author hoelzl Fri Feb 04 14:16:55 2011 +0100 (2011-02-04) changeset 41705 1100512e16d8 parent 41689 3e39b0e730d6 child 41832 27cb9113b1a0 permissions -rw-r--r--
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. integral\<^isup>P M 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 "integral\<^isup>P M ?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 "integral\<^isup>P M ?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'" and [simp]: "sets M' = sets M" "space M' = space M"
79     and "absolutely_continuous (measure M')" "AE x. P x"
80   shows "measure_space.almost_everywhere M' P"
81 proof -
82   interpret \<nu>: measure_space M' by fact
83   from `AE x. P x` obtain N where N: "N \<in> null_sets" and "{x\<in>space M. \<not> P x} \<subseteq> N"
84     unfolding almost_everywhere_def by auto
85   show "\<nu>.almost_everywhere P"
86   proof (rule \<nu>.AE_I')
87     show "{x\<in>space M'. \<not> P x} \<subseteq> N" by simp fact
88     from `absolutely_continuous (measure M')` show "N \<in> \<nu>.null_sets"
89       using N unfolding absolutely_continuous_def by auto
90   qed
91 qed
93 lemma (in finite_measure_space) absolutely_continuousI:
94   assumes "finite_measure_space (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure_space ?\<nu>")
95   assumes v: "\<And>x. \<lbrakk> x \<in> space M ; \<mu> {x} = 0 \<rbrakk> \<Longrightarrow> \<nu> {x} = 0"
96   shows "absolutely_continuous \<nu>"
97 proof (unfold absolutely_continuous_def sets_eq_Pow, safe)
98   fix N assume "\<mu> N = 0" "N \<subseteq> space M"
99   interpret v: finite_measure_space ?\<nu> by fact
100   have "\<nu> N = measure ?\<nu> (\<Union>x\<in>N. {x})" by simp
101   also have "\<dots> = (\<Sum>x\<in>N. measure ?\<nu> {x})"
103     show "finite N" using `N \<subseteq> space M` finite_space by (auto intro: finite_subset)
104     show "disjoint_family_on (\<lambda>i. {i}) N" unfolding disjoint_family_on_def by auto
105     fix x assume "x \<in> N" thus "{x} \<in> sets ?\<nu>" using `N \<subseteq> space M` sets_eq_Pow by auto
106   qed
107   also have "\<dots> = 0"
108   proof (safe intro!: setsum_0')
109     fix x assume "x \<in> N"
110     hence "\<mu> {x} \<le> \<mu> N" using sets_eq_Pow `N \<subseteq> space M` by (auto intro!: measure_mono)
111     hence "\<mu> {x} = 0" using `\<mu> N = 0` by simp
112     thus "measure ?\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
113   qed
114   finally show "\<nu> N = 0" by simp
115 qed
117 lemma (in measure_space) density_is_absolutely_continuous:
118   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
119   shows "absolutely_continuous \<nu>"
120   using assms unfolding absolutely_continuous_def
123 subsection "Existence of the Radon-Nikodym derivative"
126   fixes e :: real assumes "0 < e"
127   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)"
128   shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (\<nu> (space M)) \<le>
129                     real (\<mu> A) - real (\<nu> A) \<and>
130                     (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < real (\<mu> B) - real (\<nu> B))"
131 proof -
132   let "?d A" = "real (\<mu> A) - real (\<nu> A)"
133   interpret M': finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
134   let "?A A" = "if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B)
135     then {}
136     else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)"
137   def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}"
138   have A_simps[simp]:
139     "A 0 = {}"
140     "\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all
141   { fix A assume "A \<in> sets M"
142     have "?A A \<in> sets M"
143       by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) }
144   note A'_in_sets = this
145   { fix n have "A n \<in> sets M"
146     proof (induct n)
147       case (Suc n) thus "A (Suc n) \<in> sets M"
148         using A'_in_sets[of "A n"] by (auto split: split_if_asm)
149     qed (simp add: A_def) }
150   note A_in_sets = this
151   hence "range A \<subseteq> sets M" by auto
152   { fix n B
153     assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> -e"
154     hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B)" by (auto simp: not_less)
155     have "?d (A (Suc n)) \<le> ?d (A n) - e" unfolding A_simps if_not_P[OF False]
156     proof (rule someI2_ex[OF Ex])
157       fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
158       hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto
159       hence "?d (A n \<union> B) = ?d (A n) + ?d B"
160         using `A n \<in> sets M` real_finite_measure_Union M'.real_finite_measure_Union by simp
161       also have "\<dots> \<le> ?d (A n) - e" using dB by simp
162       finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
163     qed }
164   note dA_epsilon = this
165   { fix n have "?d (A (Suc n)) \<le> ?d (A n)"
166     proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e")
167       case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp
168     next
169       case False
170       hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le)
171       thus ?thesis by simp
172     qed }
173   note dA_mono = this
174   show ?thesis
175   proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B")
176     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
177     show ?thesis
178     proof (safe intro!: bexI[of _ "space M - A n"])
179       fix B assume "B \<in> sets M" "B \<subseteq> space M - A n"
180       from B[OF this] show "-e < ?d B" .
181     next
182       show "space M - A n \<in> sets M" by (rule compl_sets) fact
183     next
184       show "?d (space M) \<le> ?d (space M - A n)"
185       proof (induct n)
186         fix n assume "?d (space M) \<le> ?d (space M - A n)"
187         also have "\<dots> \<le> ?d (space M - A (Suc n))"
188           using A_in_sets sets_into_space dA_mono[of n]
189             real_finite_measure_Diff[of "space M"]
190             real_finite_measure_Diff[of "space M"]
191             M'.real_finite_measure_Diff[of "space M"]
192             M'.real_finite_measure_Diff[of "space M"]
193           by (simp del: A_simps)
194         finally show "?d (space M) \<le> ?d (space M - A (Suc n))" .
195       qed simp
196     qed
197   next
198     case False hence B: "\<And>n. \<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
199       by (auto simp add: not_less)
200     { fix n have "?d (A n) \<le> - real n * e"
201       proof (induct n)
202         case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps)
203       qed simp } note dA_less = this
204     have decseq: "decseq (\<lambda>n. ?d (A n))" unfolding decseq_eq_incseq
205     proof (rule incseq_SucI)
206       fix n show "- ?d (A n) \<le> - ?d (A (Suc n))" using dA_mono[of n] by auto
207     qed
208     from real_finite_continuity_from_below[of A] `range A \<subseteq> sets M`
209       M'.real_finite_continuity_from_below[of A]
210     have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
211       by (auto intro!: LIMSEQ_diff)
212     obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
213     moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less]
214     have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps)
215     ultimately show ?thesis by auto
216   qed
217 qed
220   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
221   shows "\<exists>A\<in>sets M. real (\<mu> (space M)) - real (\<nu> (space M)) \<le>
222                     real (\<mu> A) - real (\<nu> A) \<and>
223                     (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> real (\<mu> B) - real (\<nu> B))"
224 proof -
225   let "?d A" = "real (\<mu> A) - real (\<nu> A)"
226   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)"
227   interpret M': finite_measure ?M' where
228     "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>" by fact auto
229   let "?r S" = "restricted_space S"
230   { fix S n
231     assume S: "S \<in> sets M"
232     hence **: "\<And>X. X \<in> op \<inter> S ` sets M \<longleftrightarrow> X \<in> sets M \<and> X \<subseteq> S" by auto
233     have [simp]: "(restricted_space S\<lparr>measure := \<nu>\<rparr>) = M'.restricted_space S"
234       by (cases M) simp
235     from M'.restricted_finite_measure[of S] restricted_finite_measure[OF S] S
236     have "finite_measure (?r S)" "0 < 1 / real (Suc n)"
237       "finite_measure (?r S\<lparr>measure := \<nu>\<rparr>)" by auto
238     from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X ..
