author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 46731 5302e932d1e5 child 47694 05663f75964c permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
1 (*  Title:      HOL/Probability/Radon_Nikodym.thy
2     Author:     Johannes Hölzl, TU München
3 *)
8 imports Lebesgue_Integration
9 begin
11 lemma (in sigma_finite_measure) Ex_finite_integrable_function:
12   shows "\<exists>h\<in>borel_measurable M. integral\<^isup>P M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>) \<and> (\<forall>x. 0 \<le> h x)"
13 proof -
14   obtain A :: "nat \<Rightarrow> 'a set" where
15     range: "range A \<subseteq> sets M" and
16     space: "(\<Union>i. A i) = space M" and
17     measure: "\<And>i. \<mu> (A i) \<noteq> \<infinity>" and
18     disjoint: "disjoint_family A"
19     using disjoint_sigma_finite by auto
20   let ?B = "\<lambda>i. 2^Suc i * \<mu> (A i)"
21   have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)"
22   proof
23     fix i have Ai: "A i \<in> sets M" using range by auto
24     from measure positive_measure[OF this]
25     show "\<exists>x. 0 < x \<and> x < inverse (?B i)"
26       by (auto intro!: ereal_dense simp: ereal_0_gt_inverse)
27   qed
28   from choice[OF this] obtain n where n: "\<And>i. 0 < n i"
29     "\<And>i. n i < inverse (2^Suc i * \<mu> (A i))" by auto
30   { fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this
31   let ?h = "\<lambda>x. \<Sum>i. n i * indicator (A i) x"
32   show ?thesis
33   proof (safe intro!: bexI[of _ ?h] del: notI)
34     have "\<And>i. A i \<in> sets M"
35       using range by fastforce+
36     then have "integral\<^isup>P M ?h = (\<Sum>i. n i * \<mu> (A i))" using pos
37       by (simp add: positive_integral_suminf positive_integral_cmult_indicator)
38     also have "\<dots> \<le> (\<Sum>i. (1 / 2)^Suc i)"
39     proof (rule suminf_le_pos)
40       fix N
41       have "n N * \<mu> (A N) \<le> inverse (2^Suc N * \<mu> (A N)) * \<mu> (A N)"
42         using positive_measure[OF `A N \<in> sets M`] n[of N]
43         by (intro ereal_mult_right_mono) auto
44       also have "\<dots> \<le> (1 / 2) ^ Suc N"
45         using measure[of N] n[of N]
46         by (cases rule: ereal2_cases[of "n N" "\<mu> (A N)"])
47            (simp_all add: inverse_eq_divide power_divide one_ereal_def ereal_power_divide)
48       finally show "n N * \<mu> (A N) \<le> (1 / 2) ^ Suc N" .
49       show "0 \<le> n N * \<mu> (A N)" using n[of N] `A N \<in> sets M` by simp
50     qed
51     finally show "integral\<^isup>P M ?h \<noteq> \<infinity>" unfolding suminf_half_series_ereal by auto
52   next
53     { fix x assume "x \<in> space M"
54       then obtain i where "x \<in> A i" using space[symmetric] by auto
55       with disjoint n have "?h x = n i"
56         by (auto intro!: suminf_cmult_indicator intro: less_imp_le)
57       then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by auto }
58     note pos = this
59     fix x show "0 \<le> ?h x"
60     proof cases
61       assume "x \<in> space M" then show "0 \<le> ?h x" using pos by (auto intro: less_imp_le)
62     next
63       assume "x \<notin> space M" then have "\<And>i. x \<notin> A i" using space by auto
64       then show "0 \<le> ?h x" by auto
65     qed
66   next
67     show "?h \<in> borel_measurable M" using range n
68       by (auto intro!: borel_measurable_psuminf borel_measurable_ereal_times ereal_0_le_mult intro: less_imp_le)
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 :: ereal))"
77 lemma (in measure_space) 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 "AE x in M'. P x"
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 "AE x in M'. P x"
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})"
102   proof (rule v.measure_setsum[symmetric])
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" "0 \<le> \<mu> {x}"
111       using sets_eq_Pow `N \<subseteq> space M` positive_measure[of "{x}"]
112       by (auto intro!: measure_mono)
113     then have "\<mu> {x} = 0" using `\<mu> N = 0` by simp
114     thus "measure ?\<nu> {x} = 0" using v[of x] `x \<in> N` `N \<subseteq> space M` by auto
115   qed
116   finally show "\<nu> N = 0" by simp
117 qed
119 lemma (in measure_space) density_is_absolutely_continuous:
120   assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
121   shows "absolutely_continuous \<nu>"
122   using assms unfolding absolutely_continuous_def
123   by (simp add: positive_integral_null_set)
125 subsection "Existence of the Radon-Nikodym derivative"
127 lemma (in finite_measure) Radon_Nikodym_aux_epsilon:
128   fixes e :: real assumes "0 < e"
129   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)"
130   shows "\<exists>A\<in>sets M. \<mu>' (space M) - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) (space M) \<le>
131                     \<mu>' A - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) A \<and>
132                     (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> - e < \<mu>' B - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) B)"
133 proof -
134   interpret M': finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
135   let ?d = "\<lambda>A. \<mu>' A - M'.\<mu>' A"
136   let ?A = "\<lambda>A. if (\<forall>B\<in>sets M. B \<subseteq> space M - A \<longrightarrow> -e < ?d B)
137     then {}
138     else (SOME B. B \<in> sets M \<and> B \<subseteq> space M - A \<and> ?d B \<le> -e)"
139   def A \<equiv> "\<lambda>n. ((\<lambda>B. B \<union> ?A B) ^^ n) {}"
140   have A_simps[simp]:
141     "A 0 = {}"
142     "\<And>n. A (Suc n) = (A n \<union> ?A (A n))" unfolding A_def by simp_all
143   { fix A assume "A \<in> sets M"
144     have "?A A \<in> sets M"
145       by (auto intro!: someI2[of _ _ "\<lambda>A. A \<in> sets M"] simp: not_less) }
146   note A'_in_sets = this
147   { fix n have "A n \<in> sets M"
148     proof (induct n)
149       case (Suc n) thus "A (Suc n) \<in> sets M"
150         using A'_in_sets[of "A n"] by (auto split: split_if_asm)
151     qed (simp add: A_def) }
152   note A_in_sets = this
153   hence "range A \<subseteq> sets M" by auto
154   { fix n B
155     assume Ex: "\<exists>B. B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> -e"
156     hence False: "\<not> (\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B)" by (auto simp: not_less)
157     have "?d (A (Suc n)) \<le> ?d (A n) - e" unfolding A_simps if_not_P[OF False]
158     proof (rule someI2_ex[OF Ex])
159       fix B assume "B \<in> sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
160       hence "A n \<inter> B = {}" "B \<in> sets M" and dB: "?d B \<le> -e" by auto
161       hence "?d (A n \<union> B) = ?d (A n) + ?d B"
162         using `A n \<in> sets M` finite_measure_Union M'.finite_measure_Union by simp
163       also have "\<dots> \<le> ?d (A n) - e" using dB by simp
164       finally show "?d (A n \<union> B) \<le> ?d (A n) - e" .
165     qed }
166   note dA_epsilon = this
167   { fix n have "?d (A (Suc n)) \<le> ?d (A n)"
168     proof (cases "\<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e")
169       case True from dA_epsilon[OF this] show ?thesis using `0 < e` by simp
170     next
171       case False
172       hence "\<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B" by (auto simp: not_le)
173       thus ?thesis by simp
174     qed }
175   note dA_mono = this
176   show ?thesis
177   proof (cases "\<exists>n. \<forall>B\<in>sets M. B \<subseteq> space M - A n \<longrightarrow> -e < ?d B")
178     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
179     show ?thesis
180     proof (safe intro!: bexI[of _ "space M - A n"])
181       fix B assume "B \<in> sets M" "B \<subseteq> space M - A n"
182       from B[OF this] show "-e < ?d B" .
183     next
184       show "space M - A n \<in> sets M" by (rule compl_sets) fact
185     next
186       show "?d (space M) \<le> ?d (space M - A n)"
187       proof (induct n)
188         fix n assume "?d (space M) \<le> ?d (space M - A n)"
189         also have "\<dots> \<le> ?d (space M - A (Suc n))"
190           using A_in_sets sets_into_space dA_mono[of n]
191           by (simp del: A_simps add: finite_measure_Diff M'.finite_measure_Diff)
192         finally show "?d (space M) \<le> ?d (space M - A (Suc n))" .
193       qed simp
194     qed
195   next
196     case False hence B: "\<And>n. \<exists>B. B\<in>sets M \<and> B \<subseteq> space M - A n \<and> ?d B \<le> - e"
197       by (auto simp add: not_less)
198     { fix n have "?d (A n) \<le> - real n * e"
199       proof (induct n)
200         case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps)
201       next
202         case 0 with M'.empty_measure show ?case by (simp add: zero_ereal_def)
203       qed } 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     have A: "incseq A" by (auto intro!: incseq_SucI)
209     from finite_continuity_from_below[OF _ A] `range A \<subseteq> sets M`
210       M'.finite_continuity_from_below[OF _ A]
211     have convergent: "(\<lambda>i. ?d (A i)) ----> ?d (\<Union>i. A i)"
212       by (auto intro!: tendsto_diff)
213     obtain n :: nat where "- ?d (\<Union>i. A i) / e < real n" using reals_Archimedean2 by auto
214     moreover from order_trans[OF decseq_le[OF decseq convergent] dA_less]
215     have "real n \<le> - ?d (\<Union>i. A i) / e" using `0<e` by (simp add: field_simps)
216     ultimately show ?thesis by auto
217   qed
218 qed
220 lemma (in finite_measure) restricted_measure_subset:
221   assumes S: "S \<in> sets M" and X: "X \<subseteq> S"
222   shows "finite_measure.\<mu>' (restricted_space S) X = \<mu>' X"
223 proof cases
224   note r = restricted_finite_measure[OF S]
225   { assume "X \<in> sets M" with S X show ?thesis
226       unfolding finite_measure.\<mu>'_def[OF r] \<mu>'_def by auto }
227   { assume "X \<notin> sets M"
228     moreover then have "S \<inter> X \<notin> sets M"
229       using X by (simp add: Int_absorb1)
230     ultimately show ?thesis
231       unfolding finite_measure.\<mu>'_def[OF r] \<mu>'_def using S by auto }
232 qed
234 lemma (in finite_measure) restricted_measure:
235   assumes X: "S \<in> sets M" "X \<in> sets (restricted_space S)"
236   shows "finite_measure.\<mu>' (restricted_space S) X = \<mu>' X"
237 proof -
238   from X have "S \<in> sets M" "X \<subseteq> S" by auto
239   from restricted_measure_subset[OF this] show ?thesis .
