author hoelzl Wed Dec 08 19:32:11 2010 +0100 (2010-12-08) changeset 41097 a1abfa4e2b44 parent 41095 c335d880ff82 child 41544 c3b977fee8a3 permissions -rw-r--r--
use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
2 imports Lebesgue_Integration
3 begin
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
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
32       then have "inverse (?B i) \<noteq> 0" unfolding pextreal_inverse_eq_0 by simp
33       then show ?thesis using measure[of i] not_0
34         by (auto intro!: exI[of _ "inverse (?B i) / 2"]
35                  simp: pextreal_noteq_omega_Ex zero_le_mult_iff zero_less_mult_iff mult_le_0_iff power_le_zero_eq)
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)
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)
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]
51         by (cases "n N")
52            (auto simp: pextreal_noteq_omega_Ex field_simps zero_le_mult_iff
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)
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
68       by (auto intro!: borel_measurable_psuminf borel_measurable_pextreal_times)
69   qed
70 qed
72 subsection "Absolutely continuous"
74 definition (in measure_space)
75   "absolutely_continuous \<nu> = (\<forall>N\<in>null_sets. \<nu> N = (0 :: pextreal))"
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
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
116 lemma (in measure_space) density_is_absolutely_continuous:
117   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
118   shows "absolutely_continuous \<nu>"
119   using assms unfolding absolutely_continuous_def
120   by (simp add: positive_integral_null_set)
122 subsection "Existence of the Radon-Nikodym derivative"
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
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
227   let "?r S" = "restricted_space S"
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
289 lemma (in finite_measure) Radon_Nikodym_finite_measure:
290   assumes "finite_measure M \<nu>"
291   assumes "absolutely_continuous \<nu>"
292   shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
293 proof -
294   interpret M': finite_measure M \<nu> using assms(1) .
295   def G \<equiv> "{g \<in> borel_measurable M. \<forall>A\<in>sets M. positive_integral (\<lambda>x. g x * indicator A x) \<le> \<nu> A}"
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)
311       hence "positive_integral (\<lambda>x. max (g x) (f x) * indicator A x) =
312         positive_integral (\<lambda>x. g x * indicator (?A \<inter> A) x) +
313         positive_integral (\<lambda>x. f x * indicator ((space M - ?A) \<inter> A) x)"
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
320       finally show "positive_integral (\<lambda>x. max (g x) (f x) * indicator A x) \<le> \<nu> A" .
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
326       from `f \<up> g` have [simp]: "g = (\<lambda>x. SUP i. f i x)"
327         unfolding isoton_def fun_eq_iff SUPR_apply by simp
328       have f_borel: "\<And>i. f i \<in> borel_measurable M" using f unfolding G_def by simp
329       thus "g \<in> borel_measurable M" by auto
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]
334       have SUP: "positive_integral (\<lambda>x. g x * indicator A x) =
335           (SUP i. positive_integral (\<lambda>x. f i x * indicator A x))"
336         unfolding isoton_def by simp
337       show "positive_integral (\<lambda>x. g x * indicator A x) \<le> \<nu> A" unfolding SUP
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)
344     fix g assume "\<forall>A\<in>sets M. positive_integral (\<lambda>x. g x * indicator A x) \<le> \<nu> A"
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" .
387   let "?t A" = "\<nu> A - positive_integral (\<lambda>x. f x * indicator A x)"
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"
395       have "(\<Sum>\<^isub>\<infinity> n. positive_integral (\<lambda>x. f x * indicator (A n) x))
396         = positive_integral (\<lambda>x. (\<Sum>\<^isub>\<infinity>n. f x * indicator (A n) x))"
397         using `range A \<subseteq> sets M`
398         by (rule_tac positive_integral_psuminf[symmetric]) (auto intro!: borel_measurable_indicator)
399       also have "\<dots> = positive_integral (\<lambda>x. f x * indicator (\<Union>n. A n) x)"
400         apply (rule positive_integral_cong)
401         apply (subst psuminf_cmult_right)
402         unfolding psuminf_indicator[OF `disjoint_family A`] ..