239     hence "?P X S n"
240     proof (simp add: **, safe)
241       fix C assume C: "C \<in> sets M" "C \<subseteq> X" "X \<subseteq> S" and
242         *: "\<forall>B\<in>sets M. S \<inter> B \<subseteq> X \<longrightarrow> - (1 / real (Suc n)) < ?d (S \<inter> B)"
243       hence "C \<subseteq> S" "C \<subseteq> X" "S \<inter> C = C" by auto
244       with *[THEN bspec, OF `C \<in> sets M`]
245       show "- (1 / real (Suc n)) < ?d C" by auto
246     qed
247     hence "\<exists>A. ?P A S n" by auto }
248   note Ex_P = this
249   def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)"
250   have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def)
251   have A_0[simp]: "A 0 = space M" unfolding A_def by simp
252   { fix i have "A i \<in> sets M" unfolding A_def
253     proof (induct i)
254       case (Suc i)
255       from Ex_P[OF this, of i] show ?case unfolding nat_rec_Suc
256         by (rule someI2_ex) simp
257     qed simp }
258   note A_in_sets = this
259   { fix n have "?P (A (Suc n)) (A n) n"
260       using Ex_P[OF A_in_sets] unfolding A_Suc
261       by (rule someI2_ex) simp }
262   note P_A = this
263   have "range A \<subseteq> sets M" using A_in_sets by auto
264   have A_mono: "\<And>i. A (Suc i) \<subseteq> A i" using P_A by simp
265   have mono_dA: "mono (\<lambda>i. ?d (A i))" using P_A by (simp add: mono_iff_le_Suc)
266   have epsilon: "\<And>C i. \<lbrakk> C \<in> sets M; C \<subseteq> A (Suc i) \<rbrakk> \<Longrightarrow> - 1 / real (Suc i) < ?d C"
267       using P_A by auto
268   show ?thesis
269   proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
270     show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto
271     from `range A \<subseteq> sets M` A_mono
272       real_finite_continuity_from_above[of A]
273       M'.real_finite_continuity_from_above[of A]
274     have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (auto intro!: LIMSEQ_diff)
275     thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _]
276       by (rule_tac LIMSEQ_le_const) (auto intro!: exI)
277   next
278     fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)"
279     show "0 \<le> ?d B"
280     proof (rule ccontr)
281       assume "\<not> 0 \<le> ?d B"
282       hence "0 < - ?d B" by auto
283       from ex_inverse_of_nat_Suc_less[OF this]
284       obtain n where *: "?d B < - 1 / real (Suc n)"
285         by (auto simp: real_eq_of_nat inverse_eq_divide field_simps)
286       have "B \<subseteq> A (Suc n)" using B by (auto simp del: nat_rec_Suc)
287       from epsilon[OF B(1) this] *
288       show False by auto
289     qed
290   qed
291 qed
294   assumes "finite_measure (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
295   assumes "absolutely_continuous \<nu>"
296   shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
297 proof -
298   interpret M': finite_measure ?M'
299     where "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>"
300     using assms(1) by auto
301   def G \<equiv> "{g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A}"
302   have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
303   hence "G \<noteq> {}" by auto
304   { fix f g assume f: "f \<in> G" and g: "g \<in> G"
305     have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def
306     proof safe
307       show "?max \<in> borel_measurable M" using f g unfolding G_def by auto
308       let ?A = "{x \<in> space M. f x \<le> g x}"
309       have "?A \<in> sets M" using f g unfolding G_def by auto
310       fix A assume "A \<in> sets M"
311       hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto
312       have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
313         using sets_into_space[OF `A \<in> sets M`] by auto
314       have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
315         g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
316         by (auto simp: indicator_def max_def)
317       hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) =
318         (\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x \<partial>M) +
319         (\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)"
320         using f g sets unfolding G_def
321         by (auto cong: positive_integral_cong intro!: positive_integral_add borel_measurable_indicator)
322       also have "\<dots> \<le> \<nu> (?A \<inter> A) + \<nu> ((space M - ?A) \<inter> A)"
323         using f g sets unfolding G_def by (auto intro!: add_mono)
324       also have "\<dots> = \<nu> A"
325         using M'.measure_additive[OF sets] union by auto
326       finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> \<nu> A" .
327     qed }
328   note max_in_G = this
329   { fix f g assume  "f \<up> g" and f: "\<And>i. f i \<in> G"
330     have "g \<in> G" unfolding G_def
331     proof safe
332       from `f \<up> g` have [simp]: "g = (\<lambda>x. SUP i. f i x)"
333         unfolding isoton_def fun_eq_iff SUPR_apply by simp
334       have f_borel: "\<And>i. f i \<in> borel_measurable M" using f unfolding G_def by simp
335       thus "g \<in> borel_measurable M" by auto
336       fix A assume "A \<in> sets M"
337       hence "\<And>i. (\<lambda>x. f i x * indicator A x) \<in> borel_measurable M"
338         using f_borel by (auto intro!: borel_measurable_indicator)
339       from positive_integral_isoton[OF isoton_indicator[OF `f \<up> g`] this]
340       have SUP: "(\<integral>\<^isup>+x. g x * indicator A x \<partial>M) =
341           (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x \<partial>M))"
342         unfolding isoton_def by simp
343       show "(\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A" unfolding SUP
344         using f `A \<in> sets M` unfolding G_def by (auto intro!: SUP_leI)
345     qed }
346   note SUP_in_G = this
347   let ?y = "SUP g : G. integral\<^isup>P M g"
348   have "?y \<le> \<nu> (space M)" unfolding G_def
349   proof (safe intro!: SUP_leI)
350     fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A"
351     from this[THEN bspec, OF top] show "integral\<^isup>P M g \<le> \<nu> (space M)"
352       by (simp cong: positive_integral_cong)
353   qed
354   hence "?y \<noteq> \<omega>" using M'.finite_measure_of_space by auto
355   from SUPR_countable_SUPR[OF this `G \<noteq> {}`] guess ys .. note ys = this
356   hence "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n"
357   proof safe
358     fix n assume "range ys \<subseteq> integral\<^isup>P M ` G"
359     hence "ys n \<in> integral\<^isup>P M ` G" by auto
360     thus "\<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n" by auto
361   qed
362   from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^isup>P M (gs n) = ys n" by auto
363   hence y_eq: "?y = (SUP i. integral\<^isup>P M (gs i))" using ys by auto
364   let "?g i x" = "Max ((\<lambda>n. gs n x) ` {..i})"
365   def f \<equiv> "SUP i. ?g i"
366   have gs_not_empty: "\<And>i. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
367   { fix i have "?g i \<in> G"
368     proof (induct i)
369       case 0 thus ?case by simp fact
370     next
371       case (Suc i)
372       with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case
373         by (auto simp add: atMost_Suc intro!: max_in_G)
374     qed }
375   note g_in_G = this
376   have "\<And>x. \<forall>i. ?g i x \<le> ?g (Suc i) x"
377     using gs_not_empty by (simp add: atMost_Suc)
378   hence isoton_g: "?g \<up> f" by (simp add: isoton_def le_fun_def f_def)
379   from SUP_in_G[OF this g_in_G] have "f \<in> G" .
