noteq> g x}" |
|
844 |
by auto |
|
845 |
finally show "AE x. f x = g x" |
|
846 |
unfolding almost_everywhere_def by auto |
|
847 |
qed |
|
848 |
||
849 |
lemma (in finite_measure) density_unique_finite_measure: |
|
850 |
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M" |
|
41544 | 851 |
assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. f' x * indicator A x)" |
40859 | 852 |
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A") |
853 |
shows "AE x. f x = f' x" |
|
854 |
proof - |
|
855 |
let "?\<nu> A" = "?P f A" and "?\<nu>' A" = "?P f' A" |
|
856 |
let "?f A x" = "f x * indicator A x" and "?f' A x" = "f' x * indicator A x" |
|
857 |
interpret M: measure_space M ?\<nu> |
|
858 |
using borel(1) by (rule measure_space_density) |
|
859 |
have ac: "absolutely_continuous ?\<nu>" |
|
860 |
using f by (rule density_is_absolutely_continuous) |
|
861 |
from split_space_into_finite_sets_and_rest[OF `measure_space M ?\<nu>` ac] |
|
862 |
obtain Q0 and Q :: "nat \<Rightarrow> 'a set" |
|
863 |
where Q: "disjoint_family Q" "range Q \<subseteq> sets M" |
|
864 |
and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)" |
|
865 |
and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<subseteq> Q0 \<Longrightarrow> \<mu> A = 0 \<and> ?\<nu> A = 0 \<or> 0 < \<mu> A \<and> ?\<nu> A = \<omega>" |
|
866 |
and Q_fin: "\<And>i. ?\<nu> (Q i) \<noteq> \<omega>" by force |
|
867 |
from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto |
|
868 |
let ?N = "{x\<in>space M. f x \<noteq> f' x}" |
|
869 |
have "?N \<in> sets M" using borel by auto |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
870 |
have *: "\<And>i x A. \<And>y::pextreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x" |
40859 | 871 |
unfolding indicator_def by auto |
872 |
have 1: "\<forall>i. AE x. ?f (Q i) x = ?f' (Q i) x" |
|
873 |
using borel Q_fin Q |
|
874 |
by (intro finite_density_unique[THEN iffD1] allI) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
875 |
(auto intro!: borel_measurable_pextreal_times f Int simp: *) |
40859 | 876 |
have 2: "AE x. ?f Q0 x = ?f' Q0 x" |
877 |
proof (rule AE_I') |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
878 |
{ fix f :: "'a \<Rightarrow> pextreal" assume borel: "f \<in> borel_measurable M" |
41544 | 879 |
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)" |
40859 | 880 |
let "?A i" = "Q0 \<inter> {x \<in> space M. f x < of_nat i}" |
881 |
have "(\<Union>i. ?A i) \<in> null_sets" |
|
882 |
proof (rule null_sets_UN) |
|
883 |
fix i have "?A i \<in> sets M" |
|
884 |
using borel Q0(1) by auto |
|
41544 | 885 |
have "?\<nu> (?A i) \<le> (\<integral>\<^isup>+x. of_nat i * indicator (?A i) x)" |
40859 | 886 |
unfolding eq[OF `?A i \<in> sets M`] |
887 |
by (auto intro!: positive_integral_mono simp: indicator_def) |
|
888 |
also have "\<dots> = of_nat i * \<mu> (?A i)" |
|
889 |
using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator) |
|
890 |
also have "\<dots> < \<omega>" |
|
891 |
using `?A i \<in> sets M`[THEN finite_measure] by auto |
|
892 |
finally have "?\<nu> (?A i) \<noteq> \<omega>" by simp |
|
893 |
then show "?A i \<in> null_sets" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto |
|
894 |
qed |
|
895 |
also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x < \<omega>}" |
|
896 |
by (auto simp: less_\<omega>_Ex_of_nat) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
897 |
finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>} \<in> null_sets" by (simp add: pextreal_less_\<omega>) } |
40859 | 898 |
from this[OF borel(1) refl] this[OF borel(2) f] |
899 |
