src/HOL/Analysis/Measure_Space.thy
 author wenzelm Mon Mar 25 17:21:26 2019 +0100 (2 months ago) changeset 69981 3dced198b9ec parent 69861 62e47f06d22c child 70136 f03a01a18c6e permissions -rw-r--r--
more strict AFP properties;
1 (*  Title:      HOL/Analysis/Measure_Space.thy
2     Author:     Lawrence C Paulson
3     Author:     Johannes Hölzl, TU München
4     Author:     Armin Heller, TU München
5 *)
7 section \<open>Measure Spaces\<close>
9 theory Measure_Space
10 imports
11   Measurable "HOL-Library.Extended_Nonnegative_Real"
12 begin
14 subsection%unimportant "Relate extended reals and the indicator function"
16 lemma suminf_cmult_indicator:
17   fixes f :: "nat \<Rightarrow> ennreal"
18   assumes "disjoint_family A" "x \<in> A i"
19   shows "(\<Sum>n. f n * indicator (A n) x) = f i"
20 proof -
21   have **: "\<And>n. f n * indicator (A n) x = (if n = i then f n else 0 :: ennreal)"
22     using \<open>x \<in> A i\<close> assms unfolding disjoint_family_on_def indicator_def by auto
23   then have "\<And>n. (\<Sum>j<n. f j * indicator (A j) x) = (if i < n then f i else 0 :: ennreal)"
24     by (auto simp: sum.If_cases)
25   moreover have "(SUP n. if i < n then f i else 0) = (f i :: ennreal)"
26   proof (rule SUP_eqI)
27     fix y :: ennreal assume "\<And>n. n \<in> UNIV \<Longrightarrow> (if i < n then f i else 0) \<le> y"
28     from this[of "Suc i"] show "f i \<le> y" by auto
29   qed (insert assms, simp)
30   ultimately show ?thesis using assms
31     by (subst suminf_eq_SUP) (auto simp: indicator_def)
32 qed
34 lemma suminf_indicator:
35   assumes "disjoint_family A"
36   shows "(\<Sum>n. indicator (A n) x :: ennreal) = indicator (\<Union>i. A i) x"
37 proof cases
38   assume *: "x \<in> (\<Union>i. A i)"
39   then obtain i where "x \<in> A i" by auto
40   from suminf_cmult_indicator[OF assms(1), OF \<open>x \<in> A i\<close>, of "\<lambda>k. 1"]
41   show ?thesis using * by simp
42 qed simp
44 lemma sum_indicator_disjoint_family:
45   fixes f :: "'d \<Rightarrow> 'e::semiring_1"
46   assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
47   shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
48 proof -
49   have "P \<inter> {i. x \<in> A i} = {j}"
50     using d \<open>x \<in> A j\<close> \<open>j \<in> P\<close> unfolding disjoint_family_on_def
51     by auto
52   thus ?thesis
53     unfolding indicator_def
54     by (simp add: if_distrib sum.If_cases[OF \<open>finite P\<close>])
55 qed
57 text \<open>
58   The type for emeasure spaces is already defined in \<^theory>\<open>HOL-Analysis.Sigma_Algebra\<close>, as it
59   is also used to represent sigma algebras (with an arbitrary emeasure).
60 \<close>
62 subsection%unimportant "Extend binary sets"
64 lemma LIMSEQ_binaryset:
65   assumes f: "f {} = 0"
66   shows  "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"
67 proof -
68   have "(\<lambda>n. \<Sum>i < Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
69     proof
70       fix n
71       show "(\<Sum>i < Suc (Suc n). f (binaryset A B i)) = f A + f B"
72         by (induct n)  (auto simp add: binaryset_def f)
73     qed
74   moreover
75   have "... \<longlonglongrightarrow> f A + f B" by (rule tendsto_const)
76   ultimately
77   have "(\<lambda>n. \<Sum>i< Suc (Suc n). f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"
78     by metis
79   hence "(\<lambda>n. \<Sum>i< n+2. f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"
80     by simp
81   thus ?thesis by (rule LIMSEQ_offset [where k=2])
82 qed
84 lemma binaryset_sums:
85   assumes f: "f {} = 0"
86   shows  "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
87     by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f] atLeast0LessThan)
89 lemma suminf_binaryset_eq:
90   fixes f :: "'a set \<Rightarrow> 'b::{comm_monoid_add, t2_space}"
91   shows "f {} = 0 \<Longrightarrow> (\<Sum>n. f (binaryset A B n)) = f A + f B"
92   by (metis binaryset_sums sums_unique)
94 subsection%unimportant \<open>Properties of a premeasure \<^term>\<open>\<mu>\<close>\<close>
96 text \<open>
97   The definitions for \<^const>\<open>positive\<close> and \<^const>\<open>countably_additive\<close> should be here, by they are
98   necessary to define \<^typ>\<open>'a measure\<close> in \<^theory>\<open>HOL-Analysis.Sigma_Algebra\<close>.
99 \<close>
102   "subadditive M f \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
104 lemma subadditiveD: "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
109     (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (f (\<Union>i. A i) \<le> (\<Sum>i. f (A i))))"
112   fixes f :: "'a set \<Rightarrow> ennreal"
113   assumes f: "positive M f" and cs: "countably_subadditive M f"
116   fix x y
117   assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
118   hence "disjoint_family (binaryset x y)"
119     by (auto simp add: disjoint_family_on_def binaryset_def)
120   hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
121          (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
122          f (\<Union>i. binaryset x y i) \<le> (\<Sum> n. f (binaryset x y n))"
124   hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
125          f (x \<union> y) \<le> (\<Sum> n. f (binaryset x y n))"
126     by (simp add: range_binaryset_eq UN_binaryset_eq)
127   thus "f (x \<union> y) \<le>  f x + f y" using f x y
128     by (auto simp add: Un o_def suminf_binaryset_eq positive_def)
129 qed
132   "additive M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)"
134 definition increasing where
135   "increasing M \<mu> \<longleftrightarrow> (\<forall>x\<in>M. \<forall>y\<in>M. x \<subseteq> y \<longrightarrow> \<mu> x \<le> \<mu> y)"
137 lemma positiveD1: "positive M f \<Longrightarrow> f {} = 0" by (auto simp: positive_def)
139 lemma positiveD_empty:
140   "positive M f \<Longrightarrow> f {} = 0"
141   by (auto simp add: positive_def)
144   "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x \<in> M \<Longrightarrow> y \<in> M \<Longrightarrow> f (x \<union> y) = f x + f y"
147 lemma increasingD:
148   "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>M \<Longrightarrow> y\<in>M \<Longrightarrow> f x \<le> f y"
149   by (auto simp add: increasing_def)
152   "(\<And>A. range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i. A i) \<in> M \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>i. A i))
157   assumes f: "positive M f" "additive M f" and A: "range A \<subseteq> M" "incseq A"
158   shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
159 proof (induct n)
160   case (Suc n)
161   then have "(\<Sum>i\<le>Suc n. f (disjointed A i)) = f (A n) + f (disjointed A (Suc n))"
162     by simp
163   also have "\<dots> = f (A n \<union> disjointed A (Suc n))"
164     using A by (subst f(2)[THEN additiveD]) (auto simp: disjointed_mono)
165   also have "A n \<union> disjointed A (Suc n) = A (Suc n)"
166     using \<open>incseq A\<close> by (auto dest: incseq_SucD simp: disjointed_mono)
167   finally show ?case .