240 qed
242 lemma (in finite_measure) Radon_Nikodym_aux:
243   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
244   shows "\<exists>A\<in>sets M. \<mu>' (space M) - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) (space M) \<le>
245                     \<mu>' A - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) A \<and>
246                     (\<forall>B\<in>sets M. B \<subseteq> A \<longrightarrow> 0 \<le> \<mu>' B - finite_measure.\<mu>' (M\<lparr>measure := \<nu>\<rparr>) B)"
247 proof -
248   interpret M': finite_measure ?M' where
249     "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>" by fact auto
250   let ?d = "\<lambda>A. \<mu>' A - M'.\<mu>' A"
251   let ?P = "\<lambda>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)"
252   let ?r = "\<lambda>S. restricted_space S"
253   { fix S n assume S: "S \<in> sets M"
254     note r = M'.restricted_finite_measure[of S] restricted_finite_measure[OF S] S
255     then have "finite_measure (?r S)" "0 < 1 / real (Suc n)"
256       "finite_measure (?r S\<lparr>measure := \<nu>\<rparr>)" by auto
257     from finite_measure.Radon_Nikodym_aux_epsilon[OF this] guess X .. note X = this
258     have "?P X S n"
259     proof (intro conjI ballI impI)
260       show "X \<in> sets M" "X \<subseteq> S" using X(1) `S \<in> sets M` by auto
261       have "S \<in> op \<inter> S ` sets M" using `S \<in> sets M` by auto
262       then show "?d S \<le> ?d X"
263         using S X restricted_measure[OF S] M'.restricted_measure[OF S] by simp
264       fix C assume "C \<in> sets M" "C \<subseteq> X"
265       then have "C \<in> sets (restricted_space S)" "C \<subseteq> X"
266         using `S \<in> sets M` `X \<subseteq> S` by auto
267       with X(2) show "- 1 / real (Suc n) < ?d C"
268         using S X restricted_measure[OF S] M'.restricted_measure[OF S] by auto
269     qed
270     hence "\<exists>A. ?P A S n" by auto }
271   note Ex_P = this
272   def A \<equiv> "nat_rec (space M) (\<lambda>n A. SOME B. ?P B A n)"
273   have A_Suc: "\<And>n. A (Suc n) = (SOME B. ?P B (A n) n)" by (simp add: A_def)
274   have A_0[simp]: "A 0 = space M" unfolding A_def by simp
275   { fix i have "A i \<in> sets M" unfolding A_def
276     proof (induct i)
277       case (Suc i)
278       from Ex_P[OF this, of i] show ?case unfolding nat_rec_Suc
279         by (rule someI2_ex) simp
280     qed simp }
281   note A_in_sets = this
282   { fix n have "?P (A (Suc n)) (A n) n"
283       using Ex_P[OF A_in_sets] unfolding A_Suc
284       by (rule someI2_ex) simp }
285   note P_A = this
286   have "range A \<subseteq> sets M" using A_in_sets by auto
287   have A_mono: "\<And>i. A (Suc i) \<subseteq> A i" using P_A by simp
288   have mono_dA: "mono (\<lambda>i. ?d (A i))" using P_A by (simp add: mono_iff_le_Suc)
289   have epsilon: "\<And>C i. \<lbrakk> C \<in> sets M; C \<subseteq> A (Suc i) \<rbrakk> \<Longrightarrow> - 1 / real (Suc i) < ?d C"
290       using P_A by auto
291   show ?thesis
292   proof (safe intro!: bexI[of _ "\<Inter>i. A i"])
293     show "(\<Inter>i. A i) \<in> sets M" using A_in_sets by auto
294     have A: "decseq A" using A_mono by (auto intro!: decseq_SucI)
295     from
296       finite_continuity_from_above[OF `range A \<subseteq> sets M` A]
297       M'.finite_continuity_from_above[OF `range A \<subseteq> sets M` A]
298     have "(\<lambda>i. ?d (A i)) ----> ?d (\<Inter>i. A i)" by (intro tendsto_diff)
299     thus "?d (space M) \<le> ?d (\<Inter>i. A i)" using mono_dA[THEN monoD, of 0 _]
300       by (rule_tac LIMSEQ_le_const) (auto intro!: exI)
301   next
302     fix B assume B: "B \<in> sets M" "B \<subseteq> (\<Inter>i. A i)"
303     show "0 \<le> ?d B"
304     proof (rule ccontr)
305       assume "\<not> 0 \<le> ?d B"
306       hence "0 < - ?d B" by auto
307       from ex_inverse_of_nat_Suc_less[OF this]
308       obtain n where *: "?d B < - 1 / real (Suc n)"
309         by (auto simp: real_eq_of_nat inverse_eq_divide field_simps)
310       have "B \<subseteq> A (Suc n)" using B by (auto simp del: nat_rec_Suc)
311       from epsilon[OF B(1) this] *
312       show False by auto
313     qed
314   qed
315 qed
317 lemma (in finite_measure) Radon_Nikodym_finite_measure:
318   assumes "finite_measure (M\<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?M'")
319   assumes "absolutely_continuous \<nu>"
320   shows "\<exists>f \<in> borel_measurable M. \<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
321 proof -
322   interpret M': finite_measure ?M'
323     where "space ?M' = space M" and "sets ?M' = sets M" and "measure ?M' = \<nu>"
324     using assms(1) by auto
325   def G \<equiv> "{g \<in> borel_measurable M. (\<forall>x. 0 \<le> g x) \<and> (\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A)}"
326   have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
327   hence "G \<noteq> {}" by auto
328   { fix f g assume f: "f \<in> G" and g: "g \<in> G"
329     have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def
330     proof safe
331       show "?max \<in> borel_measurable M" using f g unfolding G_def by auto
332       let ?A = "{x \<in> space M. f x \<le> g x}"
333       have "?A \<in> sets M" using f g unfolding G_def by auto
334       fix A assume "A \<in> sets M"
335       hence sets: "?A \<inter> A \<in> sets M" "(space M - ?A) \<inter> A \<in> sets M" using `?A \<in> sets M` by auto
336       have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
337         using sets_into_space[OF `A \<in> sets M`] by auto
338       have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
339         g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
340         by (auto simp: indicator_def max_def)
341       hence "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) =
342         (\<integral>\<^isup>+x. g x * indicator (?A \<inter> A) x \<partial>M) +
343         (\<integral>\<^isup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)"
344         using f g sets unfolding G_def
345         by (auto cong: positive_integral_cong intro!: positive_integral_add)
346       also have "\<dots> \<le> \<nu> (?A \<inter> A) + \<nu> ((space M - ?A) \<inter> A)"
347         using f g sets unfolding G_def by (auto intro!: add_mono)
348       also have "\<dots> = \<nu> A"
349         using M'.measure_additive[OF sets] union by auto
350       finally show "(\<integral>\<^isup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> \<nu> A" .
351     next
352       fix x show "0 \<le> max (g x) (f x)" using f g by (auto simp: G_def split: split_max)
353     qed }
354   note max_in_G = this
355   { fix f assume  "incseq f" and f: "\<And>i. f i \<in> G"
356     have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def
357     proof safe
358       show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M"
359         using f by (auto simp: G_def)
360       { fix x show "0 \<le> (SUP i. f i x)"
361           using f by (auto simp: G_def intro: SUP_upper2) }
362     next
363       fix A assume "A \<in> sets M"
364       have "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) =
365         (\<integral>\<^isup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
366         by (intro positive_integral_cong) (simp split: split_indicator)
367       also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f i x * indicator A x \<partial>M))"
368         using `incseq f` f `A \<in> sets M`
369         by (intro positive_integral_monotone_convergence_SUP)
370            (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
371       finally show "(\<integral>\<^isup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> \<nu> A"
372         using f `A \<in> sets M` by (auto intro!: SUP_least simp: G_def)
373     qed }
374   note SUP_in_G = this
375   let ?y = "SUP g : G. integral\<^isup>P M g"
376   have "?y \<le> \<nu> (space M)" unfolding G_def
377   proof (safe intro!: SUP_least)
378     fix g assume "\<forall>A\<in>sets M. (\<integral>\<^isup>+x. g x * indicator A x \<partial>M) \<le> \<nu> A"
379     from this[THEN bspec, OF top] show "integral\<^isup>P M g \<le> \<nu> (space M)"
380       by (simp cong: positive_integral_cong)
381   qed
382   from SUPR_countable_SUPR[OF `G \<noteq> {}`, of "integral\<^isup>P M"] guess ys .. note ys = this
383   then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n"
384   proof safe
385     fix n assume "range ys \<subseteq> integral\<^isup>P M ` G"
386     hence "ys n \<in> integral\<^isup>P M ` G" by auto
387     thus "\<exists>g. g\<in>G \<and> integral\<^isup>P M g = ys n" by auto
388   qed
389   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
390   hence y_eq: "?y = (SUP i. integral\<^isup>P M (gs i))" using ys by auto
391   let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})"
392   def f \<equiv> "\<lambda>x. SUP i. ?g i x"
393   let ?F = "\<lambda>A x. f x * indicator A x"
394   have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
395   { fix i have "?g i \<in> G"
396     proof (induct i)
397       case 0 thus ?case by simp fact
398     next
399       case (Suc i)
400       with Suc gs_not_empty `gs (Suc i) \<in> G` show ?case
401         by (auto simp add: atMost_Suc intro!: max_in_G)
402     qed }
403   note g_in_G = this
404   have "incseq ?g" using gs_not_empty
405     by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc)
406   from SUP_in_G[OF this g_in_G] have "f \<in> G" unfolding f_def .