380   hence [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
381   have "(\<lambda>i. integral\<^isup>P M (?g i)) \<up> integral\<^isup>P M f"
382     using isoton_g g_in_G by (auto intro!: positive_integral_isoton simp: G_def f_def)
383   hence "integral\<^isup>P M f = (SUP i. integral\<^isup>P M (?g i))"
384     unfolding isoton_def by simp
385   also have "\<dots> = ?y"
386   proof (rule antisym)
387     show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y"
388       using g_in_G by (auto intro!: exI Sup_mono simp: SUPR_def)
389     show "?y \<le> (SUP i. integral\<^isup>P M (?g i))" unfolding y_eq
390       by (auto intro!: SUP_mono positive_integral_mono Max_ge)
391   qed
392   finally have int_f_eq_y: "integral\<^isup>P M f = ?y" .
393   let "?t A" = "\<nu> A - (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
394   let ?M = "M\<lparr> measure := ?t\<rparr>"
395   interpret M: sigma_algebra ?M
396     by (intro sigma_algebra_cong) auto
397   have fmM: "finite_measure ?M"
399     fix A :: "nat \<Rightarrow> 'a set"  assume A: "range A \<subseteq> sets M" "disjoint_family A"
400     have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x \<partial>M))
401       = (\<integral>\<^isup>+x. (\<Sum>\<^isub>\<infinity>n. f x * indicator (A n) x) \<partial>M)"
402       using `range A \<subseteq> sets M`
403       by (rule_tac positive_integral_psuminf[symmetric]) (auto intro!: borel_measurable_indicator)
404     also have "\<dots> = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x \<partial>M)"
405       apply (rule positive_integral_cong)
406       apply (subst psuminf_cmult_right)
407       unfolding psuminf_indicator[OF `disjoint_family A`] ..
408     finally have "(\<Sum>\<^isub>\<infinity> n. (\<integral>\<^isup>+x. f x * indicator (A n) x \<partial>M))
409       = (\<integral>\<^isup>+x. f x * indicator (\<Union>n. A n) x \<partial>M)" .
410     moreover have "(\<Sum>\<^isub>\<infinity>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
412     moreover have "\<And>i. (\<integral>\<^isup>+x. f x * indicator (A i) x \<partial>M) \<le> \<nu> (A i)"
413         using A `f \<in> G` unfolding G_def by auto
414     moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<omega>" using M'.finite_measure A by (simp add: countable_UN)
415     moreover {
416       have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x \<partial>M) \<le> \<nu> (\<Union>i. A i)"
417         using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
418       also have "\<nu> (\<Union>i. A i) < \<omega>" using v_fin by (simp add: pextreal_less_\<omega>)
419       finally have "(\<integral>\<^isup>+x. f x * indicator (\<Union>i. A i) x \<partial>M) \<noteq> \<omega>"
420         by (simp add: pextreal_less_\<omega>) }
421     ultimately
422     show "(\<Sum>\<^isub>\<infinity> n. ?t (A n)) = ?t (\<Union>i. A i)"
423       apply (subst psuminf_minus) by simp_all
424   qed
425   then interpret M: finite_measure ?M
426     where "space ?M = space M" and "sets ?M = sets M" and "measure ?M = ?t"
428   have ac: "absolutely_continuous ?t" using `absolutely_continuous \<nu>` unfolding absolutely_continuous_def by auto
429   have upper_bound: "\<forall>A\<in>sets M. ?t A \<le> 0"
430   proof (rule ccontr)
431     assume "\<not> ?thesis"
432     then obtain A where A: "A \<in> sets M" and pos: "0 < ?t A"
433       by (auto simp: not_le)
434     note pos
435     also have "?t A \<le> ?t (space M)"
436       using M.measure_mono[of A "space M"] A sets_into_space by simp
437     finally have pos_t: "0 < ?t (space M)" by simp
438     moreover
439     hence pos_M: "0 < \<mu> (space M)"
440       using ac top unfolding absolutely_continuous_def by auto
441     moreover
442     have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> \<nu> (space M)"
443       using `f \<in> G` unfolding G_def by auto
444     hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<omega>"
445       using M'.finite_measure_of_space by auto
446     moreover
447     def b \<equiv> "?t (space M) / \<mu> (space M) / 2"
448     ultimately have b: "b \<noteq> 0" "b \<noteq> \<omega>"
449       using M'.finite_measure_of_space
450       by (auto simp: pextreal_inverse_eq_0 finite_measure_of_space)
451     let ?Mb = "?M\<lparr>measure := \<lambda>A. b * \<mu> A\<rparr>"
452     interpret b: sigma_algebra ?Mb by (intro sigma_algebra_cong) auto
453     have "finite_measure ?Mb"
454       by default
458     obtain A0 where "A0 \<in> sets M" and
459       space_less_A0: "real (?t (space M)) - real (b * \<mu> (space M)) \<le> real (?t A0) - real (b * \<mu> A0)" and
460       *: "\<And>B. \<lbrakk> B \<in> sets M ; B \<subseteq> A0 \<rbrakk> \<Longrightarrow> 0 \<le> real (?t B) - real (b * \<mu> B)" by auto
461     { fix B assume "B \<in> sets M" "B \<subseteq> A0"
462       with *[OF this] have "b * \<mu> B \<le> ?t B"
463         using M'.finite_measure b finite_measure
464         by (cases "b * \<mu> B", cases "?t B") (auto simp: field_simps) }
465     note bM_le_t = this
466     let "?f0 x" = "f x + b * indicator A0 x"
467     { fix A assume A: "A \<in> sets M"
468       hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
469       have "(\<integral>\<^isup>+x. ?f0 x  * indicator A x \<partial>M) =
470         (\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)"
471         by (auto intro!: positive_integral_cong simp: field_simps indicator_inter_arith)
472       hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) =
473           (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0)"
474         using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A
476     note f0_eq = this
477     { fix A assume A: "A \<in> sets M"
478       hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
479       have f_le_v: "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
480         using `f \<in> G` A unfolding G_def by auto
481       note f0_eq[OF A]
482       also have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0) \<le>
483           (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t (A \<inter> A0)"
484         using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
486       also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t A"
487         using M.measure_mono[simplified, OF _ `A \<inter> A0 \<in> sets M` `A \<in> sets M`]
489       also have "\<dots> \<le> \<nu> A"
490         using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`]
491         by (cases "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M)", cases "\<nu> A", auto)
492       finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> \<nu> A" . }
493     hence "?f0 \<in> G" using `A0 \<in> sets M` unfolding G_def
494       by (auto intro!: borel_measurable_indicator borel_measurable_pextreal_add borel_measurable_pextreal_times)
495     have real: "?t (space M) \<noteq> \<omega>" "?t A0 \<noteq> \<omega>"
496       "b * \<mu> (space M) \<noteq> \<omega>" "b * \<mu> A0 \<noteq> \<omega>"
497       using `A0 \<in> sets M` b
498         finite_measure[of A0] M.finite_measure[of A0]
499         finite_measure_of_space M.finite_measure_of_space
500       by auto
501     have int_f_finite: "integral\<^isup>P M f \<noteq> \<omega>"
502       using M'.finite_measure_of_space pos_t unfolding pextreal_zero_less_diff_iff
503       by (auto cong: positive_integral_cong)
504     have "?t (space M) > b * \<mu> (space M)" unfolding b_def
506       apply (subst mult_assoc[symmetric])
507       apply (subst pextreal_mult_inverse)
508       using finite_measure_of_space M'.finite_measure_of_space pos_t pos_M
509       using pextreal_mult_strict_right_mono[of "Real (1/2)" 1 "?t (space M)"]
510       by simp_all
511     hence  "0 < ?t (space M) - b * \<mu> (space M)"
513     also have "\<dots> \<le> ?t A0 - b * \<mu> A0"
514       using space_less_A0 pos_M pos_t b real[unfolded pextreal_noteq_omega_Ex] by auto
515     finally have "b * \<mu> A0 < ?t A0" unfolding pextreal_zero_less_diff_iff .