403       finally have "(\<Sum>\<^isub>\<infinity> n. positive_integral (\<lambda>x. f x * indicator (A n) x))
404         = positive_integral (\<lambda>x. f x * indicator (\<Union>n. A n) x)" .
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)
407       moreover have "\<And>i. positive_integral (\<lambda>x. f x * indicator (A i) x) \<le> \<nu> (A i)"
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 {
411         have "positive_integral (\<lambda>x. f x * indicator (\<Union>i. A i) x) \<le> \<nu> (\<Union>i. A i)"
412           using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
413         also have "\<nu> (\<Union>i. A i) < \<omega>" using v_fin by (simp add: pextreal_less_\<omega>)
414         finally have "positive_integral (\<lambda>x. f x * indicator (\<Union>i. A i) x) \<noteq> \<omega>"
415           by (simp add: pextreal_less_\<omega>) }
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
436     have "positive_integral (\<lambda>x. f x * indicator (space M) x) \<le> \<nu> (space M)"
437       using `f \<in> G` unfolding G_def by auto
438     hence "positive_integral (\<lambda>x. f x * indicator (space M) x) \<noteq> \<omega>"
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
444       by (auto simp: pextreal_inverse_eq_0 finite_measure_of_space)
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
465       have "positive_integral (\<lambda>x. ?f0 x  * indicator A x) =
466         positive_integral (\<lambda>x. f x * indicator A x + b * indicator (A \<inter> A0) x)"
467         by (auto intro!: positive_integral_cong simp: field_simps indicator_inter_arith)
468       hence "positive_integral (\<lambda>x. ?f0 x * indicator A x) =
469           positive_integral (\<lambda>x. f x * indicator A x) + b * \<mu> (A \<inter> A0)"
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
475       have f_le_v: "positive_integral (\<lambda>x. f x * indicator A x) \<le> \<nu> A"
476         using `f \<in> G` A unfolding G_def by auto
477       note f0_eq[OF A]
478       also have "positive_integral (\<lambda>x. f x * indicator A x) + b * \<mu> (A \<inter> A0) \<le>
479           positive_integral (\<lambda>x. f x * indicator A x) + ?t (A \<inter> A0)"
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)
482       also have "\<dots> \<le> positive_integral (\<lambda>x. f x * indicator A x) + ?t A"
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`]
487         by (cases "positive_integral (\<lambda>x. f x * indicator A x)", cases "\<nu> A", auto)
488       finally have "positive_integral (\<lambda>x. ?f0 x * indicator A x) \<le> \<nu> A" . }
489     hence "?f0 \<in> G" using `A0 \<in> sets M` unfolding G_def
490       by (auto intro!: borel_measurable_indicator borel_measurable_pextreal_add borel_measurable_pextreal_times)
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>"
498       using M'.finite_measure_of_space pos_t unfolding pextreal_zero_less_diff_iff
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])
503       apply (subst pextreal_mult_inverse)
504       using finite_measure_of_space M'.finite_measure_of_space pos_t pos_M
505       using pextreal_mult_strict_right_mono[of "Real (1/2)" 1 "?t (space M)"]
506       by simp_all
507     hence  "0 < ?t (space M) - b * \<mu> (space M)"
508       by (simp add: pextreal_zero_less_diff_iff)
509     also have "\<dots> \<le> ?t A0 - b * \<mu> A0"
510       using space_less_A0 pos_M pos_t b real[unfolded pextreal_noteq_omega_Ex] by auto
511     finally have "b * \<mu> A0 < ?t A0" unfolding pextreal_zero_less_diff_iff .
