src/HOL/Probability/Measure.thy
 author hoelzl Tue Mar 22 18:53:05 2011 +0100 (2011-03-22) changeset 42066 6db76c88907a parent 42065 2b98b4c2e2f1 child 42067 66c8281349ec permissions -rw-r--r--
generalized Caratheodory from algebra to ring_of_sets
1 (* Author: Lawrence C Paulson; Armin Heller, Johannes Hoelzl, TU Muenchen *)
3 theory Measure
4   imports Caratheodory
5 begin
7 lemma measure_algebra_more[simp]:
8   "\<lparr> space = A, sets = B, \<dots> = algebra.more M \<rparr> \<lparr> measure := m \<rparr> =
9    \<lparr> space = A, sets = B, \<dots> = algebra.more (M \<lparr> measure := m \<rparr>) \<rparr>"
10   by (cases M) simp
12 lemma measure_algebra_more_eq[simp]:
13   "\<And>X. measure \<lparr> space = T, sets = A, \<dots> = algebra.more X \<rparr> = measure X"
14   unfolding measure_space.splits by simp
16 lemma measure_sigma[simp]: "measure (sigma A) = measure A"
17   unfolding sigma_def by simp
19 lemma algebra_measure_update[simp]:
20   "algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> algebra M'"
21   unfolding algebra_iff_Un by simp
23 lemma sigma_algebra_measure_update[simp]:
24   "sigma_algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> sigma_algebra M'"
25   unfolding sigma_algebra_def sigma_algebra_axioms_def by simp
27 lemma finite_sigma_algebra_measure_update[simp]:
28   "finite_sigma_algebra (M'\<lparr>measure := m\<rparr>) \<longleftrightarrow> finite_sigma_algebra M'"
29   unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
31 lemma measurable_cancel_measure[simp]:
32   "measurable M1 (M2\<lparr>measure := m2\<rparr>) = measurable M1 M2"
33   "measurable (M2\<lparr>measure := m1\<rparr>) M1 = measurable M2 M1"
34   unfolding measurable_def by auto
36 lemma inj_on_image_eq_iff:
37   assumes "inj_on f S"
38   assumes "A \<subseteq> S" "B \<subseteq> S"
39   shows "(f ` A = f ` B) \<longleftrightarrow> (A = B)"
40 proof -
41   have "inj_on f (A \<union> B)"
42     using assms by (auto intro: subset_inj_on)
43   from inj_on_Un_image_eq_iff[OF this]
44   show ?thesis .
45 qed
47 lemma image_vimage_inter_eq:
48   assumes "f ` S = T" "X \<subseteq> T"
49   shows "f ` (f -` X \<inter> S) = X"
50 proof (intro antisym)
51   have "f ` (f -` X \<inter> S) \<subseteq> f ` (f -` X)" by auto
52   also have "\<dots> = X \<inter> range f" by simp
53   also have "\<dots> = X" using assms by auto
54   finally show "f ` (f -` X \<inter> S) \<subseteq> X" by auto
55 next
56   show "X \<subseteq> f ` (f -` X \<inter> S)"
57   proof
58     fix x assume "x \<in> X"
59     then have "x \<in> T" using `X \<subseteq> T` by auto
60     then obtain y where "x = f y" "y \<in> S"
61       using assms by auto
62     then have "{y} \<subseteq> f -` X \<inter> S" using `x \<in> X` by auto
63     moreover have "x \<in> f ` {y}" using `x = f y` by auto
64     ultimately show "x \<in> f ` (f -` X \<inter> S)" by auto
65   qed
66 qed
68 text {*
69   This formalisation of measure theory is based on the work of Hurd/Coble wand
70   was later translated by Lawrence Paulson to Isabelle/HOL. Later it was
71   modified to use the positive infinite reals and to prove the uniqueness of
72   cut stable measures.
73 *}
75 section {* Equations for the measure function @{text \<mu>} *}
77 lemma (in measure_space) measure_countably_additive:
78   assumes "range A \<subseteq> sets M" "disjoint_family A"
79   shows "(\<Sum>i. \<mu> (A i)) = \<mu> (\<Union>i. A i)"
80 proof -
81   have "(\<Union> i. A i) \<in> sets M" using assms(1) by (rule countable_UN)
82   with ca assms show ?thesis by (simp add: countably_additive_def)
83 qed
85 lemma (in sigma_algebra) sigma_algebra_cong:
86   assumes "space N = space M" "sets N = sets M"
87   shows "sigma_algebra N"
88   by default (insert sets_into_space, auto simp: assms)
90 lemma (in measure_space) measure_space_cong:
91   assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "space N = space M" "sets N = sets M"
92   shows "measure_space N"
93 proof -
94   interpret N: sigma_algebra N by (intro sigma_algebra_cong assms)
95   show ?thesis
96   proof
97     show "positive N (measure N)" using assms by (auto simp: positive_def)
98     show "countably_additive N (measure N)" unfolding countably_additive_def
99     proof safe
100       fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets N" "disjoint_family A"
101       then have "\<And>i. A i \<in> sets M" "(UNION UNIV A) \<in> sets M" unfolding assms by auto
102       from measure_countably_additive[of A] A this[THEN assms(1)]
103       show "(\<Sum>n. measure N (A n)) = measure N (UNION UNIV A)"
104         unfolding assms by simp
105     qed
106   qed
107 qed
109 lemma (in measure_space) additive: "additive M \<mu>"
110   using ca by (auto intro!: countably_additive_additive simp: positive_def)
112 lemma (in measure_space) measure_additive:
113      "a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {}
114       \<Longrightarrow> \<mu> a + \<mu> b = \<mu> (a \<union> b)"
117 lemma (in measure_space) measure_mono:
118   assumes "a \<subseteq> b" "a \<in> sets M" "b \<in> sets M"
119   shows "\<mu> a \<le> \<mu> b"
120 proof -
121   have "b = a \<union> (b - a)" using assms by auto
122   moreover have "{} = a \<inter> (b - a)" by auto
123   ultimately have "\<mu> b = \<mu> a + \<mu> (b - a)"
124     using measure_additive[of a "b - a"] Diff[of b a] assms by auto
125   moreover have "\<mu> a + 0 \<le> \<mu> a + \<mu> (b - a)" using assms by (intro add_mono) auto
126   ultimately show "\<mu> a \<le> \<mu> b" by auto
127 qed
129 lemma (in measure_space) measure_compl:
130   assumes s: "s \<in> sets M" and fin: "\<mu> s \<noteq> \<infinity>"
131   shows "\<mu> (space M - s) = \<mu> (space M) - \<mu> s"
132 proof -
133   have s_less_space: "\<mu> s \<le> \<mu> (space M)"
134     using s by (auto intro!: measure_mono sets_into_space)
135   from s have "0 \<le> \<mu> s" by auto
136   have "\<mu> (space M) = \<mu> (s \<union> (space M - s))" using s
137     by (metis Un_Diff_cancel Un_absorb1 s sets_into_space)
138   also have "... = \<mu> s + \<mu> (space M - s)"
139     by (rule additiveD [OF additive]) (auto simp add: s)
140   finally have "\<mu> (space M) = \<mu> s + \<mu> (space M - s)" .
141   then show ?thesis
142     using fin `0 \<le> \<mu> s`
143     unfolding extreal_eq_minus_iff by (auto simp: ac_simps)
144 qed
146 lemma (in measure_space) measure_Diff:
147   assumes finite: "\<mu> B \<noteq> \<infinity>"
148   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
149   shows "\<mu> (A - B) = \<mu> A - \<mu> B"
150 proof -
151   have "0 \<le> \<mu> B" using assms by auto
152   have "(A - B) \<union> B = A" using `B \<subseteq> A` by auto
153   then have "\<mu> A = \<mu> ((A - B) \<union> B)" by simp
154   also have "\<dots> = \<mu> (A - B) + \<mu> B"
155     using measurable by (subst measure_additive[symmetric]) auto
156   finally show "\<mu> (A - B) = \<mu> A - \<mu> B"
157     unfolding extreal_eq_minus_iff
158     using finite `0 \<le> \<mu> B` by auto
159 qed
161 lemma (in measure_space) measure_countable_increasing:
162   assumes A: "range A \<subseteq> sets M"
163       and A0: "A 0 = {}"
164       and ASuc: "\<And>n. A n \<subseteq> A (Suc n)"
165   shows "(SUP n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
166 proof -
167   { fix n
168     have "\<mu> (A n) = (\<Sum>i<n. \<mu> (A (Suc i) - A i))"
169       proof (induct n)
170         case 0 thus ?case by (auto simp add: A0)
171       next
172         case (Suc m)
173         have "A (Suc m) = A m \<union> (A (Suc m) - A m)"
174           by (metis ASuc Un_Diff_cancel Un_absorb1)
175         hence "\<mu> (A (Suc m)) =
176                \<mu> (A m) + \<mu> (A (Suc m) - A m)"
177           by (subst measure_additive)
178              (auto simp add: measure_additive range_subsetD [OF A])
179         with Suc show ?case
180           by simp
181       qed }
182   note Meq = this
183   have Aeq: "(\<Union>i. A (Suc i) - A i) = (\<Union>i. A i)"
184     proof (rule UN_finite2_eq [where k=1], simp)
185       fix i
186       show "(\<Union>i\<in>{0..<i}. A (Suc i) - A i) = (\<Union>i\<in>{0..<Suc i}. A i)"
187         proof (induct i)
188           case 0 thus ?case by (simp add: A0)
189         next
190           case (Suc i)
191           thus ?case
192             by (auto simp add: atLeastLessThanSuc intro: subsetD [OF ASuc])
193         qed
194     qed
195   have A1: "\<And>i. A (Suc i) - A i \<in> sets M"
196     by (metis A Diff range_subsetD)
197   have A2: "(\<Union>i. A (Suc i) - A i) \<in> sets M"
198     by (blast intro: range_subsetD [OF A])
199   have "(SUP n. \<Sum>i<n. \<mu> (A (Suc i) - A i)) = (\<Sum>i. \<mu> (A (Suc i) - A i))"
200     using A by (auto intro!: suminf_extreal_eq_SUPR[symmetric])
201   also have "\<dots> = \<mu> (\<Union>i. A (Suc i) - A i)"
202     by (rule measure_countably_additive)
203        (auto simp add: disjoint_family_Suc ASuc A1 A2)
204   also have "... =  \<mu> (\<Union>i. A i)"
205     by (simp add: Aeq)
206   finally have "(SUP n. \<Sum>i<n. \<mu> (A (Suc i) - A i)) = \<mu> (\<Union>i. A i)" .