407   then have [simp, intro]: "f \<in> borel_measurable M" unfolding G_def by auto
408   have "integral\<^isup>P M f = (SUP i. integral\<^isup>P M (?g i))" unfolding f_def
409     using g_in_G `incseq ?g`
410     by (auto intro!: positive_integral_monotone_convergence_SUP simp: G_def)
411   also have "\<dots> = ?y"
412   proof (rule antisym)
413     show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y"
414       using g_in_G
415       by (auto intro!: exI Sup_mono simp: SUP_def)
416     show "?y \<le> (SUP i. integral\<^isup>P M (?g i))" unfolding y_eq
417       by (auto intro!: SUP_mono positive_integral_mono Max_ge)
418   qed
419   finally have int_f_eq_y: "integral\<^isup>P M f = ?y" .
420   have "\<And>x. 0 \<le> f x"
421     unfolding f_def using `\<And>i. gs i \<in> G`
422     by (auto intro!: SUP_upper2 Max_ge_iff[THEN iffD2] simp: G_def)
423   let ?t = "\<lambda>A. \<nu> A - (\<integral>\<^isup>+x. ?F A x \<partial>M)"
424   let ?M = "M\<lparr> measure := ?t\<rparr>"
425   interpret M: sigma_algebra ?M
426     by (intro sigma_algebra_cong) auto
427   have f_le_\<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+x. ?F A x \<partial>M) \<le> \<nu> A"
428     using `f \<in> G` unfolding G_def by auto
429   have fmM: "finite_measure ?M"
430   proof
431     show "measure_space ?M"
432     proof (default, simp_all add: countably_additive_def positive_def, safe del: notI)
433       fix A :: "nat \<Rightarrow> 'a set"  assume A: "range A \<subseteq> sets M" "disjoint_family A"
434       have "(\<Sum>n. (\<integral>\<^isup>+x. ?F (A n) x \<partial>M)) = (\<integral>\<^isup>+x. (\<Sum>n. ?F (A n) x) \<partial>M)"
435         using `range A \<subseteq> sets M` `\<And>x. 0 \<le> f x`
436         by (intro positive_integral_suminf[symmetric]) auto
437       also have "\<dots> = (\<integral>\<^isup>+x. ?F (\<Union>n. A n) x \<partial>M)"
438         using `\<And>x. 0 \<le> f x`
439         by (intro positive_integral_cong) (simp add: suminf_cmult_ereal suminf_indicator[OF `disjoint_family A`])
440       finally have "(\<Sum>n. (\<integral>\<^isup>+x. ?F (A n) x \<partial>M)) = (\<integral>\<^isup>+x. ?F (\<Union>n. A n) x \<partial>M)" .
441       moreover have "(\<Sum>n. \<nu> (A n)) = \<nu> (\<Union>n. A n)"
442         using M'.measure_countably_additive A by (simp add: comp_def)
443       moreover have v_fin: "\<nu> (\<Union>i. A i) \<noteq> \<infinity>" using M'.finite_measure A by (simp add: countable_UN)
444       moreover {
445         have "(\<integral>\<^isup>+x. ?F (\<Union>i. A i) x \<partial>M) \<le> \<nu> (\<Union>i. A i)"
446           using A `f \<in> G` unfolding G_def by (auto simp: countable_UN)
447         also have "\<nu> (\<Union>i. A i) < \<infinity>" using v_fin by simp
448         finally have "(\<integral>\<^isup>+x. ?F (\<Union>i. A i) x \<partial>M) \<noteq> \<infinity>" by simp }
449       moreover have "\<And>i. (\<integral>\<^isup>+x. ?F (A i) x \<partial>M) \<le> \<nu> (A i)"
450         using A by (intro f_le_\<nu>) auto
451       ultimately
452       show "(\<Sum>n. ?t (A n)) = ?t (\<Union>i. A i)"
453         by (subst suminf_ereal_minus) (simp_all add: positive_integral_positive)
454     next
455       fix A assume A: "A \<in> sets M" show "0 \<le> \<nu> A - \<integral>\<^isup>+ x. ?F A x \<partial>M"
456         using f_le_\<nu>[OF A] `f \<in> G` M'.finite_measure[OF A] by (auto simp: G_def ereal_le_minus_iff)
457     qed
458   next
459     show "measure ?M (space ?M) \<noteq> \<infinity>"
460       using positive_integral_positive[of "?F (space M)"]
461       by (cases rule: ereal2_cases[of "\<nu> (space M)" "\<integral>\<^isup>+ x. ?F (space M) x \<partial>M"]) auto
462   qed
463   then interpret M: finite_measure ?M
464     where "space ?M = space M" and "sets ?M = sets M" and "measure ?M = ?t"
465     by (simp_all add: fmM)
466   have ac: "absolutely_continuous ?t" unfolding absolutely_continuous_def
467   proof
468     fix N assume N: "N \<in> null_sets"
469     with `absolutely_continuous \<nu>` have "\<nu> N = 0" unfolding absolutely_continuous_def by auto
470     moreover with N have "(\<integral>\<^isup>+ x. ?F N x \<partial>M) \<le> \<nu> N" using `f \<in> G` by (auto simp: G_def)
471     ultimately show "\<nu> N - (\<integral>\<^isup>+ x. ?F N x \<partial>M) = 0"
472       using positive_integral_positive by (auto intro!: antisym)
473   qed
474   have upper_bound: "\<forall>A\<in>sets M. ?t A \<le> 0"
475   proof (rule ccontr)
476     assume "\<not> ?thesis"
477     then obtain A where A: "A \<in> sets M" and pos: "0 < ?t A"
478       by (auto simp: not_le)
479     note pos
480     also have "?t A \<le> ?t (space M)"
481       using M.measure_mono[of A "space M"] A sets_into_space by simp
482     finally have pos_t: "0 < ?t (space M)" by simp
483     moreover
484     then have "\<mu> (space M) \<noteq> 0"
485       using ac unfolding absolutely_continuous_def by auto
486     then have pos_M: "0 < \<mu> (space M)"
487       using positive_measure[OF top] by (simp add: le_less)
488     moreover
489     have "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<le> \<nu> (space M)"
490       using `f \<in> G` unfolding G_def by auto
491     hence "(\<integral>\<^isup>+x. f x * indicator (space M) x \<partial>M) \<noteq> \<infinity>"
492       using M'.finite_measure_of_space by auto
493     moreover
494     def b \<equiv> "?t (space M) / \<mu> (space M) / 2"
495     ultimately have b: "b \<noteq> 0 \<and> 0 \<le> b \<and> b \<noteq> \<infinity>"
496       using M'.finite_measure_of_space positive_integral_positive[of "?F (space M)"]
497       by (cases rule: ereal3_cases[of "integral\<^isup>P M (?F (space M))" "\<nu> (space M)" "\<mu> (space M)"])
498          (simp_all add: field_simps)
499     then have b: "b \<noteq> 0" "0 \<le> b" "0 < b"  "b \<noteq> \<infinity>" by auto
500     let ?Mb = "?M\<lparr>measure := \<lambda>A. b * \<mu> A\<rparr>"
501     interpret b: sigma_algebra ?Mb by (intro sigma_algebra_cong) auto
502     have Mb: "finite_measure ?Mb"
503     proof
504       show "measure_space ?Mb"
505       proof
506         show "positive ?Mb (measure ?Mb)"
507           using `0 \<le> b` by (auto simp: positive_def)
508         show "countably_additive ?Mb (measure ?Mb)"
509           using `0 \<le> b` measure_countably_additive
510           by (auto simp: countably_additive_def suminf_cmult_ereal subset_eq)
511       qed
512       show "measure ?Mb (space ?Mb) \<noteq> \<infinity>"
513         using b by auto
514     qed
515     from M.Radon_Nikodym_aux[OF this]
516     obtain A0 where "A0 \<in> sets M" and
517       space_less_A0: "real (?t (space M)) - real (b * \<mu> (space M)) \<le> real (?t A0) - real (b * \<mu> A0)" and
518       *: "\<And>B. \<lbrakk> B \<in> sets M ; B \<subseteq> A0 \<rbrakk> \<Longrightarrow> 0 \<le> real (?t B) - real (b * \<mu> B)"
519       unfolding M.\<mu>'_def finite_measure.\<mu>'_def[OF Mb] by auto
520     { fix B assume B: "B \<in> sets M" "B \<subseteq> A0"
521       with *[OF this] have "b * \<mu> B \<le> ?t B"
522         using M'.finite_measure b finite_measure M.positive_measure[OF B(1)]
523         by (cases rule: ereal2_cases[of "?t B" "b * \<mu> B"]) auto }
524     note bM_le_t = this
525     let ?f0 = "\<lambda>x. f x + b * indicator A0 x"
526     { fix A assume A: "A \<in> sets M"
527       hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
528       have "(\<integral>\<^isup>+x. ?f0 x  * indicator A x \<partial>M) =
529         (\<integral>\<^isup>+x. f x * indicator A x + b * indicator (A \<inter> A0) x \<partial>M)"
530         by (auto intro!: positive_integral_cong split: split_indicator)
531       hence "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) =
532           (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0)"
533         using `A0 \<in> sets M` `A \<inter> A0 \<in> sets M` A b `f \<in> G`
534         by (simp add: G_def positive_integral_add positive_integral_cmult_indicator) }
535     note f0_eq = this
536     { fix A assume A: "A \<in> sets M"
537       hence "A \<inter> A0 \<in> sets M" using `A0 \<in> sets M` by auto
538       have f_le_v: "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
539         using `f \<in> G` A unfolding G_def by auto
540       note f0_eq[OF A]
541       also have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + b * \<mu> (A \<inter> A0) \<le>
542           (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t (A \<inter> A0)"
543         using bM_le_t[OF `A \<inter> A0 \<in> sets M`] `A \<in> sets M` `A0 \<in> sets M`
544         by (auto intro!: add_left_mono)
545       also have "\<dots> \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M) + ?t A"
546         using M.measure_mono[simplified, OF _ `A \<inter> A0 \<in> sets M` `A \<in> sets M`]
547         by (auto intro!: add_left_mono)
548       also have "\<dots> \<le> \<nu> A"
549         using f_le_v M'.finite_measure[simplified, OF `A \<in> sets M`] positive_integral_positive[of "?F A"]
550         by (cases "\<integral>\<^isup>+x. ?F A x \<partial>M", cases "\<nu> A") auto
551       finally have "(\<integral>\<^isup>+x. ?f0 x * indicator A x \<partial>M) \<le> \<nu> A" . }
552     hence "?f0 \<in> G" using `A0 \<in> sets M` b `f \<in> G` unfolding G_def
553       by (auto intro!: ereal_add_nonneg_nonneg)
554     have real: "?t (space M) \<noteq> \<infinity>" "?t A0 \<noteq> \<infinity>"
555       "b * \<mu> (space M) \<noteq> \<infinity>" "b * \<mu> A0 \<noteq> \<infinity>"
556       using `A0 \<in> sets M` b
557         finite_measure[of A0] M.finite_measure[of A0]
558         finite_measure_of_space M.finite_measure_of_space
559       by auto
560     have int_f_finite: "integral\<^isup>P M f \<noteq> \<infinity>"
561       using M'.finite_measure_of_space pos_t unfolding ereal_less_minus_iff
562       by (auto cong: positive_integral_cong)
563     have  "0 < ?t (space M) - b * \<mu> (space M)" unfolding b_def
564       using finite_measure_of_space M'.finite_measure_of_space pos_t pos_M
565       using positive_integral_positive[of "?F (space M)"]
566       by (cases rule: ereal3_cases[of "\<mu> (space M)" "\<nu> (space M)" "integral\<^isup>P M (?F (space M))"])
567          (auto simp: field_simps mult_less_cancel_left)
568     also have "\<dots> \<le> ?t A0 - b * \<mu> A0"
569       using space_less_A0 b
570       using
571         `A0 \<in> sets M`[THEN M.real_measure]
572         top[THEN M.real_measure]
573       apply safe
574       apply simp
575       using