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
518       by (rule pextreal_less_add)
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"
528     show "\<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
529     proof (rule antisym)
530       show "positive_integral (\<lambda>x. f x * indicator A x) \<le> \<nu> A"
531         using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
532       show "\<nu> A \<le> positive_integral (\<lambda>x. f x * indicator A x)"
533         using upper_bound[THEN bspec, OF `A \<in> sets M`]
534          by (simp add: pextreal_zero_le_diff)
535     qed
536   qed simp
537 qed
539 lemma (in finite_measure) split_space_into_finite_sets_and_rest:
540   assumes "measure_space M \<nu>"
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>)"
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
577     also have "\<dots> < \<omega>" unfolding setsum_\<omega> pextreal_less_\<omega> using Q' by auto
578     finally show "\<nu> (?O i) \<noteq> \<omega>" unfolding pextreal_less_\<omega> by auto
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
596   def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n"
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
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`]
638             by (cases "\<mu> A") (auto simp: pextreal_noteq_omega_Ex)
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
669 lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
670   assumes "measure_space M \<nu>"
671   assumes "absolutely_continuous \<nu>"
672   shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
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
682   have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
683     \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f x * indicator (Q i \<inter> A) x))"
684   proof
685     fix i
686     have indicator_eq: "\<And>f x A. (f x :: pextreal) * indicator (Q i \<inter> A) x * indicator (Q i) x
687       = (f x * indicator (Q i) x) * indicator A x"
688       unfolding indicator_def by auto
689     have fm: "finite_measure (restricted_space (Q i)) \<mu>"
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>"
698         using Q_fin by simp
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"
705       and f_int: "\<And>A. A\<in>sets M \<Longrightarrow> \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. (f x * indicator (Q i) x) * indicator A x)"
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.
709       \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f x * indicator (Q i \<inter> A) x))"
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>
715       \<nu> (Q i \<inter> A) = positive_integral (\<lambda>x. f i x * indicator (Q i \<inter> A) x)"
716     by auto
717   let "?f x" =
718     "(\<Sum>\<^isub>\<infinity> i. f i x * indicator (Q i) x) + \<omega> * indicator Q0 x"
719   show ?thesis
720   proof (safe intro!: bexI[of _ ?f])
721     show "?f \<in> borel_measurable M"
722       by (safe intro!: borel_measurable_psuminf borel_measurable_pextreal_times
724         borel_measurable_const borel Q_sets Q0 Diff countable_UN)
725     fix A assume "A \<in> sets M"
726     have *:
727       "\<And>x i. indicator A x * (f i x * indicator (Q i) x) =
728         f i x * indicator (Q i \<inter> A) x"
729       "\<And>x i. (indicator A x * indicator Q0 x :: pextreal) =
730         indicator (Q0 \<inter> A) x" by (auto simp: indicator_def)
731     have "positive_integral (\<lambda>x. ?f x * indicator A x) =
732       (\<Sum>\<^isub>\<infinity> i. \<nu> (Q i \<inter> A)) + \<omega> * \<mu> (Q0 \<inter> A)"
733       unfolding f[OF `A \<in> sets M`]
734       apply (simp del: pextreal_times(2) add: field_simps *)
735       apply (subst positive_integral_add)
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`)
740       unfolding psuminf_cmult_right[symmetric]
741       apply (subst positive_integral_psuminf)
742       apply (fastsimp intro: `A \<in> sets M` Q_sets borel)
743       apply (simp add: *)
744       done
745     moreover have "(\<Sum>\<^isub>\<infinity>i. \<nu> (Q i \<inter> A)) = \<nu> ((\<Union>i. Q i) \<inter> A)"
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
753     qed
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
758     ultimately show "\<nu> A = positive_integral (\<lambda>x. ?f x * indicator A x)"
759       using v.measure_additive[simplified, of "(\<Union>i. Q i) \<inter> A" "Q0 \<inter> A"]
760       by simp
761   qed
762 qed
764 lemma (in sigma_finite_measure) Radon_Nikodym:
765   assumes "measure_space M \<nu>"
766   assumes "absolutely_continuous \<nu>"
767   shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
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
774   let "?T A" = "positive_integral (\<lambda>x. h x * indicator A x)"