168 qed simp
171   fixes A:: "'i \<Rightarrow> 'a set"
172   assumes f: "positive M f" and ad: "additive M f" and "finite S"
173       and A: "A`S \<subseteq> M"
174       and disj: "disjoint_family_on A S"
175   shows  "(\<Sum>i\<in>S. f (A i)) = f (\<Union>i\<in>S. A i)"
176   using \<open>finite S\<close> disj A
177 proof induct
178   case empty show ?case using f by (simp add: positive_def)
179 next
180   case (insert s S)
181   then have "A s \<inter> (\<Union>i\<in>S. A i) = {}"
182     by (auto simp add: disjoint_family_on_def neq_iff)
183   moreover
184   have "A s \<in> M" using insert by blast
185   moreover have "(\<Union>i\<in>S. A i) \<in> M"
186     using insert \<open>finite S\<close> by auto
187   ultimately have "f (A s \<union> (\<Union>i\<in>S. A i)) = f (A s) + f(\<Union>i\<in>S. A i)"
189   with insert show ?case using ad disjoint_family_on_mono[of S "insert s S" A]
191 qed
194   fixes f :: "'a set \<Rightarrow> ennreal"
196   shows "increasing M f"
197 proof (auto simp add: increasing_def)
198   fix x y
199   assume xy: "x \<in> M" "y \<in> M" "x \<subseteq> y"
200   then have "y - x \<in> M" by auto
201   then have "f x + 0 \<le> f x + f (y-x)" by (intro add_left_mono zero_le)
202   also have "... = f (x \<union> (y-x))" using addf
204   also have "... = f y"
205     by (metis Un_Diff_cancel Un_absorb1 xy(3))
206   finally show "f x \<le> f y" by simp
207 qed
210   fixes f :: "'a set \<Rightarrow> ennreal"
211   assumes f: "positive M f" "additive M f" and A: "A`S \<subseteq> M" and S: "finite S"
212   shows "f (\<Union>i\<in>S. A i) \<le> (\<Sum>i\<in>S. f (A i))"
213 using S A
214 proof (induct S)
215   case empty thus ?case using f by (auto simp: positive_def)
216 next
217   case (insert x F)
218   hence in_M: "A x \<in> M" "(\<Union>i\<in>F. A i) \<in> M" "(\<Union>i\<in>F. A i) - A x \<in> M" using A by force+
219   have subs: "(\<Union>i\<in>F. A i) - A x \<subseteq> (\<Union>i\<in>F. A i)" by auto
220   have "(\<Union>i\<in>(insert x F). A i) = A x \<union> ((\<Union>i\<in>F. A i) - A x)" by auto
221   hence "f (\<Union>i\<in>(insert x F). A i) = f (A x \<union> ((\<Union>i\<in>F. A i) - A x))"
222     by simp
223   also have "\<dots> = f (A x) + f ((\<Union>i\<in>F. A i) - A x)"
224     using f(2) by (rule additiveD) (insert in_M, auto)
225   also have "\<dots> \<le> f (A x) + f (\<Union>i\<in>F. A i)"
226     using additive_increasing[OF f] in_M subs by (auto simp: increasing_def intro: add_left_mono)
227   also have "\<dots> \<le> f (A x) + (\<Sum>i\<in>F. f (A i))" using insert by (auto intro: add_left_mono)
228   finally show "f (\<Union>i\<in>(insert x F). A i) \<le> (\<Sum>i\<in>(insert x F). f (A i))" using insert by simp
229 qed
232   fixes f :: "'a set \<Rightarrow> ennreal"
233   assumes posf: "positive M f" and ca: "countably_additive M f"
236   fix x y
237   assume x: "x \<in> M" and y: "y \<in> M" and "x \<inter> y = {}"
238   hence "disjoint_family (binaryset x y)"
239     by (auto simp add: disjoint_family_on_def binaryset_def)
240   hence "range (binaryset x y) \<subseteq> M \<longrightarrow>
241          (\<Union>i. binaryset x y i) \<in> M \<longrightarrow>
242          f (\<Union>i. binaryset x y i) = (\<Sum> n. f (binaryset x y n))"
243     using ca
245   hence "{x,y,{}} \<subseteq> M \<longrightarrow> x \<union> y \<in> M \<longrightarrow>
246          f (x \<union> y) = (\<Sum>n. f (binaryset x y n))"
247     by (simp add: range_binaryset_eq UN_binaryset_eq)
248   thus "f (x \<union> y) = f x + f y" using posf x y
249     by (auto simp add: Un suminf_binaryset_eq positive_def)
250 qed
253   fixes A:: "nat \<Rightarrow> 'a set" and  f :: "'a set \<Rightarrow> ennreal"
255       and inc: "increasing M f"
256       and A: "range A \<subseteq> M"
257       and disj: "disjoint_family A"
258   shows  "(\<Sum>i. f (A i)) \<le> f \<Omega>"
259 proof (safe intro!: suminf_le_const)
260   fix N
261   note disj_N = disjoint_family_on_mono[OF _ disj, of "{..<N}"]
262   have "(\<Sum>i<N. f (A i)) = f (\<Union>i\<in>{..<N}. A i)"
263     using A by (intro additive_sum [OF f ad _ _]) (auto simp: disj_N)
264   also have "... \<le> f \<Omega>" using space_closed A
265     by (intro increasingD[OF inc] finite_UN) auto
266   finally show "(\<Sum>i<N. f (A i)) \<le> f \<Omega>" by simp
267 qed (insert f A, auto simp: positive_def)
270   fixes \<mu> :: "'a set \<Rightarrow> ennreal"
271   assumes "finite \<Omega>" "positive M \<mu>" "additive M \<mu>"
274   fix F :: "nat \<Rightarrow> 'a set" assume F: "range F \<subseteq> M" "(\<Union>i. F i) \<in> M" and disj: "disjoint_family F"
276   have "\<forall>i\<in>{i. F i \<noteq> {}}. \<exists>x. x \<in> F i" by auto
277   from bchoice[OF this] obtain f where f: "\<And>i. F i \<noteq> {} \<Longrightarrow> f i \<in> F i" by auto
279   have inj_f: "inj_on f {i. F i \<noteq> {}}"
280   proof (rule inj_onI, simp)
281     fix i j a b assume *: "f i = f j" "F i \<noteq> {}" "F j \<noteq> {}"
282     then have "f i \<in> F i" "f j \<in> F j" using f by force+
283     with disj * show "i = j" by (auto simp: disjoint_family_on_def)
284   qed
285   have "finite (\<Union>i. F i)"
286     by (metis F(2) assms(1) infinite_super sets_into_space)
288   have F_subset: "{i. \<mu> (F i) \<noteq> 0} \<subseteq> {i. F i \<noteq> {}}"
289     by (auto simp: positiveD_empty[OF \<open>positive M \<mu>\<close>])
290   moreover have fin_not_empty: "finite {i. F i \<noteq> {}}"
291   proof (rule finite_imageD)
292     from f have "f`{i. F i \<noteq> {}} \<subseteq> (\<Union>i. F i)" by auto
293     then show "finite (f`{i. F i \<noteq> {}})"
294       by (rule finite_subset) fact
295   qed fact
296   ultimately have fin_not_0: "finite {i. \<mu> (F i) \<noteq> 0}"
297     by (rule finite_subset)
299   have disj_not_empty: "disjoint_family_on F {i. F i \<noteq> {}}"
300     using disj by (auto simp: disjoint_family_on_def)
302   from fin_not_0 have "(\<Sum>i. \<mu> (F i)) = (\<Sum>i | \<mu> (F i) \<noteq> 0. \<mu> (F i))"
303     by (rule suminf_finite) auto
304   also have "\<dots> = (\<Sum>i | F i \<noteq> {}. \<mu> (F i))"
305     using fin_not_empty F_subset by (rule sum.mono_neutral_left) auto
306   also have "\<dots> = \<mu> (\<Union>i\<in>{i. F i \<noteq> {}}. F i)"
307     using \<open>positive M \<mu>\<close> \<open>additive M \<mu>\<close> fin_not_empty disj_not_empty F by (intro additive_sum) auto
308   also have "\<dots> = \<mu> (\<Union>i. F i)"
309     by (rule arg_cong[where f=\<mu>]) auto
310   finally show "(\<Sum>i. \<mu> (F i)) = \<mu> (\<Union>i. F i)" .