576         `A0 \<in> sets M`[THEN real_measure]
577         `A0 \<in> sets M`[THEN M'.real_measure]
578         top[THEN real_measure]
579         top[THEN M'.real_measure]
580       by (cases b) auto
581     finally have 1: "b * \<mu> A0 < ?t A0"
582       using
583         `A0 \<in> sets M`[THEN M.real_measure]
584       apply safe
585       apply simp
586       using
587         `A0 \<in> sets M`[THEN real_measure]
588         `A0 \<in> sets M`[THEN M'.real_measure]
589       by (cases b) auto
590     have "0 < ?t A0"
591       using b `A0 \<in> sets M` by (auto intro!: le_less_trans[OF _ 1])
592     then have "\<mu> A0 \<noteq> 0" using ac unfolding absolutely_continuous_def
593       using `A0 \<in> sets M` by auto
594     then have "0 < \<mu> A0" using positive_measure[OF `A0 \<in> sets M`] by auto
595     hence "0 < b * \<mu> A0" using b by (auto simp: ereal_zero_less_0_iff)
596     with int_f_finite have "?y + 0 < integral\<^isup>P M f + b * \<mu> A0" unfolding int_f_eq_y
597       using `f \<in> G`
598       by (intro ereal_add_strict_mono) (auto intro!: SUP_upper2 positive_integral_positive)
599     also have "\<dots> = integral\<^isup>P M ?f0" using f0_eq[OF top] `A0 \<in> sets M` sets_into_space
600       by (simp cong: positive_integral_cong)
601     finally have "?y < integral\<^isup>P M ?f0" by simp
602     moreover from `?f0 \<in> G` have "integral\<^isup>P M ?f0 \<le> ?y" by (auto intro!: SUP_upper)
603     ultimately show False by auto
604   qed
605   show ?thesis
606   proof (safe intro!: bexI[of _ f])
607     fix A assume A: "A\<in>sets M"
608     show "\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
609     proof (rule antisym)
610       show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) \<le> \<nu> A"
611         using `f \<in> G` `A \<in> sets M` unfolding G_def by auto
612       show "\<nu> A \<le> (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
613         using upper_bound[THEN bspec, OF `A \<in> sets M`]
614         using M'.real_measure[OF A]
615         by (cases "integral\<^isup>P M (?F A)") auto
616     qed
617   qed simp
618 qed
620 lemma (in finite_measure) split_space_into_finite_sets_and_rest:
621   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
622   assumes ac: "absolutely_continuous \<nu>"
623   shows "\<exists>A0\<in>sets M. \<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> A0 = space M - (\<Union>i. B i) \<and>
624     (\<forall>A\<in>sets M. A \<subseteq> A0 \<longrightarrow> (\<mu> A = 0 \<and> \<nu> A = 0) \<or> (\<mu> A > 0 \<and> \<nu> A = \<infinity>)) \<and>
625     (\<forall>i. \<nu> (B i) \<noteq> \<infinity>)"
626 proof -
627   interpret v: measure_space ?N
628     where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
629     by fact auto
630   let ?Q = "{Q\<in>sets M. \<nu> Q \<noteq> \<infinity>}"
631   let ?a = "SUP Q:?Q. \<mu> Q"
632   have "{} \<in> ?Q" using v.empty_measure by auto
633   then have Q_not_empty: "?Q \<noteq> {}" by blast
634   have "?a \<le> \<mu> (space M)" using sets_into_space
635     by (auto intro!: SUP_least measure_mono)
636   then have "?a \<noteq> \<infinity>" using finite_measure_of_space
637     by auto
638   from SUPR_countable_SUPR[OF Q_not_empty, of \<mu>]
639   obtain Q'' where "range Q'' \<subseteq> \<mu> ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
640     by auto
641   then have "\<forall>i. \<exists>Q'. Q'' i = \<mu> Q' \<and> Q' \<in> ?Q" by auto
642   from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = \<mu> (Q' i)" "\<And>i. Q' i \<in> ?Q"
643     by auto
644   then have a_Lim: "?a = (SUP i::nat. \<mu> (Q' i))" using a by simp
645   let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i"
646   have Union: "(SUP i. \<mu> (?O i)) = \<mu> (\<Union>i. ?O i)"
647   proof (rule continuity_from_below[of ?O])
648     show "range ?O \<subseteq> sets M" using Q' by (auto intro!: finite_UN)
649     show "incseq ?O" by (fastforce intro!: incseq_SucI)
650   qed
651   have Q'_sets: "\<And>i. Q' i \<in> sets M" using Q' by auto
652   have O_sets: "\<And>i. ?O i \<in> sets M"
653      using Q' by (auto intro!: finite_UN Un)
654   then have O_in_G: "\<And>i. ?O i \<in> ?Q"
655   proof (safe del: notI)
656     fix i have "Q' ` {..i} \<subseteq> sets M"
657       using Q' by (auto intro: finite_UN)
658     with v.measure_finitely_subadditive[of "{.. i}" Q']
659     have "\<nu> (?O i) \<le> (\<Sum>i\<le>i. \<nu> (Q' i))" by auto
660     also have "\<dots> < \<infinity>" using Q' by (simp add: setsum_Pinfty)
661     finally show "\<nu> (?O i) \<noteq> \<infinity>" by simp
662   qed auto
663   have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce
664   have a_eq: "?a = \<mu> (\<Union>i. ?O i)" unfolding Union[symmetric]
665   proof (rule antisym)
666     show "?a \<le> (SUP i. \<mu> (?O i))" unfolding a_Lim
667       using Q' by (auto intro!: SUP_mono measure_mono finite_UN)
668     show "(SUP i. \<mu> (?O i)) \<le> ?a" unfolding SUP_def
669     proof (safe intro!: Sup_mono, unfold bex_simps)
670       fix i
671       have *: "(\<Union>Q' ` {..i}) = ?O i" by auto
672       then show "\<exists>x. (x \<in> sets M \<and> \<nu> x \<noteq> \<infinity>) \<and>
673         \<mu> (\<Union>Q' ` {..i}) \<le> \<mu> x"
674         using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
675     qed
676   qed
677   let ?O_0 = "(\<Union>i. ?O i)"
678   have "?O_0 \<in> sets M" using Q' by auto
679   def Q \<equiv> "\<lambda>i. case i of 0 \<Rightarrow> Q' 0 | Suc n \<Rightarrow> ?O (Suc n) - ?O n"
680   { fix i have "Q i \<in> sets M" unfolding Q_def using Q'[of 0] by (cases i) (auto intro: O_sets) }
681   note Q_sets = this
682   show ?thesis
683   proof (intro bexI exI conjI ballI impI allI)
684     show "disjoint_family Q"
685       by (fastforce simp: disjoint_family_on_def Q_def
686         split: nat.split_asm)
687     show "range Q \<subseteq> sets M"
688       using Q_sets by auto
689     { fix A assume A: "A \<in> sets M" "A \<subseteq> space M - ?O_0"
690       show "\<mu> A = 0 \<and> \<nu> A = 0 \<or> 0 < \<mu> A \<and> \<nu> A = \<infinity>"
691       proof (rule disjCI, simp)
692         assume *: "0 < \<mu> A \<longrightarrow> \<nu> A \<noteq> \<infinity>"
693         show "\<mu> A = 0 \<and> \<nu> A = 0"
694         proof cases
695           assume "\<mu> A = 0" moreover with ac A have "\<nu> A = 0"
696             unfolding absolutely_continuous_def by auto
697           ultimately show ?thesis by simp
698         next
699           assume "\<mu> A \<noteq> 0" with * have "\<nu> A \<noteq> \<infinity>" using positive_measure[OF A(1)] by auto
700           with A have "\<mu> ?O_0 + \<mu> A = \<mu> (?O_0 \<union> A)"
701             using Q' by (auto intro!: measure_additive countable_UN)
702           also have "\<dots> = (SUP i. \<mu> (?O i \<union> A))"
703           proof (rule continuity_from_below[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
704             show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
705               using `\<nu> A \<noteq> \<infinity>` O_sets A by auto
706           qed (fastforce intro!: incseq_SucI)
707           also have "\<dots> \<le> ?a"
708           proof (safe intro!: SUP_least)
709             fix i have "?O i \<union> A \<in> ?Q"
710             proof (safe del: notI)
711               show "?O i \<union> A \<in> sets M" using O_sets A by auto
712               from O_in_G[of i] have "\<nu> (?O i \<union> A) \<le> \<nu> (?O i) + \<nu> A"
713                 using v.measure_subadditive[of "?O i" A] A O_sets by auto
714               with O_in_G[of i] show "\<nu> (?O i \<union> A) \<noteq> \<infinity>"
715                 using `\<nu> A \<noteq> \<infinity>` by auto
716             qed
717             then show "\<mu> (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
718           qed
719           finally have "\<mu> A = 0"
720             unfolding a_eq using real_measure[OF `?O_0 \<in> sets M`] real_measure[OF A(1)] by auto
721           with `\<mu> A \<noteq> 0` show ?thesis by auto
722         qed
723       qed }
724     { fix i show "\<nu> (Q i) \<noteq> \<infinity>"
725       proof (cases i)
726         case 0 then show ?thesis
727           unfolding Q_def using Q'[of 0] by simp
728       next
729         case (Suc n)
730         then show ?thesis unfolding Q_def
731           using `?O n \<in> ?Q` `?O (Suc n) \<in> ?Q`
732           using v.measure_mono[OF O_mono, of n] v.positive_measure[of "?O n"] v.positive_measure[of "?O (Suc n)"]
733           using v.measure_Diff[of "?O n" "?O (Suc n)", OF _ _ _ O_mono]
734           by (cases rule: ereal2_cases[of "\<nu> (\<Union> x\<le>Suc n. Q' x)" "\<nu> (\<Union> i\<le>n. Q' i)"]) auto
735       qed }
736     show "space M - ?O_0 \<in> sets M" using Q'_sets by auto
737     { fix j have "(\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q i)"
738       proof (induct j)
739         case 0 then show ?case by (simp add: Q_def)
740       next
741         case (Suc j)
742         have eq: "\<And>j. (\<Union>i\<le>j. ?O i) = (\<Union>i\<le>j. Q' i)" by fastforce
743         have "{..j} \<union> {..Suc j} = {..Suc j}" by auto
744         then have "(\<Union>i\<le>Suc j. Q' i) = (\<Union>i\<le>j. Q' i) \<union> Q (Suc j)"
745           by (simp add: UN_Un[symmetric] Q_def del: UN_Un)
746         then show ?case using Suc by (auto simp add: eq atMost_Suc)
747       qed }
748     then have "(\<Union>j. (\<Union>i\<le>j. ?O i)) = (\<Union>j. (\<Union>i\<le>j. Q i))" by simp
749     then show "space M - ?O_0 = space M - (\<Union>i. Q i)" by fastforce
750   qed
751 qed
753 lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
754   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
755   assumes "absolutely_continuous \<nu>"
756   shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
757 proof -
758   interpret v: measure_space ?N
759     where "space ?N = space M" and "sets ?N = sets M" and "measure ?N = \<nu>"
760     by fact auto
761   from split_space_into_finite_sets_and_rest[OF assms]
762   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
763     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
764     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
765     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 = \<infinity>"
766     and Q_fin: "\<And>i. \<nu> (Q i) \<noteq> \<infinity>" by force
767   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
768   have "\<forall>i. \<exists>f. f\<in>borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M.