516     hence "0 < ?t A0" by auto
517     hence "0 < \<mu> A0" using ac unfolding absolutely_continuous_def
518       using `A0 \<in> sets M` by auto
519     hence "0 < b * \<mu> A0" using b by auto
520     from int_f_finite this
521     have "?y + 0 < integral\<^isup>P M f + b * \<mu> A0" unfolding int_f_eq_y
523     also have "\<dots> = integral\<^isup>P M ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
524       by (simp cong: positive_integral_cong)
525     finally have "?y < integral\<^isup>P M ?f0" by simp
526     moreover from `?f0 \<in> G` have "integral\<^isup>P M ?f0 \<le> ?y" by (auto intro!: le_SUPI)
527     ultimately show False by auto
528   qed
529   show ?thesis
530   proof (safe intro!: bexI[of _ f])
531     fix A assume "A\<in>sets M"
532     show "\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
533     proof (rule antisym)
534       show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
535         using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
536       show "\<nu> A \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
537         using upper_bound[THEN bspec, OF `A \<in> sets M`]
539     qed
540   qed simp
541 qed
543 lemma (in finite_measure) split_space_into_finite_sets_and_rest:
544   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
545   assumes ac: "absolutely_continuous \<nu>"
546   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>
547     (\<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>
548     (\<forall>i. \<nu> (\<Omega> i) \<noteq> \<omega>)"
549 proof -
550   interpret v: measure_space ?N
551     where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
552     by fact auto
553   let ?Q = "{Q\<in>sets M. \<nu> Q \<noteq> \<omega>}"
554   let ?a = "SUP Q:?Q. \<mu> Q"
555   have "{} \<in> ?Q" using v.empty_measure by auto
556   then have Q_not_empty: "?Q \<noteq> {}" by blast
557   have "?a \<le> \<mu> (space M)" using sets_into_space
558     by (auto intro!: SUP_leI measure_mono top)
559   then have "?a \<noteq> \<omega>" using finite_measure_of_space
560     by auto
561   from SUPR_countable_SUPR[OF this Q_not_empty]
562   obtain Q'' where "range Q'' \<subseteq> \<mu> ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
563     by auto
564   then have "\<forall>i. \<exists>Q'. Q'' i = \<mu> Q' \<and> Q' \<in> ?Q" by auto
565   from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = \<mu> (Q' i)" "\<And>i. Q' i \<in> ?Q"
566     by auto
567   then have a_Lim: "?a = (SUP i::nat. \<mu> (Q' i))" using a by simp
568   let "?O n" = "\<Union>i\<le>n. Q' i"
569   have Union: "(SUP i. \<mu> (?O i)) = \<mu> (\<Union>i. ?O i)"
570   proof (rule continuity_from_below[of ?O])
571     show "range ?O \<subseteq> sets M" using Q' by (auto intro!: finite_UN)
572     show "\<And>i. ?O i \<subseteq> ?O (Suc i)" by fastsimp
573   qed
574   have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto
575   have O_sets: "\<And>i. ?O i \<in> sets M"
576      using Q' by (auto intro!: finite_UN Un)
577   then have O_in_G: "\<And>i. ?O i \<in> ?Q"
578   proof (safe del: notI)
579     fix i have "Q' ` {..i} \<subseteq> sets M"
580       using Q' by (auto intro: finite_UN)
581     with v.measure_finitely_subadditive[of "{.. i}" Q']
582     have "\<nu> (?O i) \<le> (\<Sum>i\<le>i. \<nu> (Q' i))" by auto
583     also have "\<dots> < \<omega>" unfolding setsum_\<omega> pextreal_less_\<omega> using Q' by auto
584     finally show "\<nu> (?O i) \<noteq> \<omega>" unfolding pextreal_less_\<omega> by auto
585   qed auto
586   have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastsimp
587   have a_eq: "?a = \<mu> (\<Union>i. ?O i)" unfolding Union[symmetric]
588   proof (rule antisym)
589     show "?a \<le> (SUP i. \<mu> (?O i))" unfolding a_Lim
590       using Q' by (auto intro!: SUP_mono measure_mono finite_UN)
591     show "(SUP i. \<mu> (?O i)) \<le> ?a" unfolding SUPR_def
592     proof (safe intro!: Sup_mono, unfold bex_simps)
593       fix i
594       have *: "(\<Union>Q' ` {..i}) = ?O i" by auto
595       then show "\<exists>x. (x \<in> sets M \<and> \<nu> x \<noteq> \<omega>) \<and>
596         \<mu> (\<Union>Q' ` {..i}) \<le> \<mu> x"
597         using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
598     qed
599   qed
600   let "?O_0" = "(\<Union>i. ?O i)"
601   have "?O_0 \<in> sets M" using Q' by auto
602   def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n"
603   { fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) }
604   note Q_sets = this
605   show ?thesis
606   proof (intro bexI exI conjI ballI impI allI)
607     show "disjoint_family Q"
608       by (fastsimp simp: disjoint_family_on_def Q_def
609         split: nat.split_asm)
610     show "range Q \<subseteq> sets M"
611       using Q_sets by auto
612     { fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0"
613       show "\<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<omega>"
614       proof (rule disjCI, simp)
615         assume *: "0 < \<mu> A \<longrightarrow> \<nu> A \<noteq> \<omega>"
616         show "\<mu> A = 0 \<and> \<nu> A = 0"
617         proof cases
618           assume "\<mu> A = 0" moreover with ac A have "\<nu> A = 0"
619             unfolding absolutely_continuous_def by auto