311 qed
314   fixes f :: "'a set \<Rightarrow> ennreal"
315   assumes f: "positive M f" "additive M f"
316   shows "countably_additive M f \<longleftrightarrow>
317     (\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i))"
319 proof safe
320   assume count_sum: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> \<Union>(A ` UNIV) \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (\<Union>(A ` UNIV))"
321   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"
322   then have dA: "range (disjointed A) \<subseteq> M" by (auto simp: range_disjointed_sets)
323   with count_sum[THEN spec, of "disjointed A"] A(3)
324   have f_UN: "(\<Sum>i. f (disjointed A i)) = f (\<Union>i. A i)"
325     by (auto simp: UN_disjointed_eq disjoint_family_disjointed)
326   moreover have "(\<lambda>n. (\<Sum>i<n. f (disjointed A i))) \<longlonglongrightarrow> (\<Sum>i. f (disjointed A i))"
327     using f(1)[unfolded positive_def] dA
328     by (auto intro!: summable_LIMSEQ)
329   from LIMSEQ_Suc[OF this]
330   have "(\<lambda>n. (\<Sum>i\<le>n. f (disjointed A i))) \<longlonglongrightarrow> (\<Sum>i. f (disjointed A i))"
331     unfolding lessThan_Suc_atMost .
332   moreover have "\<And>n. (\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
333     using disjointed_additive[OF f A(1,2)] .
334   ultimately show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)" by simp
335 next
336   assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"
337   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "disjoint_family A" "(\<Union>i. A i) \<in> M"
338   have *: "(\<Union>n. (\<Union>i<n. A i)) = (\<Union>i. A i)" by auto
339   have "(\<lambda>n. f (\<Union>i<n. A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"
340   proof (unfold *[symmetric], intro cont[rule_format])
341     show "range (\<lambda>i. \<Union>i<i. A i) \<subseteq> M" "(\<Union>i. \<Union>i<i. A i) \<in> M"
342       using A * by auto
343   qed (force intro!: incseq_SucI)
344   moreover have "\<And>n. f (\<Union>i<n. A i) = (\<Sum>i<n. f (A i))"
345     using A
346     by (intro additive_sum[OF f, of _ A, symmetric])
347        (auto intro: disjoint_family_on_mono[where B=UNIV])
348   ultimately
349   have "(\<lambda>i. f (A i)) sums f (\<Union>i. A i)"
350     unfolding sums_def by simp
351   from sums_unique[OF this]
352   show "(\<Sum>i. f (A i)) = f (\<Union>i. A i)" by simp
353 qed
355 lemma (in ring_of_sets) continuous_from_above_iff_empty_continuous:
356   fixes f :: "'a set \<Rightarrow> ennreal"
357   assumes f: "positive M f" "additive M f"
358   shows "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Inter>i. A i))
359      \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0)"
360 proof safe
361   assume cont: "(\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) \<in> M \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Inter>i. A i))"
362   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) = {}" "\<forall>i. f (A i) \<noteq> \<infinity>"
363   with cont[THEN spec, of A] show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
364     using \<open>positive M f\<close>[unfolded positive_def] by auto
365 next
366   assume cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<forall>i. f (A i) \<noteq> \<infinity>) \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
367   fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> M" "decseq A" "(\<Inter>i. A i) \<in> M" "\<forall>i. f (A i) \<noteq> \<infinity>"
369   have f_mono: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<subseteq> b \<Longrightarrow> f a \<le> f b"
370     using additive_increasing[OF f] unfolding increasing_def by simp
372   have decseq_fA: "decseq (\<lambda>i. f (A i))"
373     using A by (auto simp: decseq_def intro!: f_mono)
374   have decseq: "decseq (\<lambda>i. A i - (\<Inter>i. A i))"
375     using A by (auto simp: decseq_def)
376   then have decseq_f: "decseq (\<lambda>i. f (A i - (\<Inter>i. A i)))"
377     using A unfolding decseq_def by (auto intro!: f_mono Diff)
378   have "f (\<Inter>x. A x) \<le> f (A 0)"
379     using A by (auto intro!: f_mono)
380   then have f_Int_fin: "f (\<Inter>x. A x) \<noteq> \<infinity>"
381     using A by (auto simp: top_unique)
382   { fix i
383     have "f (A i - (\<Inter>i. A i)) \<le> f (A i)" using A by (auto intro!: f_mono)
384     then have "f (A i - (\<Inter>i. A i)) \<noteq> \<infinity>"
385       using A by (auto simp: top_unique) }
386   note f_fin = this
387   have "(\<lambda>i. f (A i - (\<Inter>i. A i))) \<longlonglongrightarrow> 0"
388   proof (intro cont[rule_format, OF _ decseq _ f_fin])
389     show "range (\<lambda>i. A i - (\<Inter>i. A i)) \<subseteq> M" "(\<Inter>i. A i - (\<Inter>i. A i)) = {}"
390       using A by auto
391   qed
392   from INF_Lim[OF decseq_f this]
393   have "(INF n. f (A n - (\<Inter>i. A i))) = 0" .
394   moreover have "(INF n. f (\<Inter>i. A i)) = f (\<Inter>i. A i)"
395     by auto
396   ultimately have "(INF n. f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i)) = 0 + f (\<Inter>i. A i)"
397     using A(4) f_fin f_Int_fin
398     by (subst INF_ennreal_add_const) (auto simp: decseq_f)
399   moreover {
400     fix n
401     have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f ((A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i))"
402       using A by (subst f(2)[THEN additiveD]) auto
403     also have "(A n - (\<Inter>i. A i)) \<union> (\<Inter>i. A i) = A n"
404       by auto
405     finally have "f (A n - (\<Inter>i. A i)) + f (\<Inter>i. A i) = f (A n)" . }
406   ultimately have "(INF n. f (A n)) = f (\<Inter>i. A i)"
407     by simp
408   with LIMSEQ_INF[OF decseq_fA]
409   show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Inter>i. A i)" by simp
410 qed
412 lemma (in ring_of_sets) empty_continuous_imp_continuous_from_below:
413   fixes f :: "'a set \<Rightarrow> ennreal"
414   assumes f: "positive M f" "additive M f" "\<forall>A\<in>M. f A \<noteq> \<infinity>"
415   assumes cont: "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> (\<Inter>i. A i) = {} \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
416   assumes A: "range A \<subseteq> M" "incseq A" "(\<Union>i. A i) \<in> M"
417   shows "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"
418 proof -
419   from A have "(\<lambda>i. f ((\<Union>i. A i) - A i)) \<longlonglongrightarrow> 0"
420     by (intro cont[rule_format]) (auto simp: decseq_def incseq_def)
421   moreover
422   { fix i
423     have "f ((\<Union>i. A i) - A i \<union> A i) = f ((\<Union>i. A i) - A i) + f (A i)"
424       using A by (intro f(2)[THEN additiveD]) auto
425     also have "((\<Union>i. A i) - A i) \<union> A i = (\<Union>i. A i)"
426       by auto
427     finally have "f ((\<Union>i. A i) - A i) = f (\<Union>i. A i) - f (A i)"