have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>} \<in> null_sets" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<omega>} \<in> null_sets" by simp_all |
|
900 |
then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<omega>}) \<in> null_sets" by (rule null_sets_Un) |
|
901 |
show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq> |
|
902 |
(Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<omega>})" by (auto simp: indicator_def) |
|
903 |
qed |
|
904 |
have **: "\<And>x. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow> |
|
905 |
?f (space M) x = ?f' (space M) x" |
|
906 |
by (auto simp: indicator_def Q0) |
|
907 |
have 3: "AE x. ?f (space M) x = ?f' (space M) x" |
|
908 |
by (rule AE_mp[OF 1[unfolded all_AE_countable] AE_mp[OF 2]]) (simp add: **) |
|
909 |
then show "AE x. f x = f' x" |
|
910 |
by (rule AE_mp) (auto intro!: AE_cong simp: indicator_def) |
|
911 |
qed |
|
912 |
||
913 |
lemma (in sigma_finite_measure) density_unique: |
|
914 |
assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M" |
|
41544 | 915 |
assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. f' x * indicator A x)" |
40859 | 916 |
(is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A") |
917 |
shows "AE x. f x = f' x" |
|
918 |
proof - |
|
919 |
obtain h where h_borel: "h \<in> borel_measurable M" |
|
920 |
and fin: "positive_integral h \<noteq> \<omega>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>" |
|
921 |
using Ex_finite_integrable_function by auto |
|
41544 | 922 |
interpret h: measure_space M "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x)" |
40859 | 923 |
using h_borel by (rule measure_space_density) |
41544 | 924 |
interpret h: finite_measure M "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x)" |
40859 | 925 |
by default (simp cong: positive_integral_cong add: fin) |
41544 | 926 |
interpret f: measure_space M "\<lambda>A. (\<integral>\<^isup>+x. f x * indicator A x)" |
40859 | 927 |
using borel(1) by (rule measure_space_density) |
41544 | 928 |
interpret f': measure_space M "\<lambda>A. (\<integral>\<^isup>+x. f' x * indicator A x)" |
40859 | 929 |
using borel(2) by (rule measure_space_density) |
930 |
{ fix A assume "A \<in> sets M" |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
931 |
then have " {x \<in> space M. h x \<noteq> 0 \<and> indicator A x \<noteq> (0::pextreal)} = A" |
40859 | 932 |
using pos sets_into_space by (force simp: indicator_def) |
41544 | 933 |
then have "(\<integral>\<^isup>+x. h x * indicator A x) = 0 \<longleftrightarrow> A \<in> null_sets" |
40859 | 934 |
using h_borel `A \<in> sets M` by (simp add: positive_integral_0_iff) } |
935 |
note h_null_sets = this |
|
936 |
{ fix A assume "A \<in> sets M" |
|
41544 | 937 |
have "(\<integral>\<^isup>+x. h x * (f x * indicator A x)) = |
40859 | 938 |
f.positive_integral (\<lambda>x. h x * indicator A x)" |
939 |
using `A \<in> sets M` h_borel borel |
|
940 |
by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong) |
|
941 |
also have "\<dots> = f'.positive_integral (\<lambda>x. h x * indicator A x)" |
|
942 |
by (rule f'.positive_integral_cong_measure) (rule f) |
|
41544 | 943 |
also have "\<dots> = (\<integral>\<^isup>+x. h x * (f' x * indicator A x))" |
40859 | 944 |
using `A \<in> sets M` h_borel borel |
945 |
by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong) |
|
41544 | 946 |
finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x)) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x))" . } |
40859 | 947 |
then have "h.almost_everywhere (\<lambda>x. f x = f' x)" |
948 |
using h_borel borel |
|
949 |
by (intro h.density_unique_finite_measure[OF borel]) |
|
950 |
(simp add: positive_integral_translated_density) |
|
951 |