207   then show ?thesis by (auto simp add: Meq)
208 qed
210 lemma (in measure_space) continuity_from_below:
211   assumes A: "range A \<subseteq> sets M" and "incseq A"
212   shows "(SUP n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
213 proof -
214   have *: "(SUP n. \<mu> (nat_case {} A (Suc n))) = (SUP n. \<mu> (nat_case {} A n))"
215     using A by (auto intro!: SUPR_eq exI split: nat.split)
216   have ueq: "(\<Union>i. nat_case {} A i) = (\<Union>i. A i)"
217     by (auto simp add: split: nat.splits)
218   have meq: "\<And>n. \<mu> (A n) = (\<mu> \<circ> nat_case {} A) (Suc n)"
219     by simp
220   have "(SUP n. \<mu> (nat_case {} A n)) = \<mu> (\<Union>i. nat_case {} A i)"
221     using range_subsetD[OF A] incseq_SucD[OF `incseq A`]
222     by (force split: nat.splits intro!: measure_countable_increasing)
223   also have "\<mu> (\<Union>i. nat_case {} A i) = \<mu> (\<Union>i. A i)"
224     by (simp add: ueq)
225   finally have "(SUP n. \<mu> (nat_case {} A n)) = \<mu> (\<Union>i. A i)" .
226   thus ?thesis unfolding meq * comp_def .
227 qed
229 lemma (in measure_space) measure_incseq:
230   assumes "range B \<subseteq> sets M" "incseq B"
231   shows "incseq (\<lambda>i. \<mu> (B i))"
232   using assms by (auto simp: incseq_def intro!: measure_mono)
234 lemma (in measure_space) continuity_from_below_Lim:
235   assumes A: "range A \<subseteq> sets M" "incseq A"
236   shows "(\<lambda>i. (\<mu> (A i))) ----> \<mu> (\<Union>i. A i)"
237   using LIMSEQ_extreal_SUPR[OF measure_incseq, OF A]
238     continuity_from_below[OF A] by simp
240 lemma (in measure_space) measure_decseq:
241   assumes "range B \<subseteq> sets M" "decseq B"
242   shows "decseq (\<lambda>i. \<mu> (B i))"
243   using assms by (auto simp: decseq_def intro!: measure_mono)
245 lemma (in measure_space) continuity_from_above:
246   assumes A: "range A \<subseteq> sets M" and "decseq A"
247   and finite: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
248   shows "(INF n. \<mu> (A n)) = \<mu> (\<Inter>i. A i)"
249 proof -
250   have le_MI: "\<mu> (\<Inter>i. A i) \<le> \<mu> (A 0)"
251     using A by (auto intro!: measure_mono)
252   hence *: "\<mu> (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by auto
254   have A0: "0 \<le> \<mu> (A 0)" using A by auto
256   have "\<mu> (A 0) - (INF n. \<mu> (A n)) = \<mu> (A 0) + (SUP n. - \<mu> (A n))"
257     by (simp add: extreal_SUPR_uminus minus_extreal_def)
258   also have "\<dots> = (SUP n. \<mu> (A 0) - \<mu> (A n))"
259     unfolding minus_extreal_def using A0 assms
260     by (subst SUPR_extreal_add) (auto simp add: measure_decseq)
261   also have "\<dots> = (SUP n. \<mu> (A 0 - A n))"
262     using A finite `decseq A`[unfolded decseq_def] by (subst measure_Diff) auto
263   also have "\<dots> = \<mu> (\<Union>i. A 0 - A i)"
264   proof (rule continuity_from_below)
265     show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
266       using A by auto
267     show "incseq (\<lambda>n. A 0 - A n)"
268       using `decseq A` by (auto simp add: incseq_def decseq_def)
269   qed
270   also have "\<dots> = \<mu> (A 0) - \<mu> (\<Inter>i. A i)"
271     using A finite * by (simp, subst measure_Diff) auto
272   finally show ?thesis
273     unfolding extreal_minus_eq_minus_iff using finite A0 by auto
274 qed
276 lemma (in measure_space) measure_insert:
277   assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
278   shows "\<mu> (insert x A) = \<mu> {x} + \<mu> A"
279 proof -
280   have "{x} \<inter> A = {}" using `x \<notin> A` by auto
281   from measure_additive[OF sets this] show ?thesis by simp
282 qed
284 lemma (in measure_space) measure_setsum:
285   assumes "finite S" and "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M"
286   assumes disj: "disjoint_family_on A S"
287   shows "(\<Sum>i\<in>S. \<mu> (A i)) = \<mu> (\<Union>i\<in>S. A i)"
288 using assms proof induct
289   case (insert i S)
290   then have "(\<Sum>i\<in>S. \<mu> (A i)) = \<mu> (\<Union>a\<in>S. A a)"
291     by (auto intro: disjoint_family_on_mono)
292   moreover have "A i \<inter> (\<Union>a\<in>S. A a) = {}"
293     using `disjoint_family_on A (insert i S)` `i \<notin> S`
294     by (auto simp: disjoint_family_on_def)
295   ultimately show ?case using insert
296     by (auto simp: measure_additive finite_UN)
297 qed simp
299 lemma (in measure_space) measure_finite_singleton:
300   assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
301   shows "\<mu> S = (\<Sum>x\<in>S. \<mu> {x})"
302   using measure_setsum[of S "\<lambda>x. {x}", OF assms]
303   by (auto simp: disjoint_family_on_def)
306   assumes "sigma_algebra M"
307   assumes fin: "finite (space M)" and pos: "positive M (measure M)" and add: "additive M (measure M)"
308   shows "measure_space M"
309 proof -
310   interpret sigma_algebra M by fact
311   show ?thesis
312   proof
313     show [simp]: "positive M (measure M)" using pos by (simp add: positive_def)
314     show "countably_additive M (measure M)"
316       fix A :: "nat \<Rightarrow> 'a set"
317       assume A: "range A \<subseteq> sets M"
318          and disj: "disjoint_family A"
319          and UnA: "(\<Union>i. A i) \<in> sets M"
320       def I \<equiv> "{i. A i \<noteq> {}}"
321       have "Union (A ` I) \<subseteq> space M" using A
322         by auto (metis range_subsetD subsetD sets_into_space)
323       hence "finite (A ` I)"
324         by (metis finite_UnionD finite_subset fin)
325       moreover have "inj_on A I" using disj
326         by (auto simp add: I_def disjoint_family_on_def inj_on_def)
327       ultimately have finI: "finite I"
328         by (metis finite_imageD)
329       hence "\<exists>N. \<forall>m\<ge>N. A m = {}"
330         proof (cases "I = {}")
331           case True thus ?thesis by (simp add: I_def)
332         next
333           case False
334           hence "\<forall>i\<in>I. i < Suc(Max I)"
335             by (simp add: Max_less_iff [symmetric] finI)
336           hence "\<forall>m \<ge> Suc(Max I). A m = {}"
337             by (simp add: I_def) (metis less_le_not_le)
338           thus ?thesis
339             by blast
340         qed
341       then obtain N where N: "\<forall>m\<ge>N. A m = {}" by blast
342       then have "\<forall>m\<ge>N. measure M (A m) = 0" using pos[unfolded positive_def] by simp
343       then have "(\<Sum>n. measure M (A n)) = (\<Sum>m<N. measure M (A m))"
344         by (simp add: suminf_finite)
345       also have "... = measure M (\<Union>i<N. A i)"
346         proof (induct N)
347           case 0 thus ?case using pos[unfolded positive_def] by simp
348         next
349           case (Suc n)
350           have "measure M (A n \<union> (\<Union> x<n. A x)) = measure M (A n) + measure M (\<Union> i<n. A i)"
352               show "A n \<inter> (\<Union> x<n. A x) = {}" using disj
353                 by (auto simp add: disjoint_family_on_def nat_less_le) blast
354               show "A n \<in> sets M" using A
355                 by force
356               show "(\<Union>i<n. A i) \<in> sets M"
357                 proof (induct n)
358                   case 0 thus ?case by simp
359                 next
360                   case (Suc n) thus ?case using A
361                     by (simp add: lessThan_Suc Un range_subsetD)
362                 qed
363             qed
364           thus ?case using Suc
365             by (simp add: lessThan_Suc)
366         qed
367       also have "... = measure M (\<Union>i. A i)"
368         proof -
369           have "(\<Union> i<N. A i) = (\<Union>i. A i)" using N
370             by auto (metis Int_absorb N disjoint_iff_not_equal lessThan_iff not_leE)
371           thus ?thesis by simp
372         qed
373       finally show "(\<Sum>n. measure M (A n)) = measure M (\<Union>i. A i)" .