620           ultimately show ?thesis by simp
621         next
622           assume "\<mu> A \<noteq> 0" with * have "\<nu> A \<noteq> \<omega>" by auto
623           with A have "\<mu> ?O_0 + \<mu> A = \<mu> (?O_0 \<union> A)"
624             using Q' by (auto intro!: measure_additive countable_UN)
625           also have "\<dots> = (SUP i. \<mu> (?O i \<union> A))"
626           proof (rule continuity_from_below[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
627             show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
628               using `\<nu> A \<noteq> \<omega>` O_sets A by auto
629           qed fastsimp
630           also have "\<dots> \<le> ?a"
631           proof (safe intro!: SUPR_bound)
632             fix i have "?O i \<union> A \<in> ?Q"
633             proof (safe del: notI)
634               show "?O i \<union> A \<in> sets M" using O_sets A by auto
635               from O_in_G[of i]
636               moreover have "\<nu> (?O i \<union> A) \<le> \<nu> (?O i) + \<nu> A"
637                 using v.measure_subadditive[of "?O i" A] A O_sets by auto
638               ultimately show "\<nu> (?O i \<union> A) \<noteq> \<omega>"
639                 using `\<nu> A \<noteq> \<omega>` by auto
640             qed
641             then show "\<mu> (?O i \<union> A) \<le> ?a" by (rule le_SUPI)
642           qed
643           finally have "\<mu> A = 0" unfolding a_eq using finite_measure[OF `?O_0 \<in> sets M`]
644             by (cases "\<mu> A") (auto simp: pextreal_noteq_omega_Ex)
645           with `\<mu> A \<noteq> 0` show ?thesis by auto
646         qed
647       qed }
648     { fix i show "\<nu> (Q i) \<noteq> \<omega>"
649       proof (cases i)
650         case 0 then show ?thesis
651           unfolding Q_def using Q'[of 0] by simp
652       next
653         case (Suc n)
654         then show ?thesis unfolding Q_def
655           using `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q` O_mono
656           using v.measure_Diff[of "?O n" "?O (Suc n)"] by auto
657       qed }
658     show "space M - ?O_0 \<in> sets M" using Q'_sets by auto
659     { fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)"
660       proof (induct j)
661         case 0 then show ?case by (simp add: Q_def)
662       next
663         case (Suc j)
664         have eq: "\<And>j. (\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q' i)" by fastsimp
665         have "{..j} \<union> {..Suc j} = {..Suc j}" by auto
666         then have "(\<Union>i\<le>Suc j. Q' i) = (\<Union>i\<le>j. Q' i) \<union> Q (Suc j)"
667           by (simp add: UN_Un[symmetric] Q_def del: UN_Un)
668         then show ?case using Suc by (auto simp add: eq atMost_Suc)
669       qed }
670     then have "(\<Union>j. (\<Union>i\<le>j. ?O i)) = (\<Union>j. (\<Union>i\<le>j. Q i))" by simp
671     then show "space M - ?O_0 = space M - (\<Union>i. Q i)" by fastsimp
672   qed
673 qed
676   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
677   assumes "absolutely_continuous \<nu>"
678   shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
679 proof -
680   interpret v: measure_space ?N
681     where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
682     by fact auto
683   from split_space_into_finite_sets_and_rest[OF assms]
684   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
685     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
686     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
687     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>"
688     and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<omega>" by force
689   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
690   have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
691     \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
692   proof
693     fix i
694     have indicator_eq: "\<And>f x A. (f x :: pextreal) * indicator (Q i \<inter> A) x * indicator (Q i) x
695       = (f x * indicator (Q i) x) * indicator A x"
696       unfolding indicator_def by auto
697     have fm: "finite_measure (restricted_space (Q i))"
698       (is "finite_measure ?R") by (rule restricted_finite_measure[OF Q_sets[of i]])
699     then interpret R: finite_measure ?R .
700     have fmv: "finite_measure (restricted_space (Q i) \<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?Q")
701       unfolding finite_measure_def finite_measure_axioms_def
702     proof
703       show "measure_space ?Q"
704         using v.restricted_measure_space Q_sets[of i] by auto
705       show "measure ?Q (space ?Q) \<noteq> \<omega>" using Q_fin by simp
706     qed
707     have "R.absolutely_continuous \<nu>"
708       using `absolutely_continuous \<nu>` `Q i \<in> sets M`
709       by (auto simp: R.absolutely_continuous_def absolutely_continuous_def)
711     obtain f where f: "(\<lambda>x. f x * indicator (Q i) x) \<in> borel_measurable M"
712       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 \<partial>M)"
713       unfolding Bex_def borel_measurable_restricted[OF `Q i \<in> sets M`]
714         positive_integral_restricted[OF `Q i \<in> sets M`] by (auto simp: indicator_eq)
715     then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>A\<in>sets M.