then show "AE x. f x = f' x" |
|
952 |
unfolding h.almost_everywhere_def almost_everywhere_def |
|
953 |
by (auto simp add: h_null_sets) |
|
954 |
qed |
|
955 |
||
956 |
lemma (in sigma_finite_measure) sigma_finite_iff_density_finite: |
|
957 |
assumes \<nu>: "measure_space M \<nu>" and f: "f \<in> borel_measurable M" |
|
41544 | 958 |
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)" |
40859 | 959 |
shows "sigma_finite_measure M \<nu> \<longleftrightarrow> (AE x. f x \<noteq> \<omega>)" |
960 |
proof |
|
961 |
assume "sigma_finite_measure M \<nu>" |
|
962 |
then interpret \<nu>: sigma_finite_measure M \<nu> . |
|
963 |
from \<nu>.Ex_finite_integrable_function obtain h where |
|
964 |
h: "h \<in> borel_measurable M" "\<nu>.positive_integral h \<noteq> \<omega>" |
|
965 |
and fin: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>" by auto |
|
966 |
have "AE x. f x * h x \<noteq> \<omega>" |
|
967 |
proof (rule AE_I') |
|
41544 | 968 |
have "\<nu>.positive_integral h = (\<integral>\<^isup>+x. f x * h x)" |
40859 | 969 |
by (simp add: \<nu>.positive_integral_cong_measure[symmetric, OF eq[symmetric]]) |
970 |
(intro positive_integral_translated_density f h) |
|
41544 | 971 |
then have "(\<integral>\<^isup>+x. f x * h x) \<noteq> \<omega>" |
40859 | 972 |
using h(2) by simp |
973 |
then show "(\<lambda>x. f x * h x) -` {\<omega>} \<inter> space M \<in> null_sets" |
|
974 |
using f h(1) by (auto intro!: positive_integral_omega borel_measurable_vimage) |
|
975 |
qed auto |
|
976 |
then show "AE x. f x \<noteq> \<omega>" |
|
977 |
proof (rule AE_mp, intro AE_cong) |
|
978 |
fix x assume "x \<in> space M" from this[THEN fin] |
|
979 |
show "f x * h x \<noteq> \<omega> \<longrightarrow> f x \<noteq> \<omega>" by auto |
|
980 |
qed |
|
981 |
next |
|
982 |
assume AE: "AE x. f x \<noteq> \<omega>" |
|
983 |
from sigma_finite guess Q .. note Q = this |
|
984 |
interpret \<nu>: measure_space M \<nu> by fact |
|
985 |
def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<omega>} | Suc n \<Rightarrow> {.. of_nat (Suc n)}) \<inter> space M" |
|
986 |
{ fix i j have "A i \<inter> Q j \<in> sets M" |
|
987 |
unfolding A_def using f Q |
|
988 |
apply (rule_tac Int) |
|
989 |
by (cases i) (auto intro: measurable_sets[OF f]) } |
|
990 |
note A_in_sets = this |
|
991 |
let "?A n" = "case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j" |
|
992 |
show "sigma_finite_measure M \<nu>" |
|
993 |
proof (default, intro exI conjI subsetI allI) |
|
994 |
fix x assume "x \<in> range ?A" |
|
995 |
then obtain n where n: "x = ?A n" by auto |
|
996 |
then show "x \<in> sets M" using A_in_sets by (cases "prod_decode n") auto |
|
997 |
next |
|
998 |
have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)" |
|
999 |
proof safe |
|
1000 |
fix x i j assume "x \<in> A i" "x \<in> Q j" |
|
1001 |
then show "x \<in> (\<Union>i. case prod_decode i of (i, j) \<Rightarrow> A i \<inter> Q j)" |
|
1002 |
by (intro UN_I[of "prod_encode (i,j)"]) auto |
|
1003 |
qed auto |
|
1004 |
also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto |
|
1005 |
also have "(\<Union>i. A i) = space M" |
|
1006 |
proof safe |
|
1007 |
fix x assume x: "x \<in> space M" |
|
1008 |
show "x \<in> (\<Union>i. A i)" |
|
1009 |
proof (cases "f x") |
|
1010 |
case infinite then show ?thesis using x unfolding A_def by (auto intro: exI[of _ 0]) |
|
1011 |
next |
|
1012 |
case (preal r) |
|
1013 |
with less_\<omega>_Ex_of_nat[of "f x"] obtain n where "f x < of_nat n" by auto |
|
1014 |
then show ?thesis using x preal unfolding A_def by (auto intro!: exI[of _ "Suc n"]) |
|
1015 |
qed |
|
1016 |
qed (auto simp: A_def) |
|
1017 |