374     qed
375   qed
376 qed
378 lemma (in measure_space) measure_setsum_split:
379   assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
380   assumes "(\<Union>i\<in>S. B i) = space M"
381   assumes "disjoint_family_on B S"
382   shows "\<mu> A = (\<Sum>i\<in>S. \<mu> (A \<inter> (B i)))"
383 proof -
384   have *: "\<mu> A = \<mu> (\<Union>i\<in>S. A \<inter> B i)"
385     using assms by auto
386   show ?thesis unfolding *
387   proof (rule measure_setsum[symmetric])
388     show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
389       using `disjoint_family_on B S`
390       unfolding disjoint_family_on_def by auto
391   qed (insert assms, auto)
392 qed
394 lemma (in measure_space) measure_subadditive:
395   assumes measurable: "A \<in> sets M" "B \<in> sets M"
396   shows "\<mu> (A \<union> B) \<le> \<mu> A + \<mu> B"
397 proof -
398   from measure_additive[of A "B - A"]
399   have "\<mu> (A \<union> B) = \<mu> A + \<mu> (B - A)"
400     using assms by (simp add: Diff)
401   also have "\<dots> \<le> \<mu> A + \<mu> B"
402     using assms by (auto intro!: add_left_mono measure_mono)
403   finally show ?thesis .
404 qed
406 lemma (in measure_space) measure_eq_0:
407   assumes "N \<in> sets M" and "\<mu> N = 0" and "K \<subseteq> N" and "K \<in> sets M"
408   shows "\<mu> K = 0"
409   using measure_mono[OF assms(3,4,1)] assms(2) positive_measure[OF assms(4)] by auto
411 lemma (in measure_space) measure_finitely_subadditive:
412   assumes "finite I" "A ` I \<subseteq> sets M"
413   shows "\<mu> (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. \<mu> (A i))"
414 using assms proof induct
415   case (insert i I)
416   then have "(\<Union>i\<in>I. A i) \<in> sets M" by (auto intro: finite_UN)
417   then have "\<mu> (\<Union>i\<in>insert i I. A i) \<le> \<mu> (A i) + \<mu> (\<Union>i\<in>I. A i)"
418     using insert by (simp add: measure_subadditive)
419   also have "\<dots> \<le> (\<Sum>i\<in>insert i I. \<mu> (A i))"
420     using insert by (auto intro!: add_left_mono)
421   finally show ?case .
422 qed simp
424 lemma (in measure_space) measure_countably_subadditive:
425   assumes "range f \<subseteq> sets M"
426   shows "\<mu> (\<Union>i. f i) \<le> (\<Sum>i. \<mu> (f i))"
427 proof -
428   have "\<mu> (\<Union>i. f i) = \<mu> (\<Union>i. disjointed f i)"
429     unfolding UN_disjointed_eq ..
430   also have "\<dots> = (\<Sum>i. \<mu> (disjointed f i))"
431     using range_disjointed_sets[OF assms] measure_countably_additive
432     by (simp add:  disjoint_family_disjointed comp_def)
433   also have "\<dots> \<le> (\<Sum>i. \<mu> (f i))"
434     using range_disjointed_sets[OF assms] assms
435     by (auto intro!: suminf_le_pos measure_mono positive_measure disjointed_subset)
436   finally show ?thesis .
437 qed
439 lemma (in measure_space) measure_UN_eq_0:
440   assumes "\<And>i::nat. \<mu> (N i) = 0" and "range N \<subseteq> sets M"
441   shows "\<mu> (\<Union> i. N i) = 0"
442 proof -
443   have "0 \<le> \<mu> (\<Union> i. N i)" using assms by auto
444   moreover have "\<mu> (\<Union> i. N i) \<le> 0"
445     using measure_countably_subadditive[OF assms(2)] assms(1) by simp
446   ultimately show ?thesis by simp
447 qed
449 lemma (in measure_space) measure_inter_full_set:
450   assumes "S \<in> sets M" "T \<in> sets M" and fin: "\<mu> (T - S) \<noteq> \<infinity>"
451   assumes T: "\<mu> T = \<mu> (space M)"
452   shows "\<mu> (S \<inter> T) = \<mu> S"
453 proof (rule antisym)
454   show " \<mu> (S \<inter> T) \<le> \<mu> S"
455     using assms by (auto intro!: measure_mono)
457   have pos: "0 \<le> \<mu> (T - S)" using assms by auto
458   show "\<mu> S \<le> \<mu> (S \<inter> T)"
459   proof (rule ccontr)
460     assume contr: "\<not> ?thesis"
461     have "\<mu> (space M) = \<mu> ((T - S) \<union> (S \<inter> T))"
462       unfolding T[symmetric] by (auto intro!: arg_cong[where f="\<mu>"])
463     also have "\<dots> \<le> \<mu> (T - S) + \<mu> (S \<inter> T)"
464       using assms by (auto intro!: measure_subadditive)
465     also have "\<dots> < \<mu> (T - S) + \<mu> S"
466       using fin contr pos by (intro extreal_less_add) auto
467     also have "\<dots> = \<mu> (T \<union> S)"
468       using assms by (subst measure_additive) auto
469     also have "\<dots> \<le> \<mu> (space M)"
470       using assms sets_into_space by (auto intro!: measure_mono)
471     finally show False ..
472   qed
473 qed
475 lemma measure_unique_Int_stable:
476   fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set"
477   assumes "Int_stable E"
478   and A: "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E"
479   and M: "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<mu>\<rparr>" (is "measure_space ?M")
480   and N: "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<nu>\<rparr>" (is "measure_space ?N")
481   and eq: "\<And>X. X \<in> sets E \<Longrightarrow> \<mu> X = \<nu> X"
482   and finite: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
483   assumes "X \<in> sets (sigma E)"
484   shows "\<mu> X = \<nu> X"
485 proof -
486   let "?D F" = "{D. D \<in> sets (sigma E) \<and> \<mu> (F \<inter> D) = \<nu> (F \<inter> D)}"
487   interpret M: measure_space ?M
488     where "space ?M = space E" and "sets ?M = sets (sigma E)" and "measure ?M = \<mu>" by (simp_all add: M)
489   interpret N: measure_space ?N
490     where "space ?N = space E" and "sets ?N = sets (sigma E)" and "measure ?N = \<nu>" by (simp_all add: N)
491   { fix F assume "F \<in> sets E" and "\<mu> F \<noteq> \<infinity>"
492     then have [intro]: "F \<in> sets (sigma E)" by auto
493     have "\<nu> F \<noteq> \<infinity>" using `\<mu> F \<noteq> \<infinity>` `F \<in> sets E` eq by simp
494     interpret D: dynkin_system "\<lparr>space=space E, sets=?D F\<rparr>"
495     proof (rule dynkin_systemI, simp_all)
496       fix A assume "A \<in> sets (sigma E) \<and> \<mu> (F \<inter> A) = \<nu> (F \<inter> A)"
497       then show "A \<subseteq> space E" using M.sets_into_space by auto
498     next
499       have "F \<inter> space E = F" using `F \<in> sets E` by auto
500       then show "\<mu> (F \<inter> space E) = \<nu> (F \<inter> space E)"
501         using `F \<in> sets E` eq by auto
502     next
503       fix A assume *: "A \<in> sets (sigma E) \<and> \<mu> (F \<inter> A) = \<nu> (F \<inter> A)"
504       then have **: "F \<inter> (space (sigma E) - A) = F - (F \<inter> A)"
505         and [intro]: "F \<inter> A \<in> sets (sigma E)"
506         using `F \<in> sets E` M.sets_into_space by auto
507       have "\<nu> (F \<inter> A) \<le> \<nu> F" by (auto intro!: N.measure_mono)
508       then have "\<nu> (F \<inter> A) \<noteq> \<infinity>" using `\<nu> F \<noteq> \<infinity>` by auto
509       have "\<mu> (F \<inter> A) \<le> \<mu> F" by (auto intro!: M.measure_mono)
510       then have "\<mu> (F \<inter> A) \<noteq> \<infinity>" using `\<mu> F \<noteq> \<infinity>` by auto
511       then have "\<mu> (F \<inter> (space (sigma E) - A)) = \<mu> F - \<mu> (F \<inter> A)" unfolding **
512         using `F \<inter> A \<in> sets (sigma E)` by (auto intro!: M.measure_Diff)
513       also have "\<dots> = \<nu> F - \<nu> (F \<inter> A)" using eq `F \<in> sets E` * by simp
514       also have "\<dots> = \<nu> (F \<inter> (space (sigma E) - A))" unfolding **
515         using `F \<inter> A \<in> sets (sigma E)` `\<nu> (F \<inter> A) \<noteq> \<infinity>` by (auto intro!: N.measure_Diff[symmetric])
516       finally show "space E - A \<in> sets (sigma E) \<and> \<mu> (F \<inter> (space E - A)) = \<nu> (F \<inter> (space E - A))"
517         using * by auto
518     next
519       fix A :: "nat \<Rightarrow> 'a set"
520       assume "disjoint_family A" "range A \<subseteq> {X \<in> sets (sigma E). \<mu> (F \<inter> X) = \<nu> (F \<inter> X)}"
521       then have A: "range (\<lambda>i. F \<inter> A i) \<subseteq> sets (sigma E)" "F \<inter> (\<Union>x. A x) = (\<Union>x. F \<inter> A x)"
522         "disjoint_family (\<lambda>i. F \<inter> A i)" "\<And>i. \<mu> (F \<inter> A i) = \<nu> (F \<inter> A i)" "range A \<subseteq> sets (sigma E)"
523         by (auto simp: disjoint_family_on_def subset_eq)
524       then show "(\<Union>x. A x) \<in> sets (sigma E) \<and> \<mu> (F \<inter> (\<Union>x. A x)) = \<nu> (F \<inter> (\<Union>x. A x))"
525         by (auto simp: M.measure_countably_additive[symmetric]
527             simp del: UN_simps)
528     qed
529     have *: "sets (sigma E) = sets \<lparr>space = space E, sets = ?D F\<rparr>"
530       using `F \<in> sets E` `Int_stable E`
531       by (intro D.dynkin_lemma)
532          (auto simp add: sets_sigma Int_stable_def eq intro: sigma_sets.Basic)
533     have "\<And>D. D \<in> sets (sigma E) \<Longrightarrow> \<mu> (F \<inter> D) = \<nu> (F \<inter> D)"
534       by (subst (asm) *) auto }
535   note * = this
536   let "?A i" = "A i \<inter> X"
537   have A': "range ?A \<subseteq> sets (sigma E)" "incseq ?A"
538     using A(1,2) `X \<in> sets (sigma E)` by (auto simp: incseq_def)
539   { fix i have "\<mu> (?A i) = \<nu> (?A i)"
540       using *[of "A i" X] `X \<in> sets (sigma E)` A finite by auto }
541   with M.continuity_from_below[OF A'] N.continuity_from_below[OF A']
542   show ?thesis using A(3) `X \<in> sets (sigma E)` by auto
543 qed
545 section "@{text \<mu>}-null sets"
547 abbreviation (in measure_space) "null_sets \<equiv> {N\<in>sets M. \<mu> N = 0}"
549 lemma (in measure_space) null_sets_Un[intro]:
550   assumes "N \<in> null_sets" "N' \<in> null_sets"
551   shows "N \<union> N' \<in> null_sets"
552 proof (intro conjI CollectI)
553   show "N \<union> N' \<in> sets M" using assms by auto
554   then have "0 \<le> \<mu> (N \<union> N')" by simp
555   moreover have "\<mu> (N \<union> N') \<le> \<mu> N + \<mu> N'"
556     using assms by (intro measure_subadditive) auto
557   ultimately show "\<mu> (N \<union> N') = 0" using assms by auto
558 qed
560 lemma UN_from_nat: "(\<Union>i. N i) = (\<Union>i. N (Countable.from_nat i))"
561 proof -
562   have "(\<Union>i. N i) = (\<Union>i. (N \<circ> Countable.from_nat) i)"
563     unfolding SUPR_def image_compose
564     unfolding surj_from_nat ..