716       \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
717       by (fastsimp intro!: exI[of _ "\<lambda>x. f x * indicator (Q i) x"] positive_integral_cong
718           simp: indicator_def)
719   qed
720   from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M"
721     and f: "\<And>A i. A \<in> sets M \<Longrightarrow>
722       \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)"
723     by auto
724   let "?f x" =
725     "(\<Sum>\<^isub>\<infinity> i. f i x * indicator (Q i) x) + \<omega> * indicator Q0 x"
726   show ?thesis
727   proof (safe intro!: bexI[of _ ?f])
728     show "?f \<in> borel_measurable M"
729       by (safe intro!: borel_measurable_psuminf borel_measurable_pextreal_times
731         borel_measurable_const borel Q_sets Q0 Diff countable_UN)
732     fix A assume "A \<in> sets M"
733     have *:
734       "\<And>x i. indicator A x * (f i x * indicator (Q i) x) =
735         f i x * indicator (Q i \<inter> A) x"
736       "\<And>x i. (indicator A x * indicator Q0 x :: pextreal) =
737         indicator (Q0 \<inter> A) x" by (auto simp: indicator_def)
738     have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) =
739       (\<Sum>\<^isub>\<infinity> i. \<nu> (Q i \<inter> A)) + \<omega> * \<mu> (Q0 \<inter> A)"
740       unfolding f[OF `A \<in> sets M`]
741       apply (simp del: pextreal_times(2) add: field_simps *)
743       apply (fastsimp intro: Q0 `A \<in> sets M`)
744       apply (fastsimp intro: Q_sets `A \<in> sets M` borel_measurable_psuminf borel)
745       apply (subst positive_integral_cmult_indicator)
746       apply (fastsimp intro: Q0 `A \<in> sets M`)
747       unfolding psuminf_cmult_right[symmetric]
748       apply (subst positive_integral_psuminf)
749       apply (fastsimp intro: `A \<in> sets M` Q_sets borel)
751       done
752     moreover have "(\<Sum>\<^isub>\<infinity>i. \<nu> (Q i \<inter> A)) = \<nu> ((\<Union>i. Q i) \<inter> A)"
753       using Q Q_sets `A \<in> sets M`
754       by (intro v.measure_countably_additive[of "\<lambda>i. Q i \<inter> A", unfolded comp_def, simplified])
755          (auto simp: disjoint_family_on_def)
756     moreover have "\<omega> * \<mu> (Q0 \<inter> A) = \<nu> (Q0 \<inter> A)"
757     proof -
758       have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
759       from in_Q0[OF this] show ?thesis by auto
760     qed
761     moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
762       using Q_sets `A \<in> sets M` Q0(1) by (auto intro!: countable_UN)
763     moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
764       using `A \<in> sets M` sets_into_space Q0 by auto
765     ultimately show "\<nu> A = (\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M)"
766       using v.measure_additive[simplified, of "(\<Union>i. Q i) \<inter> A" "Q0 \<inter> A"]
767       by simp
768   qed
769 qed
772   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
773   assumes "absolutely_continuous \<nu>"
774   shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
775 proof -
776   from Ex_finite_integrable_function
777   obtain h where finite: "integral\<^isup>P M h \<noteq> \<omega>" and
778     borel: "h \<in> borel_measurable M" and
779     pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
780     "\<And>x. x \<in> space M \<Longrightarrow> h x < \<omega>" by auto
781   let "?T A" = "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M)"
782   let ?MT = "M\<lparr> measure := ?T \<rparr>"
783   from measure_space_density[OF borel] finite
784   interpret T: finite_measure ?MT
785     where "space ?MT = space M" and "sets ?MT = sets M" and "measure ?MT = ?T"
786     unfolding finite_measure_def finite_measure_axioms_def
787     by (simp_all cong: positive_integral_cong)
788   have "\<And>N. N \<in> sets M \<Longrightarrow> {x \<in> space M. h x \<noteq> 0 \<and> indicator N x \<noteq> (0::pextreal)} = N"
789     using sets_into_space pos by (force simp: indicator_def)
790   then have "T.absolutely_continuous \<nu>" using assms(2) borel
791     unfolding T.absolutely_continuous_def absolutely_continuous_def
792     by (fastsimp simp: borel_measurable_indicator positive_integral_0_iff)
793   from T.Radon_Nikodym_finite_measure_infinite[simplified, OF assms(1) this]
794   obtain f where f_borel: "f \<in> borel_measurable M" and
795     fT: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>?MT)"
796     by (auto simp: measurable_def)
797   show ?thesis
798   proof (safe intro!: bexI[of _ "\<lambda>x. h x * f x"])
799     show "(\<lambda>x. h x * f x) \<in> borel_measurable M"
800       using borel f_borel by (auto intro: borel_measurable_pextreal_times)
801     fix A assume "A \<in> sets M"
802     then have "(\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
803       using f_borel by (auto intro: borel_measurable_pextreal_times borel_measurable_indicator)
804     from positive_integral_translated_density[OF borel this]
805     show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x \<partial>M)"
806       unfolding fT[OF `A \<in> sets M`] by (simp add: field_simps)
807   qed
808 qed
810 section "Uniqueness of densities"
812 lemma (in measure_space) finite_density_unique:
813   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
814   and fin: "integral\<^isup>P M f < \<omega>"
815   shows "(\<forall>A\<in>sets M. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. g x * indicator A x \<partial>M))
816     \<longleftrightarrow> (AE x. f x = g x)"
817     (is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _")
818 proof (intro iffI ballI)
819   fix A assume eq: "AE x. f x = g x"
820   then show "?P f A = ?P g A"
821     by (auto intro: positive_integral_cong_AE)
822 next
823   assume eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
824   from this[THEN bspec, OF top] fin
825   have g_fin: "integral\<^isup>P M g < \<omega>" by (simp cong: positive_integral_cong)
826   { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
827       and g_fin: "integral\<^isup>P M g < \<omega>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
828     let ?N = "{x\<in>space M. g x < f x}"
829     have N: "?N \<in> sets M" using borel by simp
830     have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
831       by (auto intro!: positive_integral_cong simp: indicator_def)
832     also have "\<dots> = ?P f ?N - ?P g ?N"
833     proof (rule positive_integral_diff)
834       show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
835         using borel N by auto
836       have "?P g ?N \<le> integral\<^isup>P M g"
837         by (auto intro!: positive_integral_mono simp: indicator_def)
838       then show "?P g ?N \<noteq> \<omega>" using g_fin by auto
839       fix x assume "x \<in> space M"
840       show "g x * indicator ?N x \<le> f x * indicator ?N x"
841         by (auto simp: indicator_def)
842     qed
843     also have "\<dots> = 0"
844       using eq[THEN bspec, OF N] by simp
845     finally have "\<mu> {x\<in>space M. (f x - g x) * indicator ?N x \<noteq> 0} = 0"
846       using borel N by (subst (asm) positive_integral_0_iff) auto
847     moreover have "{x\<in>space M. (f x - g x) * indicator ?N x \<noteq> 0} = ?N"
848       by (auto simp: pextreal_zero_le_diff)
849     ultimately have "?N \<in> null_sets" using N by simp }
850   from this[OF borel g_fin eq] this[OF borel(2,1) fin]
851   have "{x\<in>space M. g x < f x} \<union> {x\<in>space M. f x < g x} \<in> null_sets"
852     using eq by (intro null_sets_Un) auto
853   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}"
854     by auto
855   finally show "AE x. f x = g x"
856     unfolding almost_everywhere_def by auto
857 qed
859 lemma (in finite_measure) density_unique_finite_measure:
860   assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
861   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
862     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
863   shows "AE x. f x = f' x"
864 proof -
865   let "?\<nu> A" = "?P f A" and "?\<nu>' A" = "?P f' A"
866   let "?f A x" = "f x * indicator A x" and "?f' A x" = "f' x * indicator A x"
867   interpret M: measure_space "M\<lparr> measure := ?\<nu>\<rparr>"
868     using borel(1) by (rule measure_space_density) simp