finally show "(\<Union>i. ?A i) = space M" by simp |
|
1018 |
next |
|
1019 |
fix n obtain i j where |
|
1020 |
[simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto |
|
41544 | 1021 |
have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x) \<noteq> \<omega>" |
40859 | 1022 |
proof (cases i) |
1023 |
case 0 |
|
1024 |
have "AE x. f x * indicator (A i \<inter> Q j) x = 0" |
|
1025 |
using AE by (rule AE_mp) (auto intro!: AE_cong simp: A_def `i = 0`) |
|
41544 | 1026 |
then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x) = 0" |
40859 | 1027 |
using A_in_sets f |
1028 |
apply (subst positive_integral_0_iff) |
|
1029 |
apply fast |
|
1030 |
apply (subst (asm) AE_iff_null_set) |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
1031 |
apply (intro borel_measurable_pextreal_neq_const) |
40859 | 1032 |
apply fast |
1033 |
by simp |
|
1034 |
then show ?thesis by simp |
|
1035 |
next |
|
1036 |
case (Suc n) |
|
41544 | 1037 |
then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x) \<le> |
1038 |
(\<integral>\<^isup>+x. of_nat (Suc n) * indicator (Q j) x)" |
|
40859 | 1039 |
by (auto intro!: positive_integral_mono simp: indicator_def A_def) |
1040 |
also have "\<dots> = of_nat (Suc n) * \<mu> (Q j)" |
|
1041 |
using Q by (auto intro!: positive_integral_cmult_indicator) |
|
1042 |
also have "\<dots> < \<omega>" |
|
1043 |
using Q by auto |
|
1044 |
finally show ?thesis by simp |
|
1045 |
qed |
|
1046 |
then show "\<nu> (?A n) \<noteq> \<omega>" |
|
1047 |
using A_in_sets Q eq by auto |
|
1048 |
qed |
|
1049 |
qed |
|
1050 |
||
40871 | 1051 |
section "Radon-Nikodym derivative" |
38656 | 1052 |
|
1053 |
definition (in sigma_finite_measure) |
|
1054 |
"RN_deriv \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and> |
|
41544 | 1055 |
(\<forall>A \<in> sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x))" |
38656 | 1056 |
|
40859 | 1057 |
lemma (in sigma_finite_measure) RN_deriv_cong: |
1058 |
assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> \<mu>' A = \<mu> A" "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A" |
|
1059 |
shows "sigma_finite_measure.RN_deriv M \<mu>' \<nu>' x = RN_deriv \<nu> x" |
|
1060 |
proof - |
|
1061 |
interpret \<mu>': sigma_finite_measure M \<mu>' |
|
1062 |
using cong(1) by (rule sigma_finite_measure_cong) |
|
1063 |
show ?thesis |
|
1064 |
unfolding RN_deriv_def \<mu>'.RN_deriv_def |
|
1065 |
by (simp add: cong positive_integral_cong_measure[OF cong(1)]) |
|
1066 |
qed |
|
1067 |
||
38656 | 1068 |
lemma (in sigma_finite_measure) RN_deriv: |
1069 |
assumes "measure_space M \<nu>" |
|
1070 |
assumes "absolutely_continuous \<nu>" |
|
1071 |
shows "RN_deriv \<nu> \<in> borel_measurable M" (is ?borel) |
|
41544 | 1072 |
and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. RN_deriv \<nu> x * indicator A x)" |
38656 | 1073 |
(is "\<And>A. _ \<Longrightarrow> ?int A") |
1074 |
proof - |
|
1075 |
note Ex = Radon_Nikodym[OF assms, unfolded Bex_def] |
|
1076 |
thus ?borel unfolding RN_deriv_def by (rule someI2_ex) auto |
|
1077 |
fix A assume "A \<in> sets M" |
|
1078 |
from Ex show "?int A" unfolding RN_deriv_def |
|
1079 |
by (rule someI2_ex) (simp add: `A \<in> sets M`) |
|
1080 |
qed |
|
1081 |
||
40859 | 1082 |
lemma (in sigma_finite_measure) RN_deriv_positive_integral: |
1083 |
assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>" |
|
1084 |
and f: "f \<in> borel_measurable M" |
|
41544 | 1085 |
shows "measure_space.positive_integral M \<nu> f = (\<integral>\<^isup>+x. RN_deriv \<nu> x * f x)" |
40859 | 1086 |
proof - |
1087 |
interpret \<nu>: measure_space M \<nu> by fact |
|
1088 |
have "\<nu>.positive_integral f = |
|
41544 | 1089 |