565   then show ?thesis by simp
566 qed
568 lemma (in measure_space) null_sets_UN[intro]:
569   assumes "\<And>i::'i::countable. N i \<in> null_sets"
570   shows "(\<Union>i. N i) \<in> null_sets"
571 proof (intro conjI CollectI)
572   show "(\<Union>i. N i) \<in> sets M" using assms by auto
573   then have "0 \<le> \<mu> (\<Union>i. N i)" by simp
574   moreover have "\<mu> (\<Union>i. N i) \<le> (\<Sum>n. \<mu> (N (Countable.from_nat n)))"
575     unfolding UN_from_nat[of N]
576     using assms by (intro measure_countably_subadditive) auto
577   ultimately show "\<mu> (\<Union>i. N i) = 0" using assms by auto
578 qed
580 lemma (in measure_space) null_sets_finite_UN:
581   assumes "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> null_sets"
582   shows "(\<Union>i\<in>S. A i) \<in> null_sets"
583 proof (intro CollectI conjI)
584   show "(\<Union>i\<in>S. A i) \<in> sets M" using assms by (intro finite_UN) auto
585   then have "0 \<le> \<mu> (\<Union>i\<in>S. A i)" by simp
586   moreover have "\<mu> (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. \<mu> (A i))"
587     using assms by (intro measure_finitely_subadditive) auto
588   ultimately show "\<mu> (\<Union>i\<in>S. A i) = 0" using assms by auto
589 qed
591 lemma (in measure_space) null_set_Int1:
592   assumes "B \<in> null_sets" "A \<in> sets M" shows "A \<inter> B \<in> null_sets"
593 using assms proof (intro CollectI conjI)
594   show "\<mu> (A \<inter> B) = 0" using assms by (intro measure_eq_0[of B "A \<inter> B"]) auto
595 qed auto
597 lemma (in measure_space) null_set_Int2:
598   assumes "B \<in> null_sets" "A \<in> sets M" shows "B \<inter> A \<in> null_sets"
599   using assms by (subst Int_commute) (rule null_set_Int1)
601 lemma (in measure_space) measure_Diff_null_set:
602   assumes "B \<in> null_sets" "A \<in> sets M"
603   shows "\<mu> (A - B) = \<mu> A"
604 proof -
605   have *: "A - B = (A - (A \<inter> B))" by auto
606   have "A \<inter> B \<in> null_sets" using assms by (rule null_set_Int1)
607   then show ?thesis
608     unfolding * using assms
609     by (subst measure_Diff) auto
610 qed
612 lemma (in measure_space) null_set_Diff:
613   assumes "B \<in> null_sets" "A \<in> sets M" shows "B - A \<in> null_sets"
614 using assms proof (intro CollectI conjI)
615   show "\<mu> (B - A) = 0" using assms by (intro measure_eq_0[of B "B - A"]) auto
616 qed auto
618 lemma (in measure_space) measure_Un_null_set:
619   assumes "A \<in> sets M" "B \<in> null_sets"
620   shows "\<mu> (A \<union> B) = \<mu> A"
621 proof -
622   have *: "A \<union> B = A \<union> (B - A)" by auto
623   have "B - A \<in> null_sets" using assms(2,1) by (rule null_set_Diff)
624   then show ?thesis
625     unfolding * using assms
626     by (subst measure_additive[symmetric]) auto
627 qed
629 section "Formalise almost everywhere"
631 definition (in measure_space)
632   almost_everywhere :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "AE " 10) where
633   "almost_everywhere P \<longleftrightarrow> (\<exists>N\<in>null_sets. { x \<in> space M. \<not> P x } \<subseteq> N)"
635 syntax
636   "_almost_everywhere" :: "'a \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
638 translations
639   "AE x in M. P" == "CONST measure_space.almost_everywhere M (%x. P)"
641 lemma (in measure_space) AE_cong_measure:
642   assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
643   shows "(AE x in N. P x) \<longleftrightarrow> (AE x. P x)"
644 proof -
645   interpret N: measure_space N
646     by (rule measure_space_cong) fact+
647   show ?thesis
648     unfolding N.almost_everywhere_def almost_everywhere_def
649     by (auto simp: assms)
650 qed
652 lemma (in measure_space) AE_I':
653   "N \<in> null_sets \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x. P x)"
654   unfolding almost_everywhere_def by auto
656 lemma (in measure_space) AE_iff_null_set:
657   assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
658   shows "(AE x. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets"
659 proof
660   assume "AE x. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "\<mu> N = 0"
661     unfolding almost_everywhere_def by auto
662   have "0 \<le> \<mu> ?P" using assms by simp
663   moreover have "\<mu> ?P \<le> \<mu> N"
664     using assms N(1,2) by (auto intro: measure_mono)
665   ultimately have "\<mu> ?P = 0" unfolding `\<mu> N = 0` by auto
666   then show "?P \<in> null_sets" using assms by simp
667 next
668   assume "?P \<in> null_sets" with assms show "AE x. P x" by (auto intro: AE_I')
669 qed
671 lemma (in measure_space) AE_iff_measurable:
672   "N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x. P x) \<longleftrightarrow> \<mu> N = 0"
673   using AE_iff_null_set[of P] by simp
675 lemma (in measure_space) AE_True[intro, simp]: "AE x. True"
676   unfolding almost_everywhere_def by auto
678 lemma (in measure_space) AE_E[consumes 1]:
679   assumes "AE x. P x"
680   obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "\<mu> N = 0" "N \<in> sets M"
681   using assms unfolding almost_everywhere_def by auto
683 lemma (in measure_space) AE_E2:
684   assumes "AE x. P x" "{x\<in>space M. P x} \<in> sets M"
685   shows "\<mu> {x\<in>space M. \<not> P x} = 0" (is "\<mu> ?P = 0")
686 proof -
687   have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}"
688     by auto
689   with AE_iff_null_set[of P] assms show ?thesis by auto
690 qed
692 lemma (in measure_space) AE_I:
693   assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "\<mu> N = 0" "N \<in> sets M"
694   shows "AE x. P x"
695   using assms unfolding almost_everywhere_def by auto
697 lemma (in measure_space) AE_mp[elim!]:
698   assumes AE_P: "AE x. P x" and AE_imp: "AE x. P x \<longrightarrow> Q x"
699   shows "AE x. Q x"
700 proof -
701   from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"
702     and A: "A \<in> sets M" "\<mu> A = 0"
703     by (auto elim!: AE_E)
705   from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"
706     and B: "B \<in> sets M" "\<mu> B = 0"
707     by (auto elim!: AE_E)
709   show ?thesis
710   proof (intro AE_I)
711     have "0 \<le> \<mu> (A \<union> B)" using A B by auto
712     moreover have "\<mu> (A \<union> B) \<le> 0"
713       using measure_subadditive[of A B] A B by auto
714     ultimately show "A \<union> B \<in> sets M" "\<mu> (A \<union> B) = 0" using A B by auto
715     show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
716       using P imp by auto
717   qed
718 qed
720 lemma (in measure_space)
721   shows AE_iffI: "AE x. P x \<Longrightarrow> AE x. P x \<longleftrightarrow> Q x \<Longrightarrow> AE x. Q x"
722     and AE_disjI1: "AE x. P x \<Longrightarrow> AE x. P x \<or> Q x"
723     and AE_disjI2: "AE x. Q x \<Longrightarrow> AE x. P x \<or> Q x"
724     and AE_conjI: "AE x. P x \<Longrightarrow> AE x. Q x \<Longrightarrow> AE x. P x \<and> Q x"
725     and AE_conj_iff[simp]: "(AE x. P x \<and> Q x) \<longleftrightarrow> (AE x. P x) \<and> (AE x. Q x)"
726   by auto
728 lemma (in measure_space) AE_space: "AE x. x \<in> space M"
729   by (rule AE_I[where N="{}"]) auto
731 lemma (in measure_space) AE_I2[simp, intro]:
732   "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x. P x"
733   using AE_space by auto
735 lemma (in measure_space) AE_Ball_mp:
736   "\<forall>x\<in>space M. P x \<Longrightarrow> AE x. P x \<longrightarrow> Q x \<Longrightarrow> AE x. Q x"
737   by auto
739 lemma (in measure_space) AE_cong[cong]:
740   "(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x. P x) \<longleftrightarrow> (AE x. Q x)"
741   by auto
743 lemma (in measure_space) AE_all_countable:
744   "(AE x. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x. P i x)"
745 proof
746   assume "\<forall>i. AE x. P i x"
747   from this[unfolded almost_everywhere_def Bex_def, THEN choice]
748   obtain N where N: "\<And>i. N i \<in> null_sets" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto
749   have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto
750   also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
751   finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
752   moreover from N have "(\<Union>i. N i) \<in> null_sets"
753     by (intro null_sets_UN) auto
754   ultimately show "AE x. \<forall>i. P i x"
755     unfolding almost_everywhere_def by auto
756 qed auto
758 lemma (in measure_space) AE_finite_all:
759   assumes f: "finite S" shows "(AE x. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x. P i x)"
760   using f by induct auto
762 lemma (in measure_space) restricted_measure_space:
763   assumes "S \<in> sets M"
764   shows "measure_space (restricted_space S)"
765     (is "measure_space ?r")
766   unfolding measure_space_def measure_space_axioms_def
767 proof safe
768   show "sigma_algebra ?r" using restricted_sigma_algebra[OF assms] .