769     \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
770   proof
771     fix i
772     have indicator_eq: "\<And>f x A. (f x :: ereal) * indicator (Q i \<inter> A) x * indicator (Q i) x
773       = (f x * indicator (Q i) x) * indicator A x"
774       unfolding indicator_def by auto
775     have fm: "finite_measure (restricted_space (Q i))"
776       (is "finite_measure ?R") by (rule restricted_finite_measure[OF Q_sets[of i]])
777     then interpret R: finite_measure ?R .
778     have fmv: "finite_measure (restricted_space (Q i) \<lparr> measure := \<nu>\<rparr>)" (is "finite_measure ?Q")
779     proof
780       show "measure_space ?Q"
781         using v.restricted_measure_space Q_sets[of i] by auto
782       show "measure ?Q (space ?Q) \<noteq> \<infinity>" using Q_fin by simp
783     qed
784     have "R.absolutely_continuous \<nu>"
785       using `absolutely_continuous \<nu>` `Q i \<in> sets M`
786       by (auto simp: R.absolutely_continuous_def absolutely_continuous_def)
787     from R.Radon_Nikodym_finite_measure[OF fmv this]
788     obtain f where f: "(\<lambda>x. f x * indicator (Q i) x) \<in> borel_measurable M"
789       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)"
790       unfolding Bex_def borel_measurable_restricted[OF `Q i \<in> sets M`]
791         positive_integral_restricted[OF `Q i \<in> sets M`] by (auto simp: indicator_eq)
792     then show "\<exists>f. f\<in>borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M.
793       \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f x * indicator (Q i \<inter> A) x \<partial>M))"
794       by (auto intro!: exI[of _ "\<lambda>x. max 0 (f x * indicator (Q i) x)"] positive_integral_cong_pos
795         split: split_indicator split_if_asm simp: max_def)
796   qed
797   from choice[OF this] obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
798     and f: "\<And>A i. A \<in> sets M \<Longrightarrow>
799       \<nu> (Q i \<inter> A) = (\<integral>\<^isup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)"
800     by auto
801   let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator Q0 x"
802   show ?thesis
803   proof (safe intro!: bexI[of _ ?f])
804     show "?f \<in> borel_measurable M" using Q0 borel Q_sets
805       by (auto intro!: measurable_If)
806     show "\<And>x. 0 \<le> ?f x" using borel by (auto intro!: suminf_0_le simp: indicator_def)
807     fix A assume "A \<in> sets M"
808     have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
809     have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
810       "\<And>i. AE x. 0 \<le> f i x * indicator (Q i \<inter> A) x"
811       using borel Qi Q0(1) `A \<in> sets M` by (auto intro!: borel_measurable_ereal_times)
812     have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator (Q0 \<inter> A) x \<partial>M)"
813       using borel by (intro positive_integral_cong) (auto simp: indicator_def)
814     also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * \<mu> (Q0 \<inter> A)"
815       using borel Qi Q0(1) `A \<in> sets M`
816       by (subst positive_integral_add) (auto simp del: ereal_infty_mult
817           simp add: positive_integral_cmult_indicator Int intro!: suminf_0_le)
818     also have "\<dots> = (\<Sum>i. \<nu> (Q i \<inter> A)) + \<infinity> * \<mu> (Q0 \<inter> A)"
819       by (subst f[OF `A \<in> sets M`], subst positive_integral_suminf) auto
820     finally have "(\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. \<nu> (Q i \<inter> A)) + \<infinity> * \<mu> (Q0 \<inter> A)" .
821     moreover have "(\<Sum>i. \<nu> (Q i \<inter> A)) = \<nu> ((\<Union>i. Q i) \<inter> A)"
822       using Q Q_sets `A \<in> sets M`
823       by (intro v.measure_countably_additive[of "\<lambda>i. Q i \<inter> A", unfolded comp_def, simplified])
824          (auto simp: disjoint_family_on_def)
825     moreover have "\<infinity> * \<mu> (Q0 \<inter> A) = \<nu> (Q0 \<inter> A)"
826     proof -
827       have "Q0 \<inter> A \<in> sets M" using Q0(1) `A \<in> sets M` by blast
828       from in_Q0[OF this] show ?thesis by auto
829     qed
830     moreover have "Q0 \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
831       using Q_sets `A \<in> sets M` Q0(1) by (auto intro!: countable_UN)
832     moreover have "((\<Union>i. Q i) \<inter> A) \<union> (Q0 \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> (Q0 \<inter> A) = {}"
833       using `A \<in> sets M` sets_into_space Q0 by auto
834     ultimately show "\<nu> A = (\<integral>\<^isup>+x. ?f x * indicator A x \<partial>M)"
835       using v.measure_additive[simplified, of "(\<Union>i. Q i) \<inter> A" "Q0 \<inter> A"]
836       by simp
837   qed
838 qed
840 lemma (in sigma_finite_measure) Radon_Nikodym:
841   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
842   assumes ac: "absolutely_continuous \<nu>"
843   shows "\<exists>f \<in> borel_measurable M. (\<forall>x. 0 \<le> f x) \<and> (\<forall>A\<in>sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
844 proof -
845   from Ex_finite_integrable_function
846   obtain h where finite: "integral\<^isup>P M h \<noteq> \<infinity>" and
847     borel: "h \<in> borel_measurable M" and
848     nn: "\<And>x. 0 \<le> h x" and
849     pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
850     "\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto
851   let ?T = "\<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M)"
852   let ?MT = "M\<lparr> measure := ?T \<rparr>"
853   interpret T: finite_measure ?MT
854     where "space ?MT = space M" and "sets ?MT = sets M" and "measure ?MT = ?T"
855     using borel finite nn
856     by (auto intro!: measure_space_density finite_measureI cong: positive_integral_cong)
857   have "T.absolutely_continuous \<nu>"
858   proof (unfold T.absolutely_continuous_def, safe)
859     fix N assume "N \<in> sets M" "(\<integral>\<^isup>+x. h x * indicator N x \<partial>M) = 0"
860     with borel ac pos have "AE x. x \<notin> N"
861       by (subst (asm) positive_integral_0_iff_AE) (auto split: split_indicator simp: not_le)
862     then have "N \<in> null_sets" using `N \<in> sets M` sets_into_space
863       by (subst (asm) AE_iff_measurable[OF `N \<in> sets M`]) auto
864     then show "\<nu> N = 0" using ac by (auto simp: absolutely_continuous_def)
865   qed
866   from T.Radon_Nikodym_finite_measure_infinite[simplified, OF assms(1) this]
867   obtain f where f_borel: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and
868     fT: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>?MT)"
869     by (auto simp: measurable_def)
870   show ?thesis
871   proof (safe intro!: bexI[of _ "\<lambda>x. h x * f x"])
872     show "(\<lambda>x. h x * f x) \<in> borel_measurable M"
873       using borel f_borel by auto
874     show "\<And>x. 0 \<le> h x * f x" using nn f_borel by auto
875     fix A assume "A \<in> sets M"
876     then show "\<nu> A = (\<integral>\<^isup>+x. h x * f x * indicator A x \<partial>M)"
877       unfolding fT[OF `A \<in> sets M`] mult_assoc using nn borel f_borel
878       by (intro positive_integral_translated_density) auto
879   qed
880 qed
882 section "Uniqueness of densities"
884 lemma (in measure_space) finite_density_unique:
885   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
886   assumes pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> g x"
887   and fin: "integral\<^isup>P M f \<noteq> \<infinity>"
888   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))
889     \<longleftrightarrow> (AE x. f x = g x)"
890     (is "(\<forall>A\<in>sets M. ?P f A = ?P g A) \<longleftrightarrow> _")
891 proof (intro iffI ballI)
892   fix A assume eq: "AE x. f x = g x"
893   then show "?P f A = ?P g A"
894     by (auto intro: positive_integral_cong_AE)
895 next
896   assume eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
897   from this[THEN bspec, OF top] fin
898   have g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" by (simp cong: positive_integral_cong)
899   { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
900       and pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> g x"
901       and g_fin: "integral\<^isup>P M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
902     let ?N = "{x\<in>space M. g x < f x}"
903     have N: "?N \<in> sets M" using borel by simp
904     have "?P g ?N \<le> integral\<^isup>P M g" using pos
905       by (intro positive_integral_mono_AE) (auto split: split_indicator)
906     then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by auto
907     have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^isup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
908       by (auto intro!: positive_integral_cong simp: indicator_def)
909     also have "\<dots> = ?P f ?N - ?P g ?N"
910     proof (rule positive_integral_diff)
911       show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
912         using borel N by auto
913       show "AE x. g x * indicator ?N x \<le> f x * indicator ?N x"
914            "AE x. 0 \<le> g x * indicator ?N x"
915         using pos by (auto split: split_indicator)
916     qed fact
917     also have "\<dots> = 0"
918       unfolding eq[THEN bspec, OF N] using positive_integral_positive Pg_fin by auto
919     finally have "AE x. f x \<le> g x"
920       using pos borel positive_integral_PInf_AE[OF borel(2) g_fin]
921       by (subst (asm) positive_integral_0_iff_AE)
922          (auto split: split_indicator simp: not_less ereal_minus_le_iff) }
923   from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
924   show "AE x. f x = g x" by auto
925 qed
927 lemma (in finite_measure) density_unique_finite_measure:
928   assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
929   assumes pos: "AE x. 0 \<le> f x" "AE x. 0 \<le> f' x"
930   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)"
931     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
932   shows "AE x. f x = f' x"
933 proof -
934   let ?\<nu> = "\<lambda>A. ?P f A" and ?\<nu>' = "\<lambda>A. ?P f' A"