measure_space.positive_integral M (\<lambda>A. (\<integral>\<^isup>+x. RN_deriv \<nu> x * indicator A x)) f" |
40859 | 1090 |
by (intro \<nu>.positive_integral_cong_measure[symmetric] RN_deriv(2)[OF \<nu>, symmetric]) |
41544 | 1091 |
also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv \<nu> x * f x)" |
40859 | 1092 |
by (intro positive_integral_translated_density RN_deriv[OF \<nu>] f) |
1093 |
finally show ?thesis . |
|
1094 |
qed |
|
1095 |
||
1096 |
lemma (in sigma_finite_measure) RN_deriv_unique: |
|
1097 |
assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>" |
|
1098 |
and f: "f \<in> borel_measurable M" |
|
41544 | 1099 |
and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x)" |
40859 | 1100 |
shows "AE x. f x = RN_deriv \<nu> x" |
1101 |
proof (rule density_unique[OF f RN_deriv(1)[OF \<nu>]]) |
|
1102 |
fix A assume A: "A \<in> sets M" |
|
41544 | 1103 |
show "(\<integral>\<^isup>+x. f x * indicator A x) = (\<integral>\<^isup>+x. RN_deriv \<nu> x * indicator A x)" |
40859 | 1104 |
unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] .. |
1105 |
qed |
|
1106 |
||
1107 |
lemma (in sigma_finite_measure) RN_deriv_vimage: |
|
1108 |
fixes f :: "'b \<Rightarrow> 'a" |
|
41095 | 1109 |
assumes f: "bij_inv S (space M) f g" |
40859 | 1110 |
assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>" |
1111 |
shows "AE x. |
|
41095 | 1112 |
sigma_finite_measure.RN_deriv (vimage_algebra S f) (\<lambda>A. \<mu> (f ` A)) (\<lambda>A. \<nu> (f ` A)) (g x) = RN_deriv \<nu> x" |
40859 | 1113 |
proof (rule RN_deriv_unique[OF \<nu>]) |
1114 |
interpret sf: sigma_finite_measure "vimage_algebra S f" "\<lambda>A. \<mu> (f ` A)" |
|
41095 | 1115 |
using f by (rule sigma_finite_measure_isomorphic[OF bij_inv_bij_betw(1)]) |
40859 | 1116 |
interpret \<nu>: measure_space M \<nu> using \<nu>(1) . |
1117 |
have \<nu>': "measure_space (vimage_algebra S f) (\<lambda>A. \<nu> (f ` A))" |
|
41095 | 1118 |
using f by (rule \<nu>.measure_space_isomorphic[OF bij_inv_bij_betw(1)]) |
40859 | 1119 |
{ fix A assume "A \<in> sets M" then have "f ` (f -` A \<inter> S) = A" |
41095 | 1120 |
using sets_into_space f[THEN bij_inv_bij_betw(1), unfolded bij_betw_def] |
40859 | 1121 |
by (intro image_vimage_inter_eq[where T="space M"]) auto } |
1122 |
note A_f = this |
|
1123 |
then have ac: "sf.absolutely_continuous (\<lambda>A. \<nu> (f ` A))" |
|
1124 |
using \<nu>(2) by (auto simp: sf.absolutely_continuous_def absolutely_continuous_def) |
|
41095 | 1125 |
show "(\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x)) \<in> borel_measurable M" |
40859 | 1126 |
using sf.RN_deriv(1)[OF \<nu>' ac] |
1127 |
unfolding measurable_vimage_iff_inv[OF f] comp_def . |
|
1128 |
fix A assume "A \<in> sets M" |
|
41095 | 1129 |
then have *: "\<And>x. x \<in> space M \<Longrightarrow> indicator (f -` A \<inter> S) (g x) = (indicator A x :: pextreal)" |
1130 |
using f by (auto simp: bij_inv_def indicator_def) |
|
40859 | 1131 |
have "\<nu> (f ` (f -` A \<inter> S)) = sf.positive_integral (\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) x * indicator (f -` A \<inter> S) x)" |
1132 |
using `A \<in> sets M` by (force intro!: sf.RN_deriv(2)[OF \<nu>' ac]) |
|
41544 | 1133 |
also have "\<dots> = (\<integral>\<^isup>+x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x) * indicator A x)" |
40859 | 1134 |
unfolding positive_integral_vimage_inv[OF f] |
1135 |
by (simp add: * cong: positive_integral_cong) |
|
41544 | 1136 |
finally show "\<nu> A = (\<integral>\<^isup>+x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x) * indicator A x)" |
40859 | 1137 |
unfolding A_f[OF `A \<in> sets M`] . |
1138 |
qed |
|
1139 |
||
1140 |