769   show "positive ?r (measure ?r)" using `S \<in> sets M` by (auto simp: positive_def)
771   show "countably_additive ?r (measure ?r)"
773   proof safe
774     fix A :: "nat \<Rightarrow> 'a set"
775     assume *: "range A \<subseteq> sets ?r" and **: "disjoint_family A"
776     from restriction_in_sets[OF assms *[simplified]] **
777     show "(\<Sum>n. measure ?r (A n)) = measure ?r (\<Union>i. A i)"
778       using measure_countably_additive by simp
779   qed
780 qed
782 lemma (in measure_space) AE_restricted:
783   assumes "A \<in> sets M"
784   shows "(AE x in restricted_space A. P x) \<longleftrightarrow> (AE x. x \<in> A \<longrightarrow> P x)"
785 proof -
786   interpret R: measure_space "restricted_space A"
787     by (rule restricted_measure_space[OF `A \<in> sets M`])
788   show ?thesis
789   proof
790     assume "AE x in restricted_space A. P x"
791     from this[THEN R.AE_E] guess N' .
792     then obtain N where "{x \<in> A. \<not> P x} \<subseteq> A \<inter> N" "\<mu> (A \<inter> N) = 0" "N \<in> sets M"
793       by auto
794     moreover then have "{x \<in> space M. \<not> (x \<in> A \<longrightarrow> P x)} \<subseteq> A \<inter> N"
795       using `A \<in> sets M` sets_into_space by auto
796     ultimately show "AE x. x \<in> A \<longrightarrow> P x"
797       using `A \<in> sets M` by (auto intro!: AE_I[where N="A \<inter> N"])
798   next
799     assume "AE x. x \<in> A \<longrightarrow> P x"
800     from this[THEN AE_E] guess N .
801     then show "AE x in restricted_space A. P x"
802       using null_set_Int1[OF _ `A \<in> sets M`] `A \<in> sets M`[THEN sets_into_space]
803       by (auto intro!: R.AE_I[where N="A \<inter> N"] simp: subset_eq)
804   qed
805 qed
807 lemma (in measure_space) measure_space_subalgebra:
808   assumes "sigma_algebra N" and "sets N \<subseteq> sets M" "space N = space M"
809   and measure[simp]: "\<And>X. X \<in> sets N \<Longrightarrow> measure N X = measure M X"
810   shows "measure_space N"
811 proof -
812   interpret N: sigma_algebra N by fact
813   show ?thesis
814   proof
815     from `sets N \<subseteq> sets M` have "\<And>A. range A \<subseteq> sets N \<Longrightarrow> range A \<subseteq> sets M" by blast
816     then show "countably_additive N (measure N)"
817       by (auto intro!: measure_countably_additive simp: countably_additive_def subset_eq)
818     show "positive N (measure_space.measure N)"
819       using assms(2) by (auto simp add: positive_def)
820   qed
821 qed
823 lemma (in measure_space) AE_subalgebra:
824   assumes ae: "AE x in N. P x"
825   and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
826   and sa: "sigma_algebra N"
827   shows "AE x. P x"
828 proof -
829   interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
830   from ae[THEN N.AE_E] guess N .
831   with N show ?thesis unfolding almost_everywhere_def by auto
832 qed
834 section "@{text \<sigma>}-finite Measures"
836 locale sigma_finite_measure = measure_space +
837   assumes sigma_finite: "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<infinity>)"
839 lemma (in sigma_finite_measure) restricted_sigma_finite_measure:
840   assumes "S \<in> sets M"
841   shows "sigma_finite_measure (restricted_space S)"
842     (is "sigma_finite_measure ?r")
843   unfolding sigma_finite_measure_def sigma_finite_measure_axioms_def
844 proof safe
845   show "measure_space ?r" using restricted_measure_space[OF assms] .
846 next
847   obtain A :: "nat \<Rightarrow> 'a set" where
848       "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
849     using sigma_finite by auto
850   show "\<exists>A::nat \<Rightarrow> 'a set. range A \<subseteq> sets ?r \<and> (\<Union>i. A i) = space ?r \<and> (\<forall>i. measure ?r (A i) \<noteq> \<infinity>)"
851   proof (safe intro!: exI[of _ "\<lambda>i. A i \<inter> S"] del: notI)
852     fix i
853     show "A i \<inter> S \<in> sets ?r"
854       using `range A \<subseteq> sets M` `S \<in> sets M` by auto
855   next
856     fix x i assume "x \<in> S" thus "x \<in> space ?r" by simp
857   next
858     fix x assume "x \<in> space ?r" thus "x \<in> (\<Union>i. A i \<inter> S)"
859       using `(\<Union>i. A i) = space M` `S \<in> sets M` by auto
860   next
861     fix i
862     have "\<mu> (A i \<inter> S) \<le> \<mu> (A i)"
863       using `range A \<subseteq> sets M` `S \<in> sets M` by (auto intro!: measure_mono)
864     then show "measure ?r (A i \<inter> S) \<noteq> \<infinity>" using `\<mu> (A i) \<noteq> \<infinity>` by auto
865   qed
866 qed
868 lemma (in sigma_finite_measure) sigma_finite_measure_cong:
869   assumes cong: "\<And>A. A \<in> sets M \<Longrightarrow> measure M' A = \<mu> A" "sets M' = sets M" "space M' = space M"
870   shows "sigma_finite_measure M'"
871 proof -
872   interpret M': measure_space M' by (intro measure_space_cong cong)
873   from sigma_finite guess A .. note A = this
874   then have "\<And>i. A i \<in> sets M" by auto
875   with A have fin: "\<forall>i. measure M' (A i) \<noteq> \<infinity>" using cong by auto
876   show ?thesis
877     apply default
878     using A fin cong by auto
879 qed
881 lemma (in sigma_finite_measure) disjoint_sigma_finite:
882   "\<exists>A::nat\<Rightarrow>'a set. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and>
883     (\<forall>i. \<mu> (A i) \<noteq> \<infinity>) \<and> disjoint_family A"
884 proof -
885   obtain A :: "nat \<Rightarrow> 'a set" where
886     range: "range A \<subseteq> sets M" and
887     space: "(\<Union>i. A i) = space M" and
888     measure: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
889     using sigma_finite by auto
890   note range' = range_disjointed_sets[OF range] range
891   { fix i
892     have "\<mu> (disjointed A i) \<le> \<mu> (A i)"
893       using range' disjointed_subset[of A i] by (auto intro!: measure_mono)
894     then have "\<mu> (disjointed A i) \<noteq> \<infinity>"
895       using measure[of i] by auto }
896   with disjoint_family_disjointed UN_disjointed_eq[of A] space range'
897   show ?thesis by (auto intro!: exI[of _ "disjointed A"])
898 qed
900 lemma (in sigma_finite_measure) sigma_finite_up:
901   "\<exists>F. range F \<subseteq> sets M \<and> incseq F \<and> (\<Union>i. F i) = space M \<and> (\<forall>i. \<mu> (F i) \<noteq> \<infinity>)"
902 proof -
903   obtain F :: "nat \<Rightarrow> 'a set" where
904     F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
905     using sigma_finite by auto
906   then show ?thesis
907   proof (intro exI[of _ "\<lambda>n. \<Union>i\<le>n. F i"] conjI allI)
908     from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
909     then show "(\<Union>n. \<Union> i\<le>n. F i) = space M"
910       using F by fastsimp
911   next
912     fix n
913     have "\<mu> (\<Union> i\<le>n. F i) \<le> (\<Sum>i\<le>n. \<mu> (F i))" using F
914       by (auto intro!: measure_finitely_subadditive)
915     also have "\<dots> < \<infinity>"
916       using F by (auto simp: setsum_Pinfty)