428       using f(3)[rule_format, of "A i"] A by (auto simp: ennreal_add_diff_cancel subset_eq) }
429   moreover have "\<forall>\<^sub>F i in sequentially. f (A i) \<le> f (\<Union>i. A i)"
430     using increasingD[OF additive_increasing[OF f(1, 2)], of "A _" "\<Union>i. A i"] A
431     by (auto intro!: always_eventually simp: subset_eq)
432   ultimately show "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union>i. A i)"
433     by (auto intro: ennreal_tendsto_const_minus)
434 qed
437   fixes f :: "'a set \<Rightarrow> ennreal"
438   assumes f: "positive M f" "additive M f" and fin: "\<forall>A\<in>M. f A \<noteq> \<infinity>"
439   assumes cont: "\<And>A. range A \<subseteq> M \<Longrightarrow> decseq A \<Longrightarrow> (\<Inter>i. A i) = {} \<Longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
442   using empty_continuous_imp_continuous_from_below[OF f fin] cont
443   by blast
445 subsection%unimportant \<open>Properties of \<^const>\<open>emeasure\<close>\<close>
447 lemma emeasure_positive: "positive (sets M) (emeasure M)"
448   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
450 lemma emeasure_empty[simp, intro]: "emeasure M {} = 0"
451   using emeasure_positive[of M] by (simp add: positive_def)
453 lemma emeasure_single_in_space: "emeasure M {x} \<noteq> 0 \<Longrightarrow> x \<in> space M"
454   using emeasure_notin_sets[of "{x}" M] by (auto dest: sets.sets_into_space zero_less_iff_neq_zero[THEN iffD2])
457   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
459 lemma suminf_emeasure:
460   "range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
461   using sets.countable_UN[of A UNIV M] emeasure_countably_additive[of M]
464 lemma sums_emeasure:
465   "disjoint_family F \<Longrightarrow> (\<And>i. F i \<in> sets M) \<Longrightarrow> (\<lambda>i. emeasure M (F i)) sums emeasure M (\<Union>i. F i)"
466   unfolding sums_iff by (intro conjI suminf_emeasure) auto
471 lemma plus_emeasure:
472   "a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> emeasure M a + emeasure M b = emeasure M (a \<union> b)"
475 lemma emeasure_Union:
476   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"
477   using plus_emeasure[of A M "B - A"] by auto
479 lemma sum_emeasure:
480   "F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow>
481     (\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"
484 lemma emeasure_mono:
485   "a \<subseteq> b \<Longrightarrow> b \<in> sets M \<Longrightarrow> emeasure M a \<le> emeasure M b"
488 lemma emeasure_space:
489   "emeasure M A \<le> emeasure M (space M)"
490   by (metis emeasure_mono emeasure_notin_sets sets.sets_into_space sets.top zero_le)
492 lemma emeasure_Diff:
493   assumes finite: "emeasure M B \<noteq> \<infinity>"
494   and [measurable]: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
495   shows "emeasure M (A - B) = emeasure M A - emeasure M B"
496 proof -
497   have "(A - B) \<union> B = A" using \<open>B \<subseteq> A\<close> by auto
498   then have "emeasure M A = emeasure M ((A - B) \<union> B)" by simp
499   also have "\<dots> = emeasure M (A - B) + emeasure M B"
500     by (subst plus_emeasure[symmetric]) auto
501   finally show "emeasure M (A - B) = emeasure M A - emeasure M B"
502     using finite by simp
503 qed
505 lemma emeasure_compl:
506   "s \<in> sets M \<Longrightarrow> emeasure M s \<noteq> \<infinity> \<Longrightarrow> emeasure M (space M - s) = emeasure M (space M) - emeasure M s"
507   by (rule emeasure_Diff) (auto dest: sets.sets_into_space)
509 lemma Lim_emeasure_incseq:
510   "range A \<subseteq> sets M \<Longrightarrow> incseq A \<Longrightarrow> (\<lambda>i. (emeasure M (A i))) \<longlonglongrightarrow> emeasure M (\<Union>i. A i)"
515 lemma incseq_emeasure:
516   assumes "range B \<subseteq> sets M" "incseq B"
517   shows "incseq (\<lambda>i. emeasure M (B i))"
518   using assms by (auto simp: incseq_def intro!: emeasure_mono)
520 lemma SUP_emeasure_incseq:
521   assumes A: "range A \<subseteq> sets M" "incseq A"
522   shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union>i. A i)"
523   using LIMSEQ_SUP[OF incseq_emeasure, OF A] Lim_emeasure_incseq[OF A]
526 lemma decseq_emeasure:
527   assumes "range B \<subseteq> sets M" "decseq B"
528   shows "decseq (\<lambda>i. emeasure M (B i))"
529   using assms by (auto simp: decseq_def intro!: emeasure_mono)
531 lemma INF_emeasure_decseq:
532   assumes A: "range A \<subseteq> sets M" and "decseq A"
533   and finite: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
534   shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"
535 proof -
536   have le_MI: "emeasure M (\<Inter>i. A i) \<le> emeasure M (A 0)"
537     using A by (auto intro!: emeasure_mono)
538   hence *: "emeasure M (\<Inter>i. A i) \<noteq> \<infinity>" using finite[of 0] by (auto simp: top_unique)
540   have "emeasure M (A 0) - (INF n. emeasure M (A n)) = (SUP n. emeasure M (A 0) - emeasure M (A n))"
542   also have "\<dots> = (SUP n. emeasure M (A 0 - A n))"
543     using A finite \<open>decseq A\<close>[unfolded decseq_def] by (subst emeasure_Diff) auto
544   also have "\<dots> = emeasure M (\<Union>i. A 0 - A i)"
545   proof (rule SUP_emeasure_incseq)
546     show "range (\<lambda>n. A 0 - A n) \<subseteq> sets M"
547       using A by auto
548     show "incseq (\<lambda>n. A 0 - A n)"
549       using \<open>decseq A\<close> by (auto simp add: incseq_def decseq_def)
550   qed
551   also have "\<dots> = emeasure M (A 0) - emeasure M (\<Inter>i. A i)"
552     using A finite * by (simp, subst emeasure_Diff) auto
553   finally show ?thesis
554     by (rule ennreal_minus_cancel[rotated 3])
555        (insert finite A, auto intro: INF_lower emeasure_mono)
556 qed
558 lemma INF_emeasure_decseq':
559   assumes A: "\<And>i. A i \<in> sets M" and "decseq A"
560   and finite: "\<exists>i. emeasure M (A i) \<noteq> \<infinity>"
561   shows "(INF n. emeasure M (A n)) = emeasure M (\<Inter>i. A i)"
562 proof -
563   from finite obtain i where i: "emeasure M (A i) < \<infinity>"
564     by (auto simp: less_top)
565   have fin: "i \<le> j \<Longrightarrow> emeasure M (A j) < \<infinity>" for j
566     by (rule le_less_trans[OF emeasure_mono i]) (auto intro!: decseqD[OF \<open>decseq A\<close>] A)
568   have "(INF n. emeasure M (A n)) = (INF n. emeasure M (A (n + i)))"
569   proof (rule INF_eq)
570     show "\<exists>j\<in>UNIV. emeasure M (A (j + i)) \<le> emeasure M (A i')" for i'
571       by (intro bexI[of _ i'] emeasure_mono decseqD[OF \<open>decseq A\<close>] A) auto
572   qed auto
573   also have "\<dots> = emeasure M (INF n. (A (n + i)))"
574     using A \<open>decseq A\<close> fin by (intro INF_emeasure_decseq) (auto simp: decseq_def less_top)
575   also have "(INF n. (A (n + i))) = (INF n. A n)"
576     by (meson INF_eq UNIV_I assms(2) decseqD le_add1)
577   finally show ?thesis .