869   have ac: "absolutely_continuous ?\<nu>"
870     using f by (rule density_is_absolutely_continuous)
871   from split_space_into_finite_sets_and_rest[OF `measure_space (M\<lparr> measure := ?\<nu>\<rparr>)` ac]
872   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
873     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
874     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
875     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>"
876     and Q_fin: "\<And>i. ?\<nu> (Q i) \<noteq> \<omega>" by force
877   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
878   let ?N = "{x\<in>space M. f x \<noteq> f' x}"
879   have "?N \<in> sets M" using borel by auto
880   have *: "\<And>i x A. \<And>y::pextreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
881     unfolding indicator_def by auto
882   have "\<forall>i. AE x. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q
883     by (intro finite_density_unique[THEN iffD1] allI)
884        (auto intro!: borel_measurable_pextreal_times f Int simp: *)
885   moreover have "AE x. ?f Q0 x = ?f' Q0 x"
886   proof (rule AE_I')
887     { fix f :: "'a \<Rightarrow> pextreal" assume borel: "f \<in> borel_measurable M"
888         and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
889       let "?A i" = "Q0 \<inter> {x \<in> space M. f x < of_nat i}"
890       have "(\<Union>i. ?A i) \<in> null_sets"
891       proof (rule null_sets_UN)
892         fix i have "?A i \<in> sets M"
893           using borel Q0(1) by auto
894         have "?\<nu> (?A i) \<le> (\<integral>\<^isup>+x. of_nat i * indicator (?A i) x \<partial>M)"
895           unfolding eq[OF `?A i \<in> sets M`]
896           by (auto intro!: positive_integral_mono simp: indicator_def)
897         also have "\<dots> = of_nat i * \<mu> (?A i)"
898           using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator)
899         also have "\<dots> < \<omega>"
900           using `?A i \<in> sets M`[THEN finite_measure] by auto
901         finally have "?\<nu> (?A i) \<noteq> \<omega>" by simp
902         then show "?A i \<in> null_sets" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
903       qed
904       also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x < \<omega>}"
905         by (auto simp: less_\<omega>_Ex_of_nat)
906       finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<omega>} \<in> null_sets" by (simp add: pextreal_less_\<omega>) }
907     from this[OF borel(1) refl] this[OF borel(2) f]
908     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
909     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)
910     show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq>
911       (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)
912   qed
913   moreover have "\<And>x. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
914     ?f (space M) x = ?f' (space M) x"
915     by (auto simp: indicator_def Q0)
916   ultimately have "AE x. ?f (space M) x = ?f' (space M) x"
917     by (auto simp: all_AE_countable)
918   then show "AE x. f x = f' x" by auto
919 qed
921 lemma (in sigma_finite_measure) density_unique:
922   assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
923   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)"
924     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
925   shows "AE x. f x = f' x"
926 proof -
927   obtain h where h_borel: "h \<in> borel_measurable M"
928     and fin: "integral\<^isup>P M h \<noteq> \<omega>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<omega>"
929     using Ex_finite_integrable_function by auto
930   interpret h: measure_space "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
931     using h_borel by (rule measure_space_density) simp
932   interpret h: finite_measure "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
933     by default (simp cong: positive_integral_cong add: fin)
934   let ?fM = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)\<rparr>"
935   interpret f: measure_space ?fM
936     using borel(1) by (rule measure_space_density) simp
937   let ?f'M = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)\<rparr>"
938   interpret f': measure_space ?f'M
939     using borel(2) by (rule measure_space_density) simp
940   { fix A assume "A \<in> sets M"
941     then have " {x \<in> space M. h x \<noteq> 0 \<and> indicator A x \<noteq> (0::pextreal)} = A"
942       using pos sets_into_space by (force simp: indicator_def)
943     then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets"
944       using h_borel `A \<in> sets M` by (simp add: positive_integral_0_iff) }
945   note h_null_sets = this
946   { fix A assume "A \<in> sets M"
947     have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)"
948       using `A \<in> sets M` h_borel borel
949       by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
950     also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)"
951       by (rule f'.positive_integral_cong_measure) (simp_all add: f)
952     also have "\<dots> = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)"
953       using `A \<in> sets M` h_borel borel
954       by (simp add: positive_integral_translated_density ac_simps cong: positive_integral_cong)
955     finally have "(\<integral>\<^isup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * (f' x * indicator A x) \<partial>M)" . }
956   then have "h.almost_everywhere (\<lambda>x. f x = f' x)"
957     using h_borel borel
958     apply (intro h.density_unique_finite_measure)
962   then show "AE x. f x = f' x"
963     unfolding h.almost_everywhere_def almost_everywhere_def
964     by (auto simp add: h_null_sets)
965 qed
967 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
968   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
969     and f: "f \<in> borel_measurable M"
970     and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
971   shows "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>) \<longleftrightarrow> (AE x. f x \<noteq> \<omega>)"
972 proof
973   assume "sigma_finite_measure ?N"
974   then interpret \<nu>: sigma_finite_measure ?N
975     where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
976     and "sets ?N = sets M" and "space ?N = space M" by (auto simp: measurable_def)
977   from \<nu>.Ex_finite_integrable_function obtain h where
978     h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<omega>"
979     and fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<omega>" by auto
980   have "AE x. f x * h x \<noteq> \<omega>"
981   proof (rule AE_I')
982     have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h
983       by (subst \<nu>.positive_integral_cong_measure[symmetric,
984           of "M\<lparr> measure := \<lambda> A. \<integral>\<^isup>+x. f x * indicator A x \<partial>M \<rparr>"])
985          (auto intro!: positive_integral_translated_density simp: eq)
986     then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<omega>"
987       using h(2) by simp
988     then show "(\<lambda>x. f x * h x) -` {\<omega>} \<inter> space M \<in> null_sets"
989       using f h(1) by (auto intro!: positive_integral_omega borel_measurable_vimage)
990   qed auto
991   then show "AE x. f x \<noteq> \<omega>"
992     using fin by (auto elim!: AE_Ball_mp)
993 next
994   assume AE: "AE x. f x \<noteq> \<omega>"
995   from sigma_finite guess Q .. note Q = this
996   interpret \<nu>: measure_space ?N
997     where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
998     and "sets ?N = sets M" and "space ?N = space M" using \<nu> by (auto simp: measurable_def)
999   def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<omega>} | Suc n \<Rightarrow> {.. of_nat (Suc n)}) \<inter> space M"
1000   { fix i j have "A i \<inter> Q j \<in> sets M"
1001     unfolding A_def using f Q
1002     apply (rule_tac Int)
1003     by (cases i) (auto intro: measurable_sets[OF f]) }
1004   note A_in_sets = this
1005   let "?A n" = "case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j"
1006   show "sigma_finite_measure ?N"
1007   proof (default, intro exI conjI subsetI allI)
1008     fix x assume "x \<in> range ?A"
1009     then obtain n where n: "x = ?A n" by auto
1010     then show "x \<in> sets ?N" using A_in_sets by (cases "prod_decode n") auto