917     finally show "\<mu> (\<Union> i\<le>n. F i) \<noteq> \<infinity>" by simp
918   qed (force simp: incseq_def)+
919 qed
921 section {* Measure preserving *}
923 definition "measure_preserving A B =
924     {f \<in> measurable A B. (\<forall>y \<in> sets B. measure B y = measure A (f -` y \<inter> space A))}"
926 lemma measure_preservingI[intro?]:
927   assumes "f \<in> measurable A B"
928     and "\<And>y. y \<in> sets B \<Longrightarrow> measure A (f -` y \<inter> space A) = measure B y"
929   shows "f \<in> measure_preserving A B"
930   unfolding measure_preserving_def using assms by auto
932 lemma (in measure_space) measure_space_vimage:
933   fixes M' :: "('c, 'd) measure_space_scheme"
934   assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
935   shows "measure_space M'"
936 proof -
937   interpret M': sigma_algebra M' by fact
938   show ?thesis
939   proof
940     show "positive M' (measure M')" using T
941       by (auto simp: measure_preserving_def positive_def measurable_sets)
943     show "countably_additive M' (measure M')"
944     proof (intro countably_additiveI)
945       fix A :: "nat \<Rightarrow> 'c set" assume "range A \<subseteq> sets M'" "disjoint_family A"
946       then have A: "\<And>i. A i \<in> sets M'" "(\<Union>i. A i) \<in> sets M'" by auto
947       then have *: "range (\<lambda>i. T -` (A i) \<inter> space M) \<subseteq> sets M"
948         using T by (auto simp: measurable_def measure_preserving_def)
949       moreover have "(\<Union>i. T -`  A i \<inter> space M) \<in> sets M"
950         using * by blast
951       moreover have **: "disjoint_family (\<lambda>i. T -` A i \<inter> space M)"
952         using `disjoint_family A` by (auto simp: disjoint_family_on_def)
953       ultimately show "(\<Sum>i. measure M' (A i)) = measure M' (\<Union>i. A i)"
954         using measure_countably_additive[OF _ **] A T
955         by (auto simp: comp_def vimage_UN measure_preserving_def)
956     qed
957   qed
958 qed
960 lemma (in measure_space) almost_everywhere_vimage:
961   assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
962     and AE: "measure_space.almost_everywhere M' P"
963   shows "AE x. P (T x)"
964 proof -
965   interpret M': measure_space M' using T by (rule measure_space_vimage)
966   from AE[THEN M'.AE_E] guess N .
967   then show ?thesis
968     unfolding almost_everywhere_def M'.almost_everywhere_def
969     using T(2) unfolding measurable_def measure_preserving_def
970     by (intro bexI[of _ "T -` N \<inter> space M"]) auto
971 qed
973 lemma measure_unique_Int_stable_vimage:
974   fixes A :: "nat \<Rightarrow> 'a set"
975   assumes E: "Int_stable E"
976   and A: "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E" "\<And>i. measure M (A i) \<noteq> \<infinity>"
977   and N: "measure_space N" "T \<in> measurable N M"
978   and M: "measure_space M" "sets (sigma E) = sets M" "space E = space M"
979   and eq: "\<And>X. X \<in> sets E \<Longrightarrow> measure M X = measure N (T -` X \<inter> space N)"
980   assumes X: "X \<in> sets (sigma E)"
981   shows "measure M X = measure N (T -` X \<inter> space N)"
982 proof (rule measure_unique_Int_stable[OF E A(1,2,3) _ _ eq _ X])
983   interpret M: measure_space M by fact
984   interpret N: measure_space N by fact
985   let "?T X" = "T -` X \<inter> space N"
986   show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = measure M\<rparr>"
987     by (rule M.measure_space_cong) (auto simp: M)
988   show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure = \<lambda>X. measure N (?T X)\<rparr>" (is "measure_space ?E")
989   proof (rule N.measure_space_vimage)
990     show "sigma_algebra ?E"
991       by (rule M.sigma_algebra_cong) (auto simp: M)
992     show "T \<in> measure_preserving N ?E"
993       using `T \<in> measurable N M` by (auto simp: M measurable_def measure_preserving_def)
994   qed
995   show "\<And>i. M.\<mu> (A i) \<noteq> \<infinity>" by fact
996 qed
998 lemma (in measure_space) measure_preserving_Int_stable:
999   fixes A :: "nat \<Rightarrow> 'a set"
1000   assumes E: "Int_stable E" "range A \<subseteq> sets E" "incseq A" "(\<Union>i. A i) = space E" "\<And>i. measure E (A i) \<noteq> \<infinity>"
1001   and N: "measure_space (sigma E)"
1002   and T: "T \<in> measure_preserving M E"
1003   shows "T \<in> measure_preserving M (sigma E)"
1004 proof
1005   interpret E: measure_space "sigma E" by fact
1006   show "T \<in> measurable M (sigma E)"
1007     using T E.sets_into_space
1008     by (intro measurable_sigma) (auto simp: measure_preserving_def measurable_def)
1009   fix X assume "X \<in> sets (sigma E)"
1010   show "\<mu> (T -` X \<inter> space M) = E.\<mu> X"
1011   proof (rule measure_unique_Int_stable_vimage[symmetric])
1012     show "sets (sigma E) = sets (sigma E)" "space E = space (sigma E)"
1013       "\<And>i. E.\<mu> (A i) \<noteq> \<infinity>" using E by auto
1014     show "measure_space M" by default
1015   next
1016     fix X assume "X \<in> sets E" then show "E.\<mu> X = \<mu> (T -` X \<inter> space M)"
1017       using T unfolding measure_preserving_def by auto
1018   qed fact+
1019 qed
1021 section "Real measure values"
1023 lemma (in measure_space) real_measure_Union:
1024   assumes finite: "\<mu> A \<noteq> \<infinity>" "\<mu> B \<noteq> \<infinity>"
1025   and measurable: "A \<in> sets M" "B \<in> sets M" "A \<inter> B = {}"
1026   shows "real (\<mu> (A \<union> B)) = real (\<mu> A) + real (\<mu> B)"
1027   unfolding measure_additive[symmetric, OF measurable]
1028   using measurable(1,2)[THEN positive_measure]
1029   using finite by (cases rule: extreal2_cases[of "\<mu> A" "\<mu> B"]) auto
1031 lemma (in measure_space) real_measure_finite_Union:
1032   assumes measurable:
1033     "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M" "disjoint_family_on A S"
1034   assumes finite: "\<And>i. i \<in> S \<Longrightarrow> \<mu> (A i) \<noteq> \<infinity>"
1035   shows "real (\<mu> (\<Union>i\<in>S. A i)) = (\<Sum>i\<in>S. real (\<mu> (A i)))"
1036   using finite measurable(2)[THEN positive_measure]
1037   by (force intro!: setsum_real_of_extreal[symmetric]
1038             simp: measure_setsum[OF measurable, symmetric])
1040 lemma (in measure_space) real_measure_Diff:
1041   assumes finite: "\<mu> A \<noteq> \<infinity>"
1042   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
1043   shows "real (\<mu> (A - B)) = real (\<mu> A) - real (\<mu> B)"
1044 proof -
1045   have "\<mu> (A - B) \<le> \<mu> A" "\<mu> B \<le> \<mu> A"
1046     using measurable by (auto intro!: measure_mono)
1047   hence "real (\<mu> ((A - B) \<union> B)) = real (\<mu> (A - B)) + real (\<mu> B)"
1048     using measurable finite by (rule_tac real_measure_Union) auto
1049   thus ?thesis using `B \<subseteq> A` by (auto simp: Un_absorb2)
1050 qed
1052 lemma (in measure_space) real_measure_UNION:
1053   assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