lemma (in sigma_finite_measure) RN_deriv_finite: |
|
1141 |
assumes sfm: "sigma_finite_measure M \<nu>" and ac: "absolutely_continuous \<nu>" |
|
1142 |
shows "AE x. RN_deriv \<nu> x \<noteq> \<omega>" |
|
1143 |
proof - |
|
1144 |
interpret \<nu>: sigma_finite_measure M \<nu> by fact |
|
1145 |
have \<nu>: "measure_space M \<nu>" by default |
|
1146 |
from sfm show ?thesis |
|
1147 |
using sigma_finite_iff_density_finite[OF \<nu> RN_deriv[OF \<nu> ac]] by simp |
|
1148 |
qed |
|
1149 |
||
1150 |
lemma (in sigma_finite_measure) |
|
1151 |
assumes \<nu>: "sigma_finite_measure M \<nu>" "absolutely_continuous \<nu>" |
|
1152 |
and f: "f \<in> borel_measurable M" |
|
41544 | 1153 |
shows RN_deriv_integral: "measure_space.integral M \<nu> f = (\<integral>x. real (RN_deriv \<nu> x) * f x)" (is ?integral) |
40859 | 1154 |
and RN_deriv_integrable: "measure_space.integrable M \<nu> f \<longleftrightarrow> integrable (\<lambda>x. real (RN_deriv \<nu> x) * f x)" (is ?integrable) |
1155 |
proof - |
|
1156 |
interpret \<nu>: sigma_finite_measure M \<nu> by fact |
|
1157 |
have ms: "measure_space M \<nu>" by default |
|
41023
9118eb4eb8dc
it is known as the extended reals, not the infinite reals
hoelzl
parents:
40871
diff
changeset
|
1158 |
have minus_cong: "\<And>A B A' B'::pextreal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A - real B = real A' - real B'" by simp |
40859 | 1159 |
have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto |
1160 |
{ fix f :: "'a \<Rightarrow> real" assume "f \<in> borel_measurable M" |
|
1161 |
{ fix x assume *: "RN_deriv \<nu> x \<noteq> \<omega>" |
|
1162 |
have "Real (real (RN_deriv \<nu> x)) * Real (f x) = Real (real (RN_deriv \<nu> x) * f x)" |
|
1163 |
by (simp add: mult_le_0_iff) |
|
1164 |
then have "RN_deriv \<nu> x * Real (f x) = Real (real (RN_deriv \<nu> x) * f x)" |
|
1165 |
using * by (simp add: Real_real) } |
|
1166 |
note * = this |
|
41544 | 1167 |
have "(\<integral>\<^isup>+x. RN_deriv \<nu> x * Real (f x)) = (\<integral>\<^isup>+x. Real (real (RN_deriv \<nu> x) * f x))" |
40859 | 1168 |
apply (rule positive_integral_cong_AE) |
1169 |
apply (rule AE_mp[OF RN_deriv_finite[OF \<nu>]]) |
|
1170 |
by (auto intro!: AE_cong simp: *) } |
|
1171 |
with this[OF f] this[OF f'] f f' |
|
1172 |
show ?integral ?integrable |
|
1173 |
unfolding \<nu>.integral_def integral_def \<nu>.integrable_def integrable_def |
|
1174 |
by (auto intro!: RN_deriv(1)[OF ms \<nu>(2)] minus_cong simp: RN_deriv_positive_integral[OF ms \<nu>(2)]) |
|
1175 |
qed |
|
1176 |
||
38656 | 1177 |
lemma (in sigma_finite_measure) RN_deriv_singleton: |
1178 |
assumes "measure_space M \<nu>" |
|
1179 |
and ac: "absolutely_continuous \<nu>" |
|
1180 |
and "{x} \<in> sets M" |
|
1181 |
shows "\<nu> {x} = RN_deriv \<nu> x * \<mu> {x}" |
|
1182 |
proof - |
|
1183 |
note deriv = RN_deriv[OF assms(1, 2)] |
|
1184 |
from deriv(2)[OF `{x} \<in> sets M`] |
|
41544 | 1185 |
have "\<nu> {x} = (\<integral>\<^isup>+w. RN_deriv \<nu> x * indicator {x} w)" |
38656 | 1186 |
by (auto simp: indicator_def intro!: positive_integral_cong) |
1187 |
thus ?thesis using positive_integral_cmult_indicator[OF `{x} \<in> sets M`] |
|
1188 |
by auto |
|
1189 |
qed |
|
1190 |
||
1191 |
theorem (in finite_measure_space) RN_deriv_finite_measure: |
|
1192 |
assumes "measure_space M \<nu>" |
|
1193 |
and ac: "absolutely_continuous \<nu>" |
|
1194 |
and "x \<in> space M" |
|
1195 |
shows "\<nu> {x} = RN_deriv \<nu> x * \<mu> {x}" |
|
1196 |
proof - |
|
1197 |
have "{x} \<in> sets M" using sets_eq_Pow `x \<in> space M` by auto |
|
1198 |
from RN_deriv_singleton[OF assms(1,2) this] show ?thesis . |
|
1199 |
qed |
|
1200 |
||
1201 |
end |