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)
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"
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"
790       using borel f_borel by (auto intro: borel_measurable_pextreal_times)
791     fix A assume "A \<in> sets M"
792     then have "(\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
793       using f_borel by (auto intro: borel_measurable_pextreal_times borel_measurable_indicator)
794     from positive_integral_translated_density[OF borel this]
795     show "\<nu> A = positive_integral (\<lambda>x. h x * f x * indicator A x)"
796       unfolding fT[OF `A \<in> sets M`] by (simp add: field_simps)
797   qed
798 qed
800 section "Uniqueness of densities"
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>"
805   shows "(\<forall>A\<in>sets M. positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. g x * indicator A x))
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
820     have "?P (\<lambda>x. (f x - g x)) ?N = positive_integral (\<lambda>x. f x * indicator ?N x - g x * indicator ?N x)"
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"
838       by (auto simp: pextreal_zero_le_diff)
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
849 lemma (in finite_measure) density_unique_finite_measure:
850   assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
851   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. f' x * indicator A x)"
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
870   have *: "\<And>i x A. \<And>y::pextreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
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)
875        (auto intro!: borel_measurable_pextreal_times f Int simp: *)
876   have 2: "AE x. ?f Q0 x = ?f' Q0 x"
877   proof (rule AE_I')
878     { fix f :: "'a \<Rightarrow> pextreal" assume borel: "f \<in> borel_measurable M"
879         and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
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
885         have "?\<nu> (?A i) \<le> positive_integral (\<lambda>x. of_nat i * indicator (?A i) x)"
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)
897       finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>} \<in> null_sets" by (simp add: pextreal_less_\<omega>) }
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
913 lemma (in sigma_finite_measure) density_unique:
914   assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
915   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. f' x * indicator A x)"
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
922   interpret h: measure_space M "\<lambda>A. positive_integral (\<lambda>x. h x * indicator A x)"
923     using h_borel by (rule measure_space_density)
924   interpret h: finite_measure M "\<lambda>A. positive_integral (\<lambda>x. h x * indicator A x)"
925     by default (simp cong: positive_integral_cong add: fin)
926   interpret f: measure_space M "\<lambda>A. positive_integral (\<lambda>x. f x * indicator A x)"
927     using borel(1) by (rule measure_space_density)
928   interpret f': measure_space M "\<lambda>A. positive_integral (\<lambda>x. f' x * indicator A x)"
929     using borel(2) by (rule measure_space_density)
930   { fix A assume "A \<in> sets M"
931     then have " {x \<in> space M. h x \<noteq> 0 \<and> indicator A x \<noteq> (0::pextreal)} = A"
932       using pos sets_into_space by (force simp: indicator_def)
933     then have "positive_integral (\<lambda>xa. h xa * indicator A xa) = 0 \<longleftrightarrow> A \<in> null_sets"
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"
937     have "positive_integral (\<lambda>x. h x * (f x * indicator A x)) =
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)
943     also have "\<dots> = positive_integral (\<lambda>x. h x * (f' x * indicator A x))"
944       using `A \<in> sets M` h_borel borel
945       by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
946     finally have "positive_integral (\<lambda>x. h x * (f x * indicator A x)) = positive_integral (\<lambda>x. h x * (f' x * indicator A x))" . }
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
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"
958     and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
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')
968     have "\<nu>.positive_integral h = positive_integral (\<lambda>x. f x * h x)"
969       by (simp add: \<nu>.positive_integral_cong_measure[symmetric, OF eq[symmetric]])
970          (intro positive_integral_translated_density f h)
971     then have "positive_integral (\<lambda>x. f x * h x) \<noteq> \<omega>"
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
1021     have "positive_integral (\<lambda>x. f x * indicator (A i \<inter> Q j) x) \<noteq> \<omega>"
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`)