578 qed
580 lemma emeasure_INT_decseq_subset:
581   fixes F :: "nat \<Rightarrow> 'a set"
582   assumes I: "I \<noteq> {}" and F: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> i \<le> j \<Longrightarrow> F j \<subseteq> F i"
583   assumes F_sets[measurable]: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M"
584     and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (F i) \<noteq> \<infinity>"
585   shows "emeasure M (\<Inter>i\<in>I. F i) = (INF i\<in>I. emeasure M (F i))"
586 proof cases
587   assume "finite I"
588   have "(\<Inter>i\<in>I. F i) = F (Max I)"
589     using I \<open>finite I\<close> by (intro antisym INF_lower INF_greatest F) auto
590   moreover have "(INF i\<in>I. emeasure M (F i)) = emeasure M (F (Max I))"
591     using I \<open>finite I\<close> by (intro antisym INF_lower INF_greatest F emeasure_mono) auto
592   ultimately show ?thesis
593     by simp
594 next
595   assume "infinite I"
596   define L where "L n = (LEAST i. i \<in> I \<and> i \<ge> n)" for n
597   have L: "L n \<in> I \<and> n \<le> L n" for n
598     unfolding L_def
599   proof (rule LeastI_ex)
600     show "\<exists>x. x \<in> I \<and> n \<le> x"
601       using \<open>infinite I\<close> finite_subset[of I "{..< n}"]
602       by (rule_tac ccontr) (auto simp: not_le)
603   qed
604   have L_eq[simp]: "i \<in> I \<Longrightarrow> L i = i" for i
605     unfolding L_def by (intro Least_equality) auto
606   have L_mono: "i \<le> j \<Longrightarrow> L i \<le> L j" for i j
607     using L[of j] unfolding L_def by (intro Least_le) (auto simp: L_def)
609   have "emeasure M (\<Inter>i. F (L i)) = (INF i. emeasure M (F (L i)))"
610   proof (intro INF_emeasure_decseq[symmetric])
611     show "decseq (\<lambda>i. F (L i))"
612       using L by (intro antimonoI F L_mono) auto
613   qed (insert L fin, auto)
614   also have "\<dots> = (INF i\<in>I. emeasure M (F i))"
615   proof (intro antisym INF_greatest)
616     show "i \<in> I \<Longrightarrow> (INF i. emeasure M (F (L i))) \<le> emeasure M (F i)" for i
617       by (intro INF_lower2[of i]) auto
618   qed (insert L, auto intro: INF_lower)
619   also have "(\<Inter>i. F (L i)) = (\<Inter>i\<in>I. F i)"
620   proof (intro antisym INF_greatest)
621     show "i \<in> I \<Longrightarrow> (\<Inter>i. F (L i)) \<subseteq> F i" for i
622       by (intro INF_lower2[of i]) auto
623   qed (insert L, auto)
624   finally show ?thesis .
625 qed
627 lemma Lim_emeasure_decseq:
628   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
629   shows "(\<lambda>i. emeasure M (A i)) \<longlonglongrightarrow> emeasure M (\<Inter>i. A i)"
630   using LIMSEQ_INF[OF decseq_emeasure, OF A]
631   using INF_emeasure_decseq[OF A fin] by simp
633 lemma emeasure_lfp'[consumes 1, case_names cont measurable]:
634   assumes "P M"
635   assumes cont: "sup_continuous F"
636   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
637   shows "emeasure M {x\<in>space M. lfp F x} = (SUP i. emeasure M {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
638 proof -
639   have "emeasure M {x\<in>space M. lfp F x} = emeasure M (\<Union>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
640     using sup_continuous_lfp[OF cont] by (auto simp add: bot_fun_def intro!: arg_cong2[where f=emeasure])
641   moreover { fix i from \<open>P M\<close> have "{x\<in>space M. (F ^^ i) (\<lambda>x. False) x} \<in> sets M"
642     by (induct i arbitrary: M) (auto simp add: pred_def[symmetric] intro: *) }
643   moreover have "incseq (\<lambda>i. {x\<in>space M. (F ^^ i) (\<lambda>x. False) x})"
644   proof (rule incseq_SucI)
645     fix i
646     have "(F ^^ i) (\<lambda>x. False) \<le> (F ^^ (Suc i)) (\<lambda>x. False)"
647     proof (induct i)
648       case 0 show ?case by (simp add: le_fun_def)
649     next
650       case Suc thus ?case using monoD[OF sup_continuous_mono[OF cont] Suc] by auto
651     qed
652     then show "{x \<in> space M. (F ^^ i) (\<lambda>x. False) x} \<subseteq> {x \<in> space M. (F ^^ Suc i) (\<lambda>x. False) x}"
653       by auto
654   qed
655   ultimately show ?thesis
656     by (subst SUP_emeasure_incseq) auto
657 qed
659 lemma emeasure_lfp:
660   assumes [simp]: "\<And>s. sets (M s) = sets N"
661   assumes cont: "sup_continuous F" "sup_continuous f"
662   assumes meas: "\<And>P. Measurable.pred N P \<Longrightarrow> Measurable.pred N (F P)"
663   assumes iter: "\<And>P s. Measurable.pred N P \<Longrightarrow> P \<le> lfp F \<Longrightarrow> emeasure (M s) {x\<in>space N. F P x} = f (\<lambda>s. emeasure (M s) {x\<in>space N. P x}) s"
664   shows "emeasure (M s) {x\<in>space N. lfp F x} = lfp f s"
665 proof (subst lfp_transfer_bounded[where \<alpha>="\<lambda>F s. emeasure (M s) {x\<in>space N. F x}" and g=f and f=F and P="Measurable.pred N", symmetric])
666   fix C assume "incseq C" "\<And>i. Measurable.pred N (C i)"
667   then show "(\<lambda>s. emeasure (M s) {x \<in> space N. (SUP i. C i) x}) = (SUP i. (\<lambda>s. emeasure (M s) {x \<in> space N. C i x}))"
668     unfolding SUP_apply[abs_def]
669     by (subst SUP_emeasure_incseq) (auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])
670 qed (auto simp add: iter le_fun_def SUP_apply[abs_def] intro!: meas cont)
673   "finite I \<Longrightarrow> A ` I \<subseteq> sets M \<Longrightarrow> emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
677   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
678   using emeasure_subadditive_finite[of "{True, False}" "\<lambda>True \<Rightarrow> A | False \<Rightarrow> B" M] by simp
681   assumes "range f \<subseteq> sets M"
682   shows "emeasure M (\<Union>i. f i) \<le> (\<Sum>i. emeasure M (f i))"
683 proof -
684   have "emeasure M (\<Union>i. f i) = emeasure M (\<Union>i. disjointed f i)"
685     unfolding UN_disjointed_eq ..
686   also have "\<dots> = (\<Sum>i. emeasure M (disjointed f i))"
687     using sets.range_disjointed_sets[OF assms] suminf_emeasure[of "disjointed f"]
688     by (simp add:  disjoint_family_disjointed comp_def)
689   also have "\<dots> \<le> (\<Sum>i. emeasure M (f i))"
690     using sets.range_disjointed_sets[OF assms] assms
691     by (auto intro!: suminf_le emeasure_mono disjointed_subset)
692   finally show ?thesis .
693 qed
695 lemma emeasure_insert:
696   assumes sets: "{x} \<in> sets M" "A \<in> sets M" and "x \<notin> A"
697   shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
698 proof -
699   have "{x} \<inter> A = {}" using \<open>x \<notin> A\<close> by auto
700   from plus_emeasure[OF sets this] show ?thesis by simp
701 qed
703 lemma emeasure_insert_ne:
704   "A \<noteq> {} \<Longrightarrow> {x} \<in> sets M \<Longrightarrow> A \<in> sets M \<Longrightarrow> x \<notin> A \<Longrightarrow> emeasure M (insert x A) = emeasure M {x} + emeasure M A"
705   by (rule emeasure_insert)
707 lemma emeasure_eq_sum_singleton:
708   assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
709   shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"
710   using sum_emeasure[of "\<lambda>x. {x}" S M] assms
711   by (auto simp: disjoint_family_on_def subset_eq)
713 lemma sum_emeasure_cover:
714   assumes "finite S" and "A \<in> sets M" and br_in_M: "B ` S \<subseteq> sets M"
715   assumes A: "A \<subseteq> (\<Union>i\<in>S. B i)"
716   assumes disj: "disjoint_family_on B S"
717   shows "emeasure M A = (\<Sum>i\<in>S. emeasure M (A \<inter> (B i)))"
718 proof -
719   have "(\<Sum>i\<in>S. emeasure M (A \<inter> (B i))) = emeasure M (\<Union>i\<in>S. A \<inter> (B i))"
720   proof (rule sum_emeasure)
721     show "disjoint_family_on (\<lambda>i. A \<inter> B i) S"
722       using \<open>disjoint_family_on B S\<close>
723       unfolding disjoint_family_on_def by auto
724   qed (insert assms, auto)
725   also have "(\<Union>i\<in>S. A \<inter> (B i)) = A"
726     using A by auto
727   finally show ?thesis by simp
728 qed
730 lemma emeasure_eq_0:
731   "N \<in> sets M \<Longrightarrow> emeasure M N = 0 \<Longrightarrow> K \<subseteq> N \<Longrightarrow> emeasure M K = 0"
732   by (metis emeasure_mono order_eq_iff zero_le)
734 lemma emeasure_UN_eq_0:
735   assumes "\<And>i::nat. emeasure M (N i) = 0" and "range N \<subseteq> sets M"
736   shows "emeasure M (\<Union>i. N i) = 0"
737 proof -
738   have "emeasure M (\<Union>i. N i) \<le> 0"
739     using emeasure_subadditive_countably[OF assms(2)] assms(1) by simp
740   then show ?thesis
741     by (auto intro: antisym zero_le)
742 qed
744 lemma measure_eqI_finite:
745   assumes [simp]: "sets M = Pow A" "sets N = Pow A" and "finite A"
746   assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}"
747   shows "M = N"
748 proof (rule measure_eqI)
749   fix X assume "X \<in> sets M"
750   then have X: "X \<subseteq> A" by auto
751   then have "emeasure M X = (\<Sum>a\<in>X. emeasure M {a})"
752     using \<open>finite A\<close> by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset)
753   also have "\<dots> = (\<Sum>a\<in>X. emeasure N {a})"
754     using X eq by (auto intro!: sum.cong)
755   also have "\<dots> = emeasure N X"
756     using X \<open>finite A\<close> by (subst emeasure_eq_sum_singleton) (auto dest: finite_subset)
757   finally show "emeasure M X = emeasure N X" .