1054   assumes finite: "\<mu> (\<Union>i. A i) \<noteq> \<infinity>"
1055   shows "(\<lambda>i. real (\<mu> (A i))) sums (real (\<mu> (\<Union>i. A i)))"
1056 proof -
1057   have "\<And>i. 0 \<le> \<mu> (A i)" using measurable by auto
1058   with summable_sums[OF summable_extreal_pos, of "\<lambda>i. \<mu> (A i)"]
1060   have "(\<lambda>i. \<mu> (A i)) sums (\<mu> (\<Union>i. A i))" by simp
1061   moreover
1062   { fix i
1063     have "\<mu> (A i) \<le> \<mu> (\<Union>i. A i)"
1064       using measurable by (auto intro!: measure_mono)
1065     moreover have "0 \<le> \<mu> (A i)" using measurable by auto
1066     ultimately have "\<mu> (A i) = extreal (real (\<mu> (A i)))"
1067       using finite by (cases "\<mu> (A i)") auto }
1068   moreover
1069   have "0 \<le> \<mu> (\<Union>i. A i)" using measurable by auto
1070   then have "\<mu> (\<Union>i. A i) = extreal (real (\<mu> (\<Union>i. A i)))"
1071     using finite by (cases "\<mu> (\<Union>i. A i)") auto
1072   ultimately show ?thesis
1073     unfolding sums_extreal[symmetric] by simp
1074 qed
1076 lemma (in measure_space) real_measure_subadditive:
1077   assumes measurable: "A \<in> sets M" "B \<in> sets M"
1078   and fin: "\<mu> A \<noteq> \<infinity>" "\<mu> B \<noteq> \<infinity>"
1079   shows "real (\<mu> (A \<union> B)) \<le> real (\<mu> A) + real (\<mu> B)"
1080 proof -
1081   have "0 \<le> \<mu> (A \<union> B)" using measurable by auto
1082   then show "real (\<mu> (A \<union> B)) \<le> real (\<mu> A) + real (\<mu> B)"
1083     using measure_subadditive[OF measurable] fin
1084     by (cases rule: extreal3_cases[of "\<mu> (A \<union> B)" "\<mu> A" "\<mu> B"]) auto
1085 qed
1087 lemma (in measure_space) real_measure_setsum_singleton:
1088   assumes S: "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
1089   and fin: "\<And>x. x \<in> S \<Longrightarrow> \<mu> {x} \<noteq> \<infinity>"
1090   shows "real (\<mu> S) = (\<Sum>x\<in>S. real (\<mu> {x}))"
1091   using measure_finite_singleton[OF S] fin
1092   using positive_measure[OF S(2)]
1093   by (force intro!: setsum_real_of_extreal[symmetric])
1095 lemma (in measure_space) real_continuity_from_below:
1096   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "\<mu> (\<Union>i. A i) \<noteq> \<infinity>"
1097   shows "(\<lambda>i. real (\<mu> (A i))) ----> real (\<mu> (\<Union>i. A i))"
1098 proof -
1099   have "0 \<le> \<mu> (\<Union>i. A i)" using A by auto
1100   then have "extreal (real (\<mu> (\<Union>i. A i))) = \<mu> (\<Union>i. A i)"
1101     using fin by (auto intro: extreal_real')
1102   then show ?thesis
1103     using continuity_from_below_Lim[OF A]
1104     by (intro lim_real_of_extreal) simp
1105 qed
1107 lemma (in measure_space) continuity_from_above_Lim:
1108   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
1109   shows "(\<lambda>i. (\<mu> (A i))) ----> \<mu> (\<Inter>i. A i)"
1110   using LIMSEQ_extreal_INFI[OF measure_decseq, OF A]
1111   using continuity_from_above[OF A fin] by simp
1113 lemma (in measure_space) real_continuity_from_above:
1114   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. \<mu> (A i) \<noteq> \<infinity>"
1115   shows "(\<lambda>n. real (\<mu> (A n))) ----> real (\<mu> (\<Inter>i. A i))"
1116 proof -
1117   have "0 \<le> \<mu> (\<Inter>i. A i)" using A by auto
1118   moreover
1119   have "\<mu> (\<Inter>i. A i) \<le> \<mu> (A 0)"
1120     using A by (auto intro!: measure_mono)
1121   ultimately have "extreal (real (\<mu> (\<Inter>i. A i))) = \<mu> (\<Inter>i. A i)"
1122     using fin by (auto intro: extreal_real')
1123   then show ?thesis
1124     using continuity_from_above_Lim[OF A fin]
1125     by (intro lim_real_of_extreal) simp
1126 qed
1128 lemma (in measure_space) real_measure_countably_subadditive:
1129   assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. \<mu> (A i)) \<noteq> \<infinity>"
1130   shows "real (\<mu> (\<Union>i. A i)) \<le> (\<Sum>i. real (\<mu> (A i)))"
1131 proof -
1132   { fix i
1133     have "0 \<le> \<mu> (A i)" using A by auto
1134     moreover have "\<mu> (A i) \<noteq> \<infinity>" using A by (intro suminf_PInfty[OF _ fin]) auto
1135     ultimately have "\<bar>\<mu> (A i)\<bar> \<noteq> \<infinity>" by auto }
1136   moreover have "0 \<le> \<mu> (\<Union>i. A i)" using A by auto
1137   ultimately have "extreal (real (\<mu> (\<Union>i. A i))) \<le> (\<Sum>i. extreal (real (\<mu> (A i))))"
1138     using measure_countably_subadditive[OF A] by (auto simp: extreal_real)
1139   also have "\<dots> = extreal (\<Sum>i. real (\<mu> (A i)))"
1140     using A
1141     by (auto intro!: sums_unique[symmetric] sums_extreal[THEN iffD2] summable_sums summable_real_of_extreal fin)
1142   finally show ?thesis by simp
1143 qed
1145 locale finite_measure = measure_space +
1146   assumes finite_measure_of_space: "\<mu> (space M) \<noteq> \<infinity>"
1148 sublocale finite_measure < sigma_finite_measure
1149 proof
1150   show "\<exists>A. range A \<subseteq> sets M \<and> (\<Union>i. A i) = space M \<and> (\<forall>i. \<mu> (A i) \<noteq> \<infinity>)"
1151     using finite_measure_of_space by (auto intro!: exI[of _ "\<lambda>x. space M"])
1152 qed
1154 lemma (in finite_measure) finite_measure[simp, intro]:
1155   assumes "A \<in> sets M"
1156   shows "\<mu> A \<noteq> \<infinity>"
1157 proof -
1158   from `A \<in> sets M` have "A \<subseteq> space M"
1159     using sets_into_space by blast
1160   then have "\<mu> A \<le> \<mu> (space M)"
1161     using assms top by (rule measure_mono)
1162   then show ?thesis
1163     using finite_measure_of_space by auto
1164 qed
1166 lemma (in finite_measure) measure_not_inf:
1167   assumes A: "A \<in> sets M"
1168   shows "\<bar>\<mu> A\<bar> \<noteq> \<infinity>"
1169   using finite_measure[OF A] positive_measure[OF A] by auto
1171 definition (in finite_measure)
1172   "\<mu>' A = (if A \<in> sets M then real (\<mu> A) else 0)"
1174 lemma (in finite_measure) finite_measure_eq: "A \<in> sets M \<Longrightarrow> \<mu> A = extreal (\<mu>' A)"
1175   using measure_not_inf[of A] by (auto simp: \<mu>'_def)
1177 lemma (in finite_measure) positive_measure': "0 \<le> \<mu>' A"
1178   unfolding \<mu>'_def by (auto simp: real_of_extreal_pos)
1180 lemma (in finite_measure) bounded_measure: "\<mu>' A \<le> \<mu>' (space M)"
1181 proof cases
1182   assume "A \<in> sets M"
1183   moreover then have "\<mu> A \<le> \<mu> (space M)"
1184     using sets_into_space by (auto intro!: measure_mono)
1185   ultimately show ?thesis
1186     using measure_not_inf[of A] measure_not_inf[of "space M"]
1187     by (auto simp: \<mu>'_def)
1188 qed (simp add: \<mu>'_def real_of_extreal_pos)
1190 lemma (in finite_measure) restricted_finite_measure:
1191   assumes "S \<in> sets M"
1192   shows "finite_measure (restricted_space S)"
1193     (is "finite_measure ?r")
1194   unfolding finite_measure_def finite_measure_axioms_def
1195 proof (intro conjI)
1196   show "measure_space ?r" using restricted_measure_space[OF assms] .