1011   next
1012     have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)"
1013     proof safe
1014       fix x i j assume "x \<in> A i" "x \<in> Q j"
1015       then show "x \<in> (\<Union>i. case prod_decode i of (i, j) \<Rightarrow> A i \<inter> Q j)"
1016         by (intro UN_I[of "prod_encode (i,j)"]) auto
1017     qed auto
1018     also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto
1019     also have "(\<Union>i. A i) = space M"
1020     proof safe
1021       fix x assume x: "x \<in> space M"
1022       show "x \<in> (\<Union>i. A i)"
1023       proof (cases "f x")
1024         case infinite then show ?thesis using x unfolding A_def by (auto intro: exI[of _ 0])
1025       next
1026         case (preal r)
1027         with less_\<omega>_Ex_of_nat[of "f x"] obtain n where "f x < of_nat n" by auto
1028         then show ?thesis using x preal unfolding A_def by (auto intro!: exI[of _ "Suc n"])
1029       qed
1030     qed (auto simp: A_def)
1031     finally show "(\<Union>i. ?A i) = space ?N" by simp
1032   next
1033     fix n obtain i j where
1034       [simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto
1035     have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<omega>"
1036     proof (cases i)
1037       case 0
1038       have "AE x. f x * indicator (A i \<inter> Q j) x = 0"
1039         using AE by (auto simp: A_def `i = 0`)
1040       from positive_integral_cong_AE[OF this] show ?thesis by simp
1041     next
1042       case (Suc n)
1043       then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
1044         (\<integral>\<^isup>+x. of_nat (Suc n) * indicator (Q j) x \<partial>M)"
1045         by (auto intro!: positive_integral_mono simp: indicator_def A_def)
1046       also have "\<dots> = of_nat (Suc n) * \<mu> (Q j)"
1047         using Q by (auto intro!: positive_integral_cmult_indicator)
1048       also have "\<dots> < \<omega>"
1049         using Q by auto
1050       finally show ?thesis by simp
1051     qed
1052     then show "measure ?N (?A n) \<noteq> \<omega>"
1053       using A_in_sets Q eq by auto
1054   qed
1055 qed
1059 definition
1060   "RN_deriv M \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and>
1061     (\<forall>A \<in> sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
1063 lemma (in sigma_finite_measure) RN_deriv_cong:
1064   assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> measure M' A = \<mu> A" "sets M' = sets M" "space M' = space M"
1065     and \<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A"
1066   shows "RN_deriv M' \<nu>' x = RN_deriv M \<nu> x"
1067 proof -
1068   interpret \<mu>': sigma_finite_measure M'
1069     using cong by (rule sigma_finite_measure_cong)
1070   show ?thesis
1071     unfolding RN_deriv_def
1072     by (simp add: cong \<nu> positive_integral_cong_measure[OF cong] measurable_def)
1073 qed
1075 lemma (in sigma_finite_measure) RN_deriv:
1076   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
1077   assumes "absolutely_continuous \<nu>"
1078   shows "RN_deriv M \<nu> \<in> borel_measurable M" (is ?borel)
1079   and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
1080     (is "\<And>A. _ \<Longrightarrow> ?int A")
1081 proof -
1082   note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
1083   thus ?borel unfolding RN_deriv_def by (rule someI2_ex) auto
1084   fix A assume "A \<in> sets M"
1085   from Ex show "?int A" unfolding RN_deriv_def
1086     by (rule someI2_ex) (simp add: `A \<in> sets M`)
1087 qed
1089 lemma (in sigma_finite_measure) RN_deriv_positive_integral:
1090   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
1091     and f: "f \<in> borel_measurable M"
1092   shows "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
1093 proof -
1094   interpret \<nu>: measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
1095   have "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f =
1096     integral\<^isup>P (M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)\<rparr>) f"
1097     by (intro \<nu>.positive_integral_cong_measure[symmetric])
1098        (simp_all add:  RN_deriv(2)[OF \<nu>, symmetric])
1099   also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
1100     by (intro positive_integral_translated_density)
1101        (simp_all add: RN_deriv[OF \<nu>] f)
1102   finally show ?thesis .
1103 qed
1105 lemma (in sigma_finite_measure) RN_deriv_unique:
1106   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
1107   and f: "f \<in> borel_measurable M"
1108   and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
1109   shows "AE x. f x = RN_deriv M \<nu> x"
1110 proof (rule density_unique[OF f RN_deriv(1)[OF \<nu>]])
1111   fix A assume A: "A \<in> sets M"
1112   show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
1113     unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] ..
1114 qed
1116 lemma (in sigma_finite_measure) RN_deriv_finite:
1117   assumes sfm: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" and ac: "absolutely_continuous \<nu>"
1118   shows "AE x. RN_deriv M \<nu> x \<noteq> \<omega>"
1119 proof -
1120   interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
1121   have \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
1122   from sfm show ?thesis
1123     using sigma_finite_iff_density_finite[OF \<nu> RN_deriv[OF \<nu> ac]] by simp
1124 qed
1126 lemma (in sigma_finite_measure)
1127   assumes \<nu>: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
1128     and f: "f \<in> borel_measurable M"
1129   shows RN_deriv_integrable: "integrable (M\<lparr>measure := \<nu>\<rparr>) f \<longleftrightarrow>
1130       integrable M (\<lambda>x. real (RN_deriv M \<nu> x) * f x)" (is ?integrable)
1131     and RN_deriv_integral: "integral\<^isup>L (M\<lparr>measure := \<nu>\<rparr>) f =
1132       (\<integral>x. real (RN_deriv M \<nu> x) * f x \<partial>M)" (is ?integral)
1133 proof -
1134   interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
1135   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
1136   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
1137   have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
1138   have Nf: "f \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)" using f unfolding measurable_def by auto
1139   { fix f :: "'a \<Rightarrow> real"
1140     { fix x assume *: "RN_deriv M \<nu> x \<noteq> \<omega>"
1141       have "Real (real (RN_deriv M \<nu> x)) * Real (f x) = Real (real (RN_deriv M \<nu> x) * f x)"
1143       then have "RN_deriv M \<nu> x * Real (f x) = Real (real (RN_deriv M \<nu> x) * f x)"
1144         using * by (simp add: Real_real) }
1145     then have "(\<integral>\<^isup>+x. RN_deriv M \<nu> x * Real (f x) \<partial>M) = (\<integral>\<^isup>+x. Real (real (RN_deriv M \<nu> x) * f x) \<partial>M)"
1146       using RN_deriv_finite[OF \<nu>] by (auto intro: positive_integral_cong_AE) }
1147   with this this f f' Nf
1148   show ?integral ?integrable
1149     unfolding lebesgue_integral_def integrable_def
1150     by (auto intro!: RN_deriv(1)[OF ms \<nu>(2)] minus_cong
1151              simp: RN_deriv_positive_integral[OF ms \<nu>(2)])
1152 qed
1154 lemma (in sigma_finite_measure) RN_deriv_singleton:
1155   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
1156   and ac: "absolutely_continuous \<nu>"
1157   and "{x} \<in> sets M"
1158   shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
1159 proof -
1160   note deriv = RN_deriv[OF assms(1, 2)]
1161   from deriv(2)[OF `{x} \<in> sets M`]
1162   have "\<nu> {x} = (\<integral>\<^isup>+w. RN_deriv M \<nu> x * indicator {x} w \<partial>M)"
1163     by (auto simp: indicator_def intro!: positive_integral_cong)
1164   thus ?thesis using positive_integral_cmult_indicator[OF `{x} \<in> sets M`]
1165     by auto
1166 qed
1168 theorem (in finite_measure_space) RN_deriv_finite_measure:
1169   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
1170   and ac: "absolutely_continuous \<nu>"
1171   and "x \<in> space M"
1172   shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
1173 proof -
1174   have "{x} \<in> sets M" using sets_eq_Pow `x \<in> space M` by auto
1175   from RN_deriv_singleton[OF assms(1,2) this] show ?thesis .
1176 qed
1178 end