758 qed simp
760 lemma measure_eqI_generator_eq:
761   fixes M N :: "'a measure" and E :: "'a set set" and A :: "nat \<Rightarrow> 'a set"
762   assumes "Int_stable E" "E \<subseteq> Pow \<Omega>"
763   and eq: "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
764   and M: "sets M = sigma_sets \<Omega> E"
765   and N: "sets N = sigma_sets \<Omega> E"
766   and A: "range A \<subseteq> E" "(\<Union>i. A i) = \<Omega>" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
767   shows "M = N"
768 proof -
769   let ?\<mu>  = "emeasure M" and ?\<nu> = "emeasure N"
770   interpret S: sigma_algebra \<Omega> "sigma_sets \<Omega> E" by (rule sigma_algebra_sigma_sets) fact
771   have "space M = \<Omega>"
772     using sets.top[of M] sets.space_closed[of M] S.top S.space_closed \<open>sets M = sigma_sets \<Omega> E\<close>
773     by blast
775   { fix F D assume "F \<in> E" and "?\<mu> F \<noteq> \<infinity>"
776     then have [intro]: "F \<in> sigma_sets \<Omega> E" by auto
777     have "?\<nu> F \<noteq> \<infinity>" using \<open>?\<mu> F \<noteq> \<infinity>\<close> \<open>F \<in> E\<close> eq by simp
778     assume "D \<in> sets M"
779     with \<open>Int_stable E\<close> \<open>E \<subseteq> Pow \<Omega>\<close> have "emeasure M (F \<inter> D) = emeasure N (F \<inter> D)"
780       unfolding M
781     proof (induct rule: sigma_sets_induct_disjoint)
782       case (basic A)
783       then have "F \<inter> A \<in> E" using \<open>Int_stable E\<close> \<open>F \<in> E\<close> by (auto simp: Int_stable_def)
784       then show ?case using eq by auto
785     next
786       case empty then show ?case by simp
787     next
788       case (compl A)
789       then have **: "F \<inter> (\<Omega> - A) = F - (F \<inter> A)"
790         and [intro]: "F \<inter> A \<in> sigma_sets \<Omega> E"
791         using \<open>F \<in> E\<close> S.sets_into_space by (auto simp: M)
792       have "?\<nu> (F \<inter> A) \<le> ?\<nu> F" by (auto intro!: emeasure_mono simp: M N)
793       then have "?\<nu> (F \<inter> A) \<noteq> \<infinity>" using \<open>?\<nu> F \<noteq> \<infinity>\<close> by (auto simp: top_unique)
794       have "?\<mu> (F \<inter> A) \<le> ?\<mu> F" by (auto intro!: emeasure_mono simp: M N)
795       then have "?\<mu> (F \<inter> A) \<noteq> \<infinity>" using \<open>?\<mu> F \<noteq> \<infinity>\<close> by (auto simp: top_unique)
796       then have "?\<mu> (F \<inter> (\<Omega> - A)) = ?\<mu> F - ?\<mu> (F \<inter> A)" unfolding **
797         using \<open>F \<inter> A \<in> sigma_sets \<Omega> E\<close> by (auto intro!: emeasure_Diff simp: M N)
798       also have "\<dots> = ?\<nu> F - ?\<nu> (F \<inter> A)" using eq \<open>F \<in> E\<close> compl by simp
799       also have "\<dots> = ?\<nu> (F \<inter> (\<Omega> - A))" unfolding **
800         using \<open>F \<inter> A \<in> sigma_sets \<Omega> E\<close> \<open>?\<nu> (F \<inter> A) \<noteq> \<infinity>\<close>
801         by (auto intro!: emeasure_Diff[symmetric] simp: M N)
802       finally show ?case
803         using \<open>space M = \<Omega>\<close> by auto
804     next
805       case (union A)
806       then have "?\<mu> (\<Union>x. F \<inter> A x) = ?\<nu> (\<Union>x. F \<inter> A x)"
807         by (subst (1 2) suminf_emeasure[symmetric]) (auto simp: disjoint_family_on_def subset_eq M N)
808       with A show ?case
809         by auto
810     qed }
811   note * = this
812   show "M = N"
813   proof (rule measure_eqI)
814     show "sets M = sets N"
815       using M N by simp
816     have [simp, intro]: "\<And>i. A i \<in> sets M"
817       using A(1) by (auto simp: subset_eq M)
818     fix F assume "F \<in> sets M"
819     let ?D = "disjointed (\<lambda>i. F \<inter> A i)"
820     from \<open>space M = \<Omega>\<close> have F_eq: "F = (\<Union>i. ?D i)"
821       using \<open>F \<in> sets M\<close>[THEN sets.sets_into_space] A(2)[symmetric] by (auto simp: UN_disjointed_eq)
822     have [simp, intro]: "\<And>i. ?D i \<in> sets M"
823       using sets.range_disjointed_sets[of "\<lambda>i. F \<inter> A i" M] \<open>F \<in> sets M\<close>
824       by (auto simp: subset_eq)
825     have "disjoint_family ?D"
826       by (auto simp: disjoint_family_disjointed)
827     moreover
828     have "(\<Sum>i. emeasure M (?D i)) = (\<Sum>i. emeasure N (?D i))"
829     proof (intro arg_cong[where f=suminf] ext)
830       fix i
831       have "A i \<inter> ?D i = ?D i"
832         by (auto simp: disjointed_def)
833       then show "emeasure M (?D i) = emeasure N (?D i)"
834         using *[of "A i" "?D i", OF _ A(3)] A(1) by auto
835     qed
836     ultimately show "emeasure M F = emeasure N F"
837       by (simp add: image_subset_iff \<open>sets M = sets N\<close>[symmetric] F_eq[symmetric] suminf_emeasure)
838   qed
839 qed
841 lemma space_empty: "space M = {} \<Longrightarrow> M = count_space {}"
842   by (rule measure_eqI) (simp_all add: space_empty_iff)
844 lemma measure_eqI_generator_eq_countable:
845   fixes M N :: "'a measure" and E :: "'a set set" and A :: "'a set set"
846   assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
847     and sets: "sets M = sigma_sets \<Omega> E" "sets N = sigma_sets \<Omega> E"
848   and A: "A \<subseteq> E" "(\<Union>A) = \<Omega>" "countable A" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
849   shows "M = N"
850 proof cases
851   assume "\<Omega> = {}"
852   have *: "sigma_sets \<Omega> E = sets (sigma \<Omega> E)"
853     using E(2) by simp
854   have "space M = \<Omega>" "space N = \<Omega>"
855     using sets E(2) unfolding * by (auto dest: sets_eq_imp_space_eq simp del: sets_measure_of)
856   then show "M = N"
857     unfolding \<open>\<Omega> = {}\<close> by (auto dest: space_empty)
858 next
859   assume "\<Omega> \<noteq> {}" with \<open>\<Union>A = \<Omega>\<close> have "A \<noteq> {}" by auto
860   from this \<open>countable A\<close> have rng: "range (from_nat_into A) = A"
861     by (rule range_from_nat_into)
862   show "M = N"
863   proof (rule measure_eqI_generator_eq[OF E sets])
864     show "range (from_nat_into A) \<subseteq> E"
865       unfolding rng using \<open>A \<subseteq> E\<close> .