1026       then have "positive_integral (\<lambda>x. f x * indicator (A i \<inter> Q j) x) = 0"
1027         using A_in_sets f
1028         apply (subst positive_integral_0_iff)
1029         apply fast
1030         apply (subst (asm) AE_iff_null_set)
1031         apply (intro borel_measurable_pextreal_neq_const)
1032         apply fast
1033         by simp
1034       then show ?thesis by simp
1035     next
1036       case (Suc n)
1037       then have "positive_integral (\<lambda>x. f x * indicator (A i \<inter> Q j) x) \<le>
1038         positive_integral (\<lambda>x. of_nat (Suc n) * indicator (Q j) x)"
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
1051 section "Radon-Nikodym derivative"
1053 definition (in sigma_finite_measure)
1054   "RN_deriv \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and>
1055     (\<forall>A \<in> sets M. \<nu> A = positive_integral (\<lambda>x. f x * indicator A x))"
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
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)
1072   and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. RN_deriv \<nu> x * indicator A x)"
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
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"
1085   shows "measure_space.positive_integral M \<nu> f = positive_integral (\<lambda>x. RN_deriv \<nu> x * f x)"
1086 proof -
1087   interpret \<nu>: measure_space M \<nu> by fact
1088   have "\<nu>.positive_integral f =
1089     measure_space.positive_integral M (\<lambda>A. positive_integral (\<lambda>x. RN_deriv \<nu> x * indicator A x)) f"
1090     by (intro \<nu>.positive_integral_cong_measure[symmetric] RN_deriv(2)[OF \<nu>, symmetric])
1091   also have "\<dots> = positive_integral (\<lambda>x. RN_deriv \<nu> x * f x)"
1092     by (intro positive_integral_translated_density RN_deriv[OF \<nu>] f)
1093   finally show ?thesis .
1094 qed
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"
1099   and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = positive_integral (\<lambda>x. f x * indicator A x)"
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"
1103   show "positive_integral (\<lambda>x. f x * indicator A x) = positive_integral (\<lambda>x. RN_deriv \<nu> x * indicator A x)"
1104     unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] ..
1105 qed
1107 lemma (in sigma_finite_measure) RN_deriv_vimage:
1108   fixes f :: "'b \<Rightarrow> 'a"
1109   assumes f: "bij_inv S (space M) f g"
1110   assumes \<nu>: "measure_space M \<nu>" "absolutely_continuous \<nu>"
1111   shows "AE x.
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"
1113 proof (rule RN_deriv_unique[OF \<nu>])
1114   interpret sf: sigma_finite_measure "vimage_algebra S f" "\<lambda>A. \<mu> (f ` A)"
1115     using f by (rule sigma_finite_measure_isomorphic[OF bij_inv_bij_betw(1)])
1116   interpret \<nu>: measure_space M \<nu> using \<nu>(1) .
1117   have \<nu>': "measure_space (vimage_algebra S f) (\<lambda>A. \<nu> (f ` A))"
1118     using f by (rule \<nu>.measure_space_isomorphic[OF bij_inv_bij_betw(1)])
1119   { fix A assume "A \<in> sets M" then have "f ` (f -` A \<inter> S) = A"
1120       using sets_into_space f[THEN bij_inv_bij_betw(1), unfolded bij_betw_def]
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)
1125   show "(\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x)) \<in> borel_measurable M"
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"
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)
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])
1133   also have "\<dots> = positive_integral (\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x) * indicator A x)"
1134     unfolding positive_integral_vimage_inv[OF f]
1135     by (simp add: * cong: positive_integral_cong)
1136   finally show "\<nu> A = positive_integral (\<lambda>x. sf.RN_deriv (\<lambda>A. \<nu> (f ` A)) (g x) * indicator A x)"
1137     unfolding A_f[OF `A \<in> sets M`] .
1138 qed
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
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"
1153   shows RN_deriv_integral: "measure_space.integral M \<nu> f = integral (\<lambda>x. real (RN_deriv \<nu> x) * f x)" (is ?integral)
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
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
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
1167     have "positive_integral (\<lambda>x. RN_deriv \<nu> x * Real (f x)) = positive_integral (\<lambda>x. Real (real (RN_deriv \<nu> x) * f x))"
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
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`]
1185   have "\<nu> {x} = positive_integral (\<lambda>w. RN_deriv \<nu> x * indicator {x} w)"
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
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
1201 end