935   let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x"
936   interpret M: measure_space "M\<lparr> measure := ?\<nu>\<rparr>"
937     using borel(1) pos(1) by (rule measure_space_density) simp
938   have ac: "absolutely_continuous ?\<nu>"
939     using f by (rule density_is_absolutely_continuous)
940   from split_space_into_finite_sets_and_rest[OF `measure_space (M\<lparr> measure := ?\<nu>\<rparr>)` ac]
941   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
942     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
943     and Q0: "Q0 \<in> sets M" "Q0 = space M - (\<Union>i. Q i)"
944     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 = \<infinity>"
945     and Q_fin: "\<And>i. ?\<nu> (Q i) \<noteq> \<infinity>" by force
946   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
947   let ?N = "{x\<in>space M. f x \<noteq> f' x}"
948   have "?N \<in> sets M" using borel by auto
949   have *: "\<And>i x A. \<And>y::ereal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
950     unfolding indicator_def by auto
951   have "\<forall>i. AE x. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
952     by (intro finite_density_unique[THEN iffD1] allI)
953        (auto intro!: borel_measurable_ereal_times f Int simp: *)
954   moreover have "AE x. ?f Q0 x = ?f' Q0 x"
955   proof (rule AE_I')
956     { fix f :: "'a \<Rightarrow> ereal" assume borel: "f \<in> borel_measurable M"
957         and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?\<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
958       let ?A = "\<lambda>i. Q0 \<inter> {x \<in> space M. f x < (i::nat)}"
959       have "(\<Union>i. ?A i) \<in> null_sets"
960       proof (rule null_sets_UN)
961         fix i ::nat have "?A i \<in> sets M"
962           using borel Q0(1) by auto
963         have "?\<nu> (?A i) \<le> (\<integral>\<^isup>+x. (i::ereal) * indicator (?A i) x \<partial>M)"
964           unfolding eq[OF `?A i \<in> sets M`]
965           by (auto intro!: positive_integral_mono simp: indicator_def)
966         also have "\<dots> = i * \<mu> (?A i)"
967           using `?A i \<in> sets M` by (auto intro!: positive_integral_cmult_indicator)
968         also have "\<dots> < \<infinity>"
969           using `?A i \<in> sets M`[THEN finite_measure] by auto
970         finally have "?\<nu> (?A i) \<noteq> \<infinity>" by simp
971         then show "?A i \<in> null_sets" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
972       qed
973       also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
974         by (auto simp: less_PInf_Ex_of_nat real_eq_of_nat)
975       finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets" by simp }
976     from this[OF borel(1) refl] this[OF borel(2) f]
977     have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets" by simp_all
978     then show "(Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets" by (rule nullsets.Un)
979     show "{x \<in> space M. ?f Q0 x \<noteq> ?f' Q0 x} \<subseteq>
980       (Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> (Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def)
981   qed
982   moreover have "\<And>x. (?f Q0 x = ?f' Q0 x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
983     ?f (space M) x = ?f' (space M) x"
984     by (auto simp: indicator_def Q0)
985   ultimately have "AE x. ?f (space M) x = ?f' (space M) x"
986     by (auto simp: AE_all_countable[symmetric])
987   then show "AE x. f x = f' x" by auto
988 qed
990 lemma (in sigma_finite_measure) density_unique:
991   assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
992   assumes f': "f' \<in> borel_measurable M" "AE x. 0 \<le> f' x"
993   assumes eq: "\<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)"
994     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
995   shows "AE x. f x = f' x"
996 proof -
997   obtain h where h_borel: "h \<in> borel_measurable M"
998     and fin: "integral\<^isup>P M h \<noteq> \<infinity>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<infinity>" "\<And>x. 0 \<le> h x"
999     using Ex_finite_integrable_function by auto
1000   then have h_nn: "AE x. 0 \<le> h x" by auto
1001   let ?H = "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
1002   have H: "measure_space ?H"
1003     using h_borel h_nn by (rule measure_space_density) simp
1004   then interpret h: measure_space ?H .
1005   interpret h: finite_measure "M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+x. h x * indicator A x \<partial>M) \<rparr>"
1006     by (intro H finite_measureI) (simp cong: positive_integral_cong add: fin)
1007   let ?fM = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)\<rparr>"
1008   interpret f: measure_space ?fM
1009     using f by (rule measure_space_density) simp
1010   let ?f'M = "M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. f' x * indicator A x \<partial>M)\<rparr>"
1011   interpret f': measure_space ?f'M
1012     using f' by (rule measure_space_density) simp
1013   { fix A assume "A \<in> sets M"
1014     then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
1015       using pos(1) sets_into_space by (force simp: indicator_def)
1016     then have "(\<integral>\<^isup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets"
1017       using h_borel `A \<in> sets M` h_nn by (subst positive_integral_0_iff) auto }
1018   note h_null_sets = this
1019   { fix A assume "A \<in> sets M"
1020     have "(\<integral>\<^isup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?fM)"
1021       using `A \<in> sets M` h_borel h_nn f f'
1022       by (intro positive_integral_translated_density[symmetric]) auto
1023     also have "\<dots> = (\<integral>\<^isup>+x. h x * indicator A x \<partial>?f'M)"
1024       by (rule f'.positive_integral_cong_measure) (simp_all add: eq)
1025     also have "\<dots> = (\<integral>\<^isup>+x. f' x * (h x * indicator A x) \<partial>M)"
1026       using `A \<in> sets M` h_borel h_nn f f'
1027       by (intro positive_integral_translated_density) auto
1028     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)"
1029       by (simp add: ac_simps)
1030     then have "(\<integral>\<^isup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^isup>+x. (f' x * indicator A x) \<partial>?H)"
1031       using `A \<in> sets M` h_borel h_nn f f'
1032       by (subst (asm) (1 2) positive_integral_translated_density[symmetric]) auto }
1033   then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
1034     by (intro h.density_unique_finite_measure absolutely_continuous_AE[OF H] density_is_absolutely_continuous)
1035        simp_all
1036   then show "AE x. f x = f' x"
1037     unfolding h.almost_everywhere_def almost_everywhere_def
1038     by (auto simp add: h_null_sets)
1039 qed
1041 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
1042   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" (is "measure_space ?N")
1043     and f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
1044     and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
1045   shows "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>) \<longleftrightarrow> (AE x. f x \<noteq> \<infinity>)"
1046 proof
1047   assume "sigma_finite_measure ?N"
1048   then interpret \<nu>: sigma_finite_measure ?N
1049     where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
1050     and "sets ?N = sets M" and "space ?N = space M" by (auto simp: measurable_def)
1051   from \<nu>.Ex_finite_integrable_function obtain h where
1052     h: "h \<in> borel_measurable M" "integral\<^isup>P ?N h \<noteq> \<infinity>" and
1053     h_nn: "\<And>x. 0 \<le> h x" and
1054     fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>" by auto
1055   have "AE x. f x * h x \<noteq> \<infinity>"
1056   proof (rule AE_I')
1057     have "integral\<^isup>P ?N h = (\<integral>\<^isup>+x. f x * h x \<partial>M)" using f h h_nn
1058       by (subst \<nu>.positive_integral_cong_measure[symmetric,
1059           of "M\<lparr> measure := \<lambda> A. \<integral>\<^isup>+x. f x * indicator A x \<partial>M \<rparr>"])
1060          (auto intro!: positive_integral_translated_density simp: eq)
1061     then have "(\<integral>\<^isup>+x. f x * h x \<partial>M) \<noteq> \<infinity>"
1062       using h(2) by simp
1063     then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets"
1064       using f h(1) by (auto intro!: positive_integral_PInf borel_measurable_vimage)
1065   qed auto
1066   then show "AE x. f x \<noteq> \<infinity>"
1067     using fin by (auto elim!: AE_Ball_mp)
1068 next
1069   assume AE: "AE x. f x \<noteq> \<infinity>"
1070   from sigma_finite guess Q .. note Q = this
1071   interpret \<nu>: measure_space ?N
1072     where "borel_measurable ?N = borel_measurable M" and "measure ?N = \<nu>"
1073     and "sets ?N = sets M" and "space ?N = space M" using \<nu> by (auto simp: measurable_def)
1074   def A \<equiv> "\<lambda>i. f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ereal(of_nat (Suc n))}) \<inter> space M"
1075   { fix i j have "A i \<inter> Q j \<in> sets M"
1076     unfolding A_def using f Q
1077     apply (rule_tac Int)
1078     by (cases i) (auto intro: measurable_sets[OF f(1)]) }
1079   note A_in_sets = this
1080   let ?A = "\<lambda>n. case prod_decode n of (i,j) \<Rightarrow> A i \<inter> Q j"
1081   show "sigma_finite_measure ?N"
1082   proof (default, intro exI conjI subsetI allI)
1083     fix x assume "x \<in> range ?A"
1084     then obtain n where n: "x = ?A n" by auto
1085     then show "x \<in> sets ?N" using A_in_sets by (cases "prod_decode n") auto
1086   next
1087     have "(\<Union>i. ?A i) = (\<Union>i j. A i \<inter> Q j)"
1088     proof safe
1089       fix x i j assume "x \<in> A i" "x \<in> Q j"
1090       then show "x \<in> (\<Union>i. case prod_decode i of (i, j) \<Rightarrow> A i \<inter> Q j)"
1091         by (intro UN_I[of "prod_encode (i,j)"]) auto
1092     qed auto
1093     also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto
1094     also have "(\<Union>i. A i) = space M"