866     show "(\<Union>i. from_nat_into A i) = \<Omega>"
867       unfolding rng using \<open>\<Union>A = \<Omega>\<close> .
868     show "emeasure M (from_nat_into A i) \<noteq> \<infinity>" for i
869       using rng by (intro A) auto
870   qed
871 qed
873 lemma measure_of_of_measure: "measure_of (space M) (sets M) (emeasure M) = M"
874 proof (intro measure_eqI emeasure_measure_of_sigma)
875   show "sigma_algebra (space M) (sets M)" ..
876   show "positive (sets M) (emeasure M)"
878   show "countably_additive (sets M) (emeasure M)"
880 qed simp_all
882 subsection \<open>\<open>\<mu>\<close>-null sets\<close>
884 definition%important null_sets :: "'a measure \<Rightarrow> 'a set set" where
885   "null_sets M = {N\<in>sets M. emeasure M N = 0}"
887 lemma null_setsD1[dest]: "A \<in> null_sets M \<Longrightarrow> emeasure M A = 0"
890 lemma null_setsD2[dest]: "A \<in> null_sets M \<Longrightarrow> A \<in> sets M"
891   unfolding null_sets_def by simp
893 lemma null_setsI[intro]: "emeasure M A = 0 \<Longrightarrow> A \<in> sets M \<Longrightarrow> A \<in> null_sets M"
894   unfolding null_sets_def by simp
896 interpretation null_sets: ring_of_sets "space M" "null_sets M" for M
897 proof (rule ring_of_setsI)
898   show "null_sets M \<subseteq> Pow (space M)"
899     using sets.sets_into_space by auto
900   show "{} \<in> null_sets M"
901     by auto
902   fix A B assume null_sets: "A \<in> null_sets M" "B \<in> null_sets M"
903   then have sets: "A \<in> sets M" "B \<in> sets M"
904     by auto
905   then have *: "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
906     "emeasure M (A - B) \<le> emeasure M A"
907     by (auto intro!: emeasure_subadditive emeasure_mono)
908   then have "emeasure M B = 0" "emeasure M A = 0"
909     using null_sets by auto
910   with sets * show "A - B \<in> null_sets M" "A \<union> B \<in> null_sets M"
911     by (auto intro!: antisym zero_le)
912 qed
914 lemma UN_from_nat_into:
915   assumes I: "countable I" "I \<noteq> {}"
916   shows "(\<Union>i\<in>I. N i) = (\<Union>i. N (from_nat_into I i))"
917 proof -
918   have "(\<Union>i\<in>I. N i) = \<Union>(N ` range (from_nat_into I))"
919     using I by simp
920   also have "\<dots> = (\<Union>i. (N \<circ> from_nat_into I) i)"
921     by simp
922   finally show ?thesis by simp
923 qed
925 lemma null_sets_UN':
926   assumes "countable I"
927   assumes "\<And>i. i \<in> I \<Longrightarrow> N i \<in> null_sets M"
928   shows "(\<Union>i\<in>I. N i) \<in> null_sets M"
929 proof cases
930   assume "I = {}" then show ?thesis by simp
931 next
932   assume "I \<noteq> {}"
933   show ?thesis
934   proof (intro conjI CollectI null_setsI)
935     show "(\<Union>i\<in>I. N i) \<in> sets M"
936       using assms by (intro sets.countable_UN') auto
937     have "emeasure M (\<Union>i\<in>I. N i) \<le> (\<Sum>n. emeasure M (N (from_nat_into I n)))"
938       unfolding UN_from_nat_into[OF \<open>countable I\<close> \<open>I \<noteq> {}\<close>]
939       using assms \<open>I \<noteq> {}\<close> by (intro emeasure_subadditive_countably) (auto intro: from_nat_into)
940     also have "(\<lambda>n. emeasure M (N (from_nat_into I n))) = (\<lambda>_. 0)"
941       using assms \<open>I \<noteq> {}\<close> by (auto intro: from_nat_into)
942     finally show "emeasure M (\<Union>i\<in>I. N i) = 0"
943       by (intro antisym zero_le) simp
944   qed
945 qed
947 lemma null_sets_UN[intro]:
948   "(\<And>i::'i::countable. N i \<in> null_sets M) \<Longrightarrow> (\<Union>i. N i) \<in> null_sets M"
949   by (rule null_sets_UN') auto
951 lemma null_set_Int1:
952   assumes "B \<in> null_sets M" "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"
953 proof (intro CollectI conjI null_setsI)
954   show "emeasure M (A \<inter> B) = 0" using assms
955     by (intro emeasure_eq_0[of B _ "A \<inter> B"]) auto
956 qed (insert assms, auto)
958 lemma null_set_Int2:
959   assumes "B \<in> null_sets M" "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"
960   using assms by (subst Int_commute) (rule null_set_Int1)
962 lemma emeasure_Diff_null_set:
963   assumes "B \<in> null_sets M" "A \<in> sets M"
964   shows "emeasure M (A - B) = emeasure M A"
965 proof -
966   have *: "A - B = (A - (A \<inter> B))" by auto
967   have "A \<inter> B \<in> null_sets M" using assms by (rule null_set_Int1)
968   then show ?thesis
969     unfolding * using assms
970     by (subst emeasure_Diff) auto
971 qed
973 lemma null_set_Diff:
974   assumes "B \<in> null_sets M" "A \<in> sets M" shows "B - A \<in> null_sets M"
975 proof (intro CollectI conjI null_setsI)
976   show "emeasure M (B - A) = 0" using assms by (intro emeasure_eq_0[of B _ "B - A"]) auto
977 qed (insert assms, auto)
979 lemma emeasure_Un_null_set:
980   assumes "A \<in> sets M" "B \<in> null_sets M"
981   shows "emeasure M (A \<union> B) = emeasure M A"
982 proof -
983   have *: "A \<union> B = A \<union> (B - A)" by auto
984   have "B - A \<in> null_sets M" using assms(2,1) by (rule null_set_Diff)
985   then show ?thesis
986     unfolding * using assms
987     by (subst plus_emeasure[symmetric]) auto
988 qed
990 subsection \<open>The almost everywhere filter (i.e.\ quantifier)\<close>
992 definition%important ae_filter :: "'a measure \<Rightarrow> 'a filter" where
993   "ae_filter M = (INF N\<in>null_sets M. principal (space M - N))"
995 abbreviation almost_everywhere :: "'a measure \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
996   "almost_everywhere M P \<equiv> eventually P (ae_filter M)"
998 syntax
999   "_almost_everywhere" :: "pttrn \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool" ("AE _ in _. _" [0,0,10] 10)
1001 translations
1002   "AE x in M. P" \<rightleftharpoons> "CONST almost_everywhere M (\<lambda>x. P)"
1004 abbreviation
1005   "set_almost_everywhere A M P \<equiv> AE x in M. x \<in> A \<longrightarrow> P x"
1007 syntax
1008   "_set_almost_everywhere" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool"
1009   ("AE _\<in>_ in _./ _" [0,0,0,10] 10)
1011 translations
1012   "AE x\<in>A in M. P" \<rightleftharpoons> "CONST set_almost_everywhere A M (\<lambda>x. P)"
1014 lemma eventually_ae_filter: "eventually P (ae_filter M) \<longleftrightarrow> (\<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
1015   unfolding ae_filter_def by (subst eventually_INF_base) (auto simp: eventually_principal subset_eq)
1017 lemma AE_I':
1018   "N \<in> null_sets M \<Longrightarrow> {x\<in>space M. \<not> P x} \<subseteq> N \<Longrightarrow> (AE x in M. P x)"