1197 next
1198   show "measure ?r (space ?r) \<noteq> \<infinity>" using finite_measure[OF `S \<in> sets M`] by auto
1199 qed
1201 lemma (in measure_space) restricted_to_finite_measure:
1202   assumes "S \<in> sets M" "\<mu> S \<noteq> \<infinity>"
1203   shows "finite_measure (restricted_space S)"
1204 proof -
1205   have "measure_space (restricted_space S)"
1206     using `S \<in> sets M` by (rule restricted_measure_space)
1207   then show ?thesis
1208     unfolding finite_measure_def finite_measure_axioms_def
1209     using assms by auto
1210 qed
1212 lemma (in finite_measure) finite_measure_Diff:
1213   assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
1214   shows "\<mu>' (A - B) = \<mu>' A - \<mu>' B"
1215   using sets[THEN finite_measure_eq]
1216   using Diff[OF sets, THEN finite_measure_eq]
1217   using measure_Diff[OF _ assms] by simp
1219 lemma (in finite_measure) finite_measure_Union:
1220   assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
1221   shows "\<mu>' (A \<union> B) = \<mu>' A + \<mu>' B"
1222   using measure_additive[OF assms]
1223   using sets[THEN finite_measure_eq]
1224   using Un[OF sets, THEN finite_measure_eq]
1225   by simp
1227 lemma (in finite_measure) finite_measure_finite_Union:
1228   assumes S: "finite S" "\<And>i. i \<in> S \<Longrightarrow> A i \<in> sets M"
1229   and dis: "disjoint_family_on A S"
1230   shows "\<mu>' (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. \<mu>' (A i))"
1231   using measure_setsum[OF assms]
1232   using finite_UN[of S A, OF S, THEN finite_measure_eq]
1233   using S(2)[THEN finite_measure_eq]
1234   by simp
1236 lemma (in finite_measure) finite_measure_UNION:
1237   assumes A: "range A \<subseteq> sets M" "disjoint_family A"
1238   shows "(\<lambda>i. \<mu>' (A i)) sums (\<mu>' (\<Union>i. A i))"
1239   using real_measure_UNION[OF A]
1240   using countable_UN[OF A(1), THEN finite_measure_eq]
1241   using A(1)[THEN subsetD, THEN finite_measure_eq]
1242   by auto
1244 lemma (in finite_measure) finite_measure_mono:
1245   assumes B: "B \<in> sets M" and "A \<subseteq> B" shows "\<mu>' A \<le> \<mu>' B"
1246 proof cases
1247   assume "A \<in> sets M"
1248   from this[THEN finite_measure_eq] B[THEN finite_measure_eq]
1249   show ?thesis using measure_mono[OF `A \<subseteq> B` `A \<in> sets M` `B \<in> sets M`] by simp
1250 next
1251   assume "A \<notin> sets M" then show ?thesis
1252     using positive_measure'[of B] unfolding \<mu>'_def by auto
1253 qed
1255 lemma (in finite_measure) finite_measure_subadditive:
1256   assumes m: "A \<in> sets M" "B \<in> sets M"
1257   shows "\<mu>' (A \<union> B) \<le> \<mu>' A + \<mu>' B"
1258   using measure_subadditive[OF m]
1259   using m[THEN finite_measure_eq] Un[OF m, THEN finite_measure_eq] by simp
1261 lemma (in finite_measure) finite_measure_countably_subadditive:
1262   assumes A: "range A \<subseteq> sets M" and sum: "summable (\<lambda>i. \<mu>' (A i))"
1263   shows "\<mu>' (\<Union>i. A i) \<le> (\<Sum>i. \<mu>' (A i))"
1264 proof -
1265   note A[THEN subsetD, THEN finite_measure_eq, simp]
1266   note countable_UN[OF A, THEN finite_measure_eq, simp]
1267   from `summable (\<lambda>i. \<mu>' (A i))`
1268   have "(\<lambda>i. extreal (\<mu>' (A i))) sums extreal (\<Sum>i. \<mu>' (A i))"
1269     by (simp add: sums_extreal) (rule summable_sums)
1270   from sums_unique[OF this, symmetric]
1272   show ?thesis by simp
1273 qed
1275 lemma (in finite_measure) finite_measure_finite_singleton:
1276   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
1277   shows "\<mu>' S = (\<Sum>x\<in>S. \<mu>' {x})"
1278   using real_measure_setsum_singleton[OF assms]
1279   using *[THEN finite_measure_eq]
1280   using finite_UN[of S "\<lambda>x. {x}", OF assms, THEN finite_measure_eq]
1281   by simp
1283 lemma (in finite_measure) finite_continuity_from_below:
1284   assumes A: "range A \<subseteq> sets M" and "incseq A"
1285   shows "(\<lambda>i. \<mu>' (A i)) ----> \<mu>' (\<Union>i. A i)"
1286   using real_continuity_from_below[OF A, OF `incseq A` finite_measure] assms
1287   using A[THEN subsetD, THEN finite_measure_eq]
1288   using countable_UN[OF A, THEN finite_measure_eq]
1289   by auto
1291 lemma (in finite_measure) finite_continuity_from_above:
1292   assumes A: "range A \<subseteq> sets M" and "decseq A"
1293   shows "(\<lambda>n. \<mu>' (A n)) ----> \<mu>' (\<Inter>i. A i)"
1294   using real_continuity_from_above[OF A, OF `decseq A` finite_measure] assms
1295   using A[THEN subsetD, THEN finite_measure_eq]
1296   using countable_INT[OF A, THEN finite_measure_eq]
1297   by auto
1299 lemma (in finite_measure) finite_measure_compl:
1300   assumes S: "S \<in> sets M"
1301   shows "\<mu>' (space M - S) = \<mu>' (space M) - \<mu>' S"
1302   using measure_compl[OF S, OF finite_measure, OF S]
1303   using S[THEN finite_measure_eq]
1304   using compl_sets[OF S, THEN finite_measure_eq]
1305   using top[THEN finite_measure_eq]
1306   by simp
1308 lemma (in finite_measure) finite_measure_inter_full_set:
1309   assumes S: "S \<in> sets M" "T \<in> sets M"
1310   assumes T: "\<mu>' T = \<mu>' (space M)"
1311   shows "\<mu>' (S \<inter> T) = \<mu>' S"
1312   using measure_inter_full_set[OF S finite_measure]
1313   using T Diff[OF S(2,1)] Diff[OF S, THEN finite_measure_eq]
1314   using Int[OF S, THEN finite_measure_eq]
1315   using S[THEN finite_measure_eq] top[THEN finite_measure_eq]
1316   by simp
1318 lemma (in finite_measure) empty_measure'[simp]: "\<mu>' {} = 0"
1319   unfolding \<mu>'_def by simp
1321 section "Finite spaces"
1323 locale finite_measure_space = measure_space + finite_sigma_algebra +
1324   assumes finite_single_measure[simp]: "\<And>x. x \<in> space M \<Longrightarrow> \<mu> {x} \<noteq> \<infinity>"
1326 lemma (in finite_measure_space) sum_over_space: "(\<Sum>x\<in>space M. \<mu> {x}) = \<mu> (space M)"
1327   using measure_setsum[of "space M" "\<lambda>i. {i}"]
1328   by (simp add: sets_eq_Pow disjoint_family_on_def finite_space)
1330 lemma finite_measure_spaceI:
1331   assumes "finite (space M)" "sets M = Pow(space M)" and space: "measure M (space M) \<noteq> \<infinity>"
1332     and add: "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M B"
1333     and "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A"
1334   shows "finite_measure_space M"
1335     unfolding finite_measure_space_def finite_measure_space_axioms_def
1336 proof (intro allI impI conjI)
1337   show "measure_space M"
1338   proof (rule finite_additivity_sufficient)
1339     have *: "\<lparr>space = space M, sets = Pow (space M), \<dots> = algebra.more M\<rparr> = M"
1340       unfolding assms(2)[symmetric] by (auto intro!: algebra.equality)
1341     show "sigma_algebra M"
1342       using sigma_algebra_Pow[of "space M" "algebra.more M"]
1343       unfolding * .
1344     show "finite (space M)" by fact
1345     show "positive M (measure M)" unfolding positive_def using assms by auto
1346     show "additive M (measure M)" unfolding additive_def using assms by simp
1347   qed
1348   then interpret measure_space M .
1349   show "finite_sigma_algebra M"
1350   proof
1351     show "finite (space M)" by fact
1352     show "sets M = Pow (space M)" using assms by auto
1353   qed
1354   { fix x assume *: "x \<in> space M"
1355     with add[of "{x}" "space M - {x}"] space
1356     show "\<mu> {x} \<noteq> \<infinity>" by (auto simp: insert_absorb[OF *] Diff_subset) }
1357 qed
1359 sublocale finite_measure_space \<subseteq> finite_measure
1360 proof
1361   show "\<mu> (space M) \<noteq> \<infinity>"
1362     unfolding sum_over_space[symmetric] setsum_Pinfty
1363     using finite_space finite_single_measure by auto
1364 qed
1366 lemma finite_measure_space_iff:
1367   "finite_measure_space M \<longleftrightarrow>
1368     finite (space M) \<and> sets M = Pow(space M) \<and> measure M (space M) \<noteq> \<infinity> \<and>
1369     measure M {} = 0 \<and> (\<forall>A\<subseteq>space M. 0 \<le> measure M A) \<and>
1370     (\<forall>A\<subseteq>space M. \<forall>B\<subseteq>space M. A \<inter> B = {} \<longrightarrow> measure M (A \<union> B) = measure M A + measure M B)"
1371     (is "_ = ?rhs")
1372 proof (intro iffI)
1373   assume "finite_measure_space M"
1374   then interpret finite_measure_space M .
1375   show ?rhs
1376     using finite_space sets_eq_Pow measure_additive empty_measure finite_measure
1377     by auto
1378 next
1379   assume ?rhs then show "finite_measure_space M"
1380     by (auto intro!: finite_measure_spaceI)
1381 qed
1383 lemma suminf_cmult_indicator:
1384   fixes f :: "nat \<Rightarrow> extreal"
1385   assumes "disjoint_family A" "x \<in> A i" "\<And>i. 0 \<le> f i"
1386   shows "(\<Sum>n. f n * indicator (A n) x) = f i"
1387 proof -
1388   have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: extreal)"
1389     using `x \<in> A i` assms unfolding disjoint_family_on_def indicator_def by auto
1390   then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: extreal)"
1391     by (auto simp: setsum_cases)
1392   moreover have "(SUP n. if i < n then f i else 0) = (f i :: extreal)"
1393   proof (rule extreal_SUPI)
1394     fix y :: extreal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
1395     from this[of "Suc i"] show "f i \<le> y" by auto
1396   qed (insert assms, simp)
1397   ultimately show ?thesis using assms
1398     by (subst suminf_extreal_eq_SUPR) (auto simp: indicator_def)
1399 qed
1401 lemma suminf_indicator:
1402   assumes "disjoint_family A"
1403   shows "(\<Sum>n. indicator (A n) x :: extreal) = indicator (\<Union>i. A i) x"
1404 proof cases
1405   assume *: "x \<in> (\<Union>i. A i)"
1406   then obtain i where "x \<in> A i" by auto
1407   from suminf_cmult_indicator[OF assms(1), OF `x \<in> A i`, of "\<lambda>k. 1"]
1408   show ?thesis using * by simp
1409 qed simp
1411 end