1095     proof safe
1096       fix x assume x: "x \<in> space M"
1097       show "x \<in> (\<Union>i. A i)"
1098       proof (cases "f x")
1099         case PInf with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
1100       next
1101         case (real r)
1102         with less_PInf_Ex_of_nat[of "f x"] obtain n :: nat where "f x < n" by (auto simp: real_eq_of_nat)
1103         then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"] simp: real_eq_of_nat)
1104       next
1105         case MInf with x show ?thesis
1106           unfolding A_def by (auto intro!: exI[of _ "Suc 0"])
1107       qed
1108     qed (auto simp: A_def)
1109     finally show "(\<Union>i. ?A i) = space ?N" by simp
1110   next
1111     fix n obtain i j where
1112       [simp]: "prod_decode n = (i, j)" by (cases "prod_decode n") auto
1113     have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>"
1114     proof (cases i)
1115       case 0
1116       have "AE x. f x * indicator (A i \<inter> Q j) x = 0"
1117         using AE by (auto simp: A_def `i = 0`)
1118       from positive_integral_cong_AE[OF this] show ?thesis by simp
1119     next
1120       case (Suc n)
1121       then have "(\<integral>\<^isup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
1122         (\<integral>\<^isup>+x. (Suc n :: ereal) * indicator (Q j) x \<partial>M)"
1123         by (auto intro!: positive_integral_mono simp: indicator_def A_def real_eq_of_nat)
1124       also have "\<dots> = Suc n * \<mu> (Q j)"
1125         using Q by (auto intro!: positive_integral_cmult_indicator)
1126       also have "\<dots> < \<infinity>"
1127         using Q by (auto simp: real_eq_of_nat[symmetric])
1128       finally show ?thesis by simp
1129     qed
1130     then show "measure ?N (?A n) \<noteq> \<infinity>"
1131       using A_in_sets Q eq by auto
1132   qed
1133 qed
1135 section "Radon-Nikodym derivative"
1137 definition
1138   "RN_deriv M \<nu> \<equiv> SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and>
1139     (\<forall>A \<in> sets M. \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M))"
1141 lemma (in sigma_finite_measure) RN_deriv_cong:
1142   assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> measure M' A = \<mu> A" "sets M' = sets M" "space M' = space M"
1143     and \<nu>: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu>' A = \<nu> A"
1144   shows "RN_deriv M' \<nu>' x = RN_deriv M \<nu> x"
1145 proof -
1146   interpret \<mu>': sigma_finite_measure M'
1147     using cong by (rule sigma_finite_measure_cong)
1148   show ?thesis
1149     unfolding RN_deriv_def
1150     by (simp add: cong \<nu> positive_integral_cong_measure[OF cong] measurable_def)
1151 qed
1153 lemma (in sigma_finite_measure) RN_deriv:
1154   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
1155   assumes "absolutely_continuous \<nu>"
1156   shows "RN_deriv M \<nu> \<in> borel_measurable M" (is ?borel)
1157   and "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
1158     (is "\<And>A. _ \<Longrightarrow> ?int A")
1159   and "0 \<le> RN_deriv M \<nu> x"
1160 proof -
1161   note Ex = Radon_Nikodym[OF assms, unfolded Bex_def]
1162   then show ?borel unfolding RN_deriv_def by (rule someI2_ex) auto
1163   from Ex show "0 \<le> RN_deriv M \<nu> x" unfolding RN_deriv_def
1164     by (rule someI2_ex) simp
1165   fix A assume "A \<in> sets M"
1166   from Ex show "?int A" unfolding RN_deriv_def
1167     by (rule someI2_ex) (simp add: `A \<in> sets M`)
1168 qed
1170 lemma (in sigma_finite_measure) RN_deriv_positive_integral:
1171   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
1172     and f: "f \<in> borel_measurable M"
1173   shows "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
1174 proof -
1175   interpret \<nu>: measure_space "M\<lparr>measure := \<nu>\<rparr>" by fact
1176   note RN = RN_deriv[OF \<nu>]
1177   have "integral\<^isup>P (M\<lparr>measure := \<nu>\<rparr>) f = (\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<nu>\<rparr>)"
1178     unfolding positive_integral_max_0 ..
1179   also have "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<nu>\<rparr>) =
1180     (\<integral>\<^isup>+x. max 0 (f x) \<partial>M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)\<rparr>)"
1181     by (intro \<nu>.positive_integral_cong_measure[symmetric]) (simp_all add: RN(2))
1182   also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * max 0 (f x) \<partial>M)"
1183     by (intro positive_integral_translated_density) (auto simp add: RN f)
1184   also have "\<dots> = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * f x \<partial>M)"
1185     using RN_deriv(3)[OF \<nu>]
1186     by (auto intro!: positive_integral_cong_pos split: split_if_asm
1187              simp: max_def ereal_mult_le_0_iff)
1188   finally show ?thesis .
1189 qed
1191 lemma (in sigma_finite_measure) RN_deriv_unique:
1192   assumes \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
1193   and f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
1194   and eq: "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = (\<integral>\<^isup>+x. f x * indicator A x \<partial>M)"
1195   shows "AE x. f x = RN_deriv M \<nu> x"
1196 proof (rule density_unique[OF f RN_deriv(1)[OF \<nu>]])
1197   show "AE x. 0 \<le> RN_deriv M \<nu> x" using RN_deriv[OF \<nu>] by auto
1198   fix A assume A: "A \<in> sets M"
1199   show "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * indicator A x \<partial>M)"
1200     unfolding eq[OF A, symmetric] RN_deriv(2)[OF \<nu> A, symmetric] ..
1201 qed
1203 lemma (in sigma_finite_measure) RN_deriv_vimage:
1204   assumes T: "T \<in> measure_preserving M M'"
1205     and T': "T' \<in> measure_preserving (M'\<lparr> measure := \<nu>' \<rparr>) (M\<lparr> measure := \<nu> \<rparr>)"
1206     and inv: "\<And>x. x \<in> space M \<Longrightarrow> T' (T x) = x"
1207   and \<nu>': "measure_space (M'\<lparr>measure := \<nu>'\<rparr>)" "measure_space.absolutely_continuous M' \<nu>'"
1208   shows "AE x. RN_deriv M' \<nu>' (T x) = RN_deriv M \<nu> x"
1209 proof (rule RN_deriv_unique)
1210   interpret \<nu>': measure_space "M'\<lparr>measure := \<nu>'\<rparr>" by fact
1211   show "measure_space (M\<lparr> measure := \<nu>\<rparr>)"
1212     by (rule \<nu>'.measure_space_vimage[OF _ T'], simp) default
1213   interpret M': measure_space M'
1214   proof (rule measure_space_vimage)
1215     have "sigma_algebra (M'\<lparr> measure := \<nu>'\<rparr>)" by default
1216     then show "sigma_algebra M'" by simp
1217   qed fact
1218   show "absolutely_continuous \<nu>" unfolding absolutely_continuous_def
1219   proof safe
1220     fix N assume N: "N \<in> sets M" and N_0: "\<mu> N = 0"
1221     then have N': "T' -` N \<inter> space M' \<in> sets M'"
1222       using T' by (auto simp: measurable_def measure_preserving_def)
1223     have "T -` (T' -` N \<inter> space M') \<inter> space M = N"
1224       using inv T N sets_into_space[OF N] by (auto simp: measurable_def measure_preserving_def)
1225     then have "measure M' (T' -` N \<inter> space M') = 0"
1226       using measure_preservingD[OF T N'] N_0 by auto
1227     with \<nu>'(2) N' show "\<nu> N = 0" using measure_preservingD[OF T', of N] N
1228       unfolding M'.absolutely_continuous_def measurable_def by auto
1229   qed
1230   interpret M': sigma_finite_measure M'
1231   proof
1232     from sigma_finite guess F .. note F = this
1233     show "\<exists>A::nat \<Rightarrow> 'c set. range A \<subseteq> sets M' \<and> (\<Union>i. A i) = space M' \<and> (\<forall>i. M'.\<mu> (A i) \<noteq> \<infinity>)"
1234     proof (intro exI conjI allI)
1235       show *: "range (\<lambda>i. T' -` F i \<inter> space M') \<subseteq> sets M'"
1236         using F T' by (auto simp: measurable_def measure_preserving_def)
1237       show "(\<Union>i. T' -` F i \<inter> space M') = space M'"
1238         using F T' by (force simp: measurable_def measure_preserving_def)
1239       fix i
1240       have "T' -` F i \<inter> space M' \<in> sets M'" using * by auto
1241       note measure_preservingD[OF T this, symmetric]
1242       moreover
1243       have Fi: "F i \<in> sets M" using F by auto
1244       then have "T -` (T' -` F i \<inter> space M') \<inter> space M = F i"
1245         using T inv sets_into_space[OF Fi]
1246         by (auto simp: measurable_def measure_preserving_def)
1247       ultimately show "measure M' (T' -` F i \<inter> space M') \<noteq> \<infinity>"
1248         using F by simp
1249     qed
1250   qed
1251   have "(RN_deriv M' \<nu>') \<circ> T \<in> borel_measurable M"
1252     by (intro measurable_comp[where b=M'] M'.RN_deriv(1) measure_preservingD2[OF T]) fact+
1253   then show "(\<lambda>x. RN_deriv M' \<nu>' (T x)) \<in> borel_measurable M"
1254     by (simp add: comp_def)
1255   show "AE x. 0 \<le> RN_deriv M' \<nu>' (T x)" using M'.RN_deriv(3)[OF \<nu>'] by auto
1256   fix A let ?A = "T' -` A \<inter> space M'"
1257   assume A: "A \<in> sets M"
1258   then have A': "?A \<in> sets M'" using T' unfolding measurable_def measure_preserving_def
1259     by auto
1260   from A have "\<nu> A = \<nu>' ?A" using T'[THEN measure_preservingD] by simp
1261   also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' x * indicator ?A x \<partial>M'"
1262     using A' by (rule M'.RN_deriv(2)[OF \<nu>'])
1263   also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator ?A (T x) \<partial>M"
1264   proof (rule positive_integral_vimage)
1265     show "sigma_algebra M'" by default
1266     show "(\<lambda>x. RN_deriv M' \<nu>' x * indicator (T' -` A \<inter> space M') x) \<in> borel_measurable M'"
1267       by (auto intro!: A' M'.RN_deriv(1)[OF \<nu>'])
1268   qed fact
1269   also have "\<dots> = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator A x \<partial>M"
1270     using T inv by (auto intro!: positive_integral_cong simp: measure_preserving_def measurable_def indicator_def)
1271   finally show "\<nu> A = \<integral>\<^isup>+ x. RN_deriv M' \<nu>' (T x) * indicator A x \<partial>M" .
1272 qed
1274 lemma (in sigma_finite_measure) RN_deriv_finite:
1275   assumes sfm: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" and ac: "absolutely_continuous \<nu>"
1276   shows "AE x. RN_deriv M \<nu> x \<noteq> \<infinity>"
1277 proof -
1278   interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
1279   have \<nu>: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
1280   from sfm show ?thesis
1281     using sigma_finite_iff_density_finite[OF \<nu> RN_deriv(1)[OF \<nu> ac]] RN_deriv(2,3)[OF \<nu> ac] by simp
1282 qed
1284 lemma (in sigma_finite_measure)
1285   assumes \<nu>: "sigma_finite_measure (M\<lparr>measure := \<nu>\<rparr>)" "absolutely_continuous \<nu>"
1286     and f: "f \<in> borel_measurable M"
1287   shows RN_deriv_integrable: "integrable (M\<lparr>measure := \<nu>\<rparr>) f \<longleftrightarrow>
1288       integrable M (\<lambda>x. real (RN_deriv M \<nu> x) * f x)" (is ?integrable)
1289     and RN_deriv_integral: "integral\<^isup>L (M\<lparr>measure := \<nu>\<rparr>) f =
1290       (\<integral>x. real (RN_deriv M \<nu> x) * f x \<partial>M)" (is ?integral)
1291 proof -
1292   interpret \<nu>: sigma_finite_measure "M\<lparr>measure := \<nu>\<rparr>" by fact
1293   have ms: "measure_space (M\<lparr>measure := \<nu>\<rparr>)" by default
1294   have minus_cong: "\<And>A B A' B'::ereal. A = A' \<Longrightarrow> B = B' \<Longrightarrow> real A - real B = real A' - real B'" by simp
1295   have f': "(\<lambda>x. - f x) \<in> borel_measurable M" using f by auto
1296   have Nf: "f \<in> borel_measurable (M\<lparr>measure := \<nu>\<rparr>)" using f by simp
1297   { fix f :: "'a \<Rightarrow> real"
1298     { fix x assume *: "RN_deriv M \<nu> x \<noteq> \<infinity>"
1299       have "ereal (real (RN_deriv M \<nu> x)) * ereal (f x) = ereal (real (RN_deriv M \<nu> x) * f x)"
1300         by (simp add: mult_le_0_iff)
1301       then have "RN_deriv M \<nu> x * ereal (f x) = ereal (real (RN_deriv M \<nu> x) * f x)"
1302         using RN_deriv(3)[OF ms \<nu>(2)] * by (auto simp add: ereal_real split: split_if_asm) }
1303     then have "(\<integral>\<^isup>+x. ereal (real (RN_deriv M \<nu> x) * f x) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * ereal (f x) \<partial>M)"
1304               "(\<integral>\<^isup>+x. ereal (- (real (RN_deriv M \<nu> x) * f x)) \<partial>M) = (\<integral>\<^isup>+x. RN_deriv M \<nu> x * ereal (- f x) \<partial>M)"
1305       using RN_deriv_finite[OF \<nu>] unfolding ereal_mult_minus_right uminus_ereal.simps(1)[symmetric]
1306       by (auto intro!: positive_integral_cong_AE) }
1307   note * = this
1308   show ?integral ?integrable
1309     unfolding lebesgue_integral_def integrable_def *
1310     using f RN_deriv(1)[OF ms \<nu>(2)]
1311     by (auto simp: RN_deriv_positive_integral[OF ms \<nu>(2)])
1312 qed
1314 lemma (in sigma_finite_measure) real_RN_deriv:
1315   assumes "finite_measure (M\<lparr>measure := \<nu>\<rparr>)" (is "finite_measure ?\<nu>")
1316   assumes ac: "absolutely_continuous \<nu>"
1317   obtains D where "D \<in> borel_measurable M"
1318     and "AE x. RN_deriv M \<nu> x = ereal (D x)"
1319     and "AE x in M\<lparr>measure := \<nu>\<rparr>. 0 < D x"
1320     and "\<And>x. 0 \<le> D x"
1321 proof
1322   interpret \<nu>: finite_measure ?\<nu> by fact
1323   have ms: "measure_space ?\<nu>" by default
1324   note RN = RN_deriv[OF ms ac]
1326   let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M \<nu> x = t}"
1328   show "(\<lambda>x. real (RN_deriv M \<nu> x)) \<in> borel_measurable M"
1329     using RN by auto
1331   have "\<nu> (?RN \<infinity>) = (\<integral>\<^isup>+ x. RN_deriv M \<nu> x * indicator (?RN \<infinity>) x \<partial>M)"
1332     using RN by auto
1333   also have "\<dots> = (\<integral>\<^isup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)"
1334     by (intro positive_integral_cong) (auto simp: indicator_def)
1335   also have "\<dots> = \<infinity> * \<mu> (?RN \<infinity>)"
1336     using RN by (intro positive_integral_cmult_indicator) auto
1337   finally have eq: "\<nu> (?RN \<infinity>) = \<infinity> * \<mu> (?RN \<infinity>)" .
1338   moreover
1339   have "\<mu> (?RN \<infinity>) = 0"
1340   proof (rule ccontr)
1341     assume "\<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>} \<noteq> 0"
1342     moreover from RN have "0 \<le> \<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>}" by auto
1343     ultimately have "0 < \<mu> {x \<in> space M. RN_deriv M \<nu> x = \<infinity>}" by auto
1344     with eq have "\<nu> (?RN \<infinity>) = \<infinity>" by simp
1345     with \<nu>.finite_measure[of "?RN \<infinity>"] RN show False by auto
1346   qed
1347   ultimately have "AE x. RN_deriv M \<nu> x < \<infinity>"
1348     using RN by (intro AE_iff_measurable[THEN iffD2]) auto
1349   then show "AE x. RN_deriv M \<nu> x = ereal (real (RN_deriv M \<nu> x))"
1350     using RN(3) by (auto simp: ereal_real)
1351   then have eq: "AE x in (M\<lparr>measure := \<nu>\<rparr>) . RN_deriv M \<nu> x = ereal (real (RN_deriv M \<nu> x))"
1352     using ac absolutely_continuous_AE[OF ms] by auto
1354   show "\<And>x. 0 \<le> real (RN_deriv M \<nu> x)"
1355     using RN by (auto intro: real_of_ereal_pos)
1357   have "\<nu> (?RN 0) = (\<integral>\<^isup>+ x. RN_deriv M \<nu> x * indicator (?RN 0) x \<partial>M)"
1358     using RN by auto
1359   also have "\<dots> = (\<integral>\<^isup>+ x. 0 \<partial>M)"
1360     by (intro positive_integral_cong) (auto simp: indicator_def)
1361   finally have "AE x in (M\<lparr>measure := \<nu>\<rparr>). RN_deriv M \<nu> x \<noteq> 0"
1362     using RN by (intro \<nu>.AE_iff_measurable[THEN iffD2]) auto
1363   with RN(3) eq show "AE x in (M\<lparr>measure := \<nu>\<rparr>). 0 < real (RN_deriv M \<nu> x)"
1364     by (auto simp: zero_less_real_of_ereal le_less)
1365 qed
1367 lemma (in sigma_finite_measure) RN_deriv_singleton:
1368   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
1369   and ac: "absolutely_continuous \<nu>"
1370   and "{x} \<in> sets M"
1371   shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
1372 proof -
1373   note deriv = RN_deriv[OF assms(1, 2)]
1374   from deriv(2)[OF `{x} \<in> sets M`]
1375   have "\<nu> {x} = (\<integral>\<^isup>+w. RN_deriv M \<nu> x * indicator {x} w \<partial>M)"
1376     by (auto simp: indicator_def intro!: positive_integral_cong)
1377   thus ?thesis using positive_integral_cmult_indicator[OF _ `{x} \<in> sets M`] deriv(3)
1378     by auto
1379 qed
1381 theorem (in finite_measure_space) RN_deriv_finite_measure:
1382   assumes "measure_space (M\<lparr>measure := \<nu>\<rparr>)"
1383   and ac: "absolutely_continuous \<nu>"
1384   and "x \<in> space M"
1385   shows "\<nu> {x} = RN_deriv M \<nu> x * \<mu> {x}"
1386 proof -
1387   have "{x} \<in> sets M" using sets_eq_Pow `x \<in> space M` by auto
1388   from RN_deriv_singleton[OF assms(1,2) this] show ?thesis .
1389 qed
1391 end