1019   unfolding eventually_ae_filter by auto
1021 lemma AE_iff_null:
1022   assumes "{x\<in>space M. \<not> P x} \<in> sets M" (is "?P \<in> sets M")
1023   shows "(AE x in M. P x) \<longleftrightarrow> {x\<in>space M. \<not> P x} \<in> null_sets M"
1024 proof
1025   assume "AE x in M. P x" then obtain N where N: "N \<in> sets M" "?P \<subseteq> N" "emeasure M N = 0"
1026     unfolding eventually_ae_filter by auto
1027   have "emeasure M ?P \<le> emeasure M N"
1028     using assms N(1,2) by (auto intro: emeasure_mono)
1029   then have "emeasure M ?P = 0"
1030     unfolding \<open>emeasure M N = 0\<close> by auto
1031   then show "?P \<in> null_sets M" using assms by auto
1032 next
1033   assume "?P \<in> null_sets M" with assms show "AE x in M. P x" by (auto intro: AE_I')
1034 qed
1036 lemma AE_iff_null_sets:
1037   "N \<in> sets M \<Longrightarrow> N \<in> null_sets M \<longleftrightarrow> (AE x in M. x \<notin> N)"
1038   using Int_absorb1[OF sets.sets_into_space, of N M]
1039   by (subst AE_iff_null) (auto simp: Int_def[symmetric])
1041 lemma AE_not_in:
1042   "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"
1043   by (metis AE_iff_null_sets null_setsD2)
1045 lemma AE_iff_measurable:
1046   "N \<in> sets M \<Longrightarrow> {x\<in>space M. \<not> P x} = N \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> emeasure M N = 0"
1047   using AE_iff_null[of _ P] by auto
1049 lemma AE_E[consumes 1]:
1050   assumes "AE x in M. P x"
1051   obtains N where "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
1052   using assms unfolding eventually_ae_filter by auto
1054 lemma AE_E2:
1055   assumes "AE x in M. P x" "{x\<in>space M. P x} \<in> sets M"
1056   shows "emeasure M {x\<in>space M. \<not> P x} = 0" (is "emeasure M ?P = 0")
1057 proof -
1058   have "{x\<in>space M. \<not> P x} = space M - {x\<in>space M. P x}" by auto
1059   with AE_iff_null[of M P] assms show ?thesis by auto
1060 qed
1062 lemma AE_E3:
1063   assumes "AE x in M. P x"
1064   obtains N where "\<And>x. x \<in> space M - N \<Longrightarrow> P x" "N \<in> null_sets M"
1065 using assms unfolding eventually_ae_filter by auto
1067 lemma AE_I:
1068   assumes "{x \<in> space M. \<not> P x} \<subseteq> N" "emeasure M N = 0" "N \<in> sets M"
1069   shows "AE x in M. P x"
1070   using assms unfolding eventually_ae_filter by auto
1072 lemma AE_mp[elim!]:
1073   assumes AE_P: "AE x in M. P x" and AE_imp: "AE x in M. P x \<longrightarrow> Q x"
1074   shows "AE x in M. Q x"
1075 proof -
1076   from AE_P obtain A where P: "{x\<in>space M. \<not> P x} \<subseteq> A"
1077     and A: "A \<in> sets M" "emeasure M A = 0"
1078     by (auto elim!: AE_E)
1080   from AE_imp obtain B where imp: "{x\<in>space M. P x \<and> \<not> Q x} \<subseteq> B"
1081     and B: "B \<in> sets M" "emeasure M B = 0"
1082     by (auto elim!: AE_E)
1084   show ?thesis
1085   proof (intro AE_I)
1086     have "emeasure M (A \<union> B) \<le> 0"
1087       using emeasure_subadditive[of A M B] A B by auto
1088     then show "A \<union> B \<in> sets M" "emeasure M (A \<union> B) = 0"
1089       using A B by auto
1090     show "{x\<in>space M. \<not> Q x} \<subseteq> A \<union> B"
1091       using P imp by auto
1092   qed
1093 qed
1095 text \<open>The next lemma is convenient to combine with a lemma whose conclusion is of the
1096 form \<open>AE x in M. P x = Q x\<close>: for such a lemma, there is no \<open>[symmetric]\<close> variant,
1097 but using \<open>AE_symmetric[OF...]\<close> will replace it.\<close>
1099 (* depricated replace by laws about eventually *)
1100 lemma
1101   shows AE_iffI: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> AE x in M. Q x"
1102     and AE_disjI1: "AE x in M. P x \<Longrightarrow> AE x in M. P x \<or> Q x"
1103     and AE_disjI2: "AE x in M. Q x \<Longrightarrow> AE x in M. P x \<or> Q x"
1104     and AE_conjI: "AE x in M. P x \<Longrightarrow> AE x in M. Q x \<Longrightarrow> AE x in M. P x \<and> Q x"
1105     and AE_conj_iff[simp]: "(AE x in M. P x \<and> Q x) \<longleftrightarrow> (AE x in M. P x) \<and> (AE x in M. Q x)"
1106   by auto
1108 lemma AE_symmetric:
1109   assumes "AE x in M. P x = Q x"
1110   shows "AE x in M. Q x = P x"
1111   using assms by auto
1113 lemma AE_impI:
1114   "(P \<Longrightarrow> AE x in M. Q x) \<Longrightarrow> AE x in M. P \<longrightarrow> Q x"
1115   by (cases P) auto
1117 lemma AE_measure:
1118   assumes AE: "AE x in M. P x" and sets: "{x\<in>space M. P x} \<in> sets M" (is "?P \<in> sets M")
1119   shows "emeasure M {x\<in>space M. P x} = emeasure M (space M)"
1120 proof -
1121   from AE_E[OF AE] guess N . note N = this
1122   with sets have "emeasure M (space M) \<le> emeasure M (?P \<union> N)"
1123     by (intro emeasure_mono) auto
1124   also have "\<dots> \<le> emeasure M ?P + emeasure M N"
1125     using sets N by (intro emeasure_subadditive) auto
1126   also have "\<dots> = emeasure M ?P" using N by simp
1127   finally show "emeasure M ?P = emeasure M (space M)"
1128     using emeasure_space[of M "?P"] by auto
1129 qed
1131 lemma AE_space: "AE x in M. x \<in> space M"
1132   by (rule AE_I[where N="{}"]) auto
1134 lemma AE_I2[simp, intro]:
1135   "(\<And>x. x \<in> space M \<Longrightarrow> P x) \<Longrightarrow> AE x in M. P x"
1136   using AE_space by force
1138 lemma AE_Ball_mp:
1139   "\<forall>x\<in>space M. P x \<Longrightarrow> AE x in M. P x \<longrightarrow> Q x \<Longrightarrow> AE x in M. Q x"
1140   by auto
1142 lemma AE_cong[cong]:
1143   "(\<And>x. x \<in> space M \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in M. Q x)"
1144   by auto
1146 lemma AE_cong_simp: "M = N \<Longrightarrow> (\<And>x. x \<in> space N =simp=> P x = Q x) \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> (AE x in N. Q x)"
1147   by (auto simp: simp_implies_def)
1149 lemma AE_all_countable:
1150   "(AE x in M. \<forall>i. P i x) \<longleftrightarrow> (\<forall>i::'i::countable. AE x in M. P i x)"
1151 proof
1152   assume "\<forall>i. AE x in M. P i x"
1153   from this[unfolded eventually_ae_filter Bex_def, THEN choice]
1154   obtain N where N: "\<And>i. N i \<in> null_sets M" "\<And>i. {x\<in>space M. \<not> P i x} \<subseteq> N i" by auto
1155   have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. {x\<in>space M. \<not> P i x})" by auto
1156   also have "\<dots> \<subseteq> (\<Union>i. N i)" using N by auto
1157   finally have "{x\<in>space M. \<not> (\<forall>i. P i x)} \<subseteq> (\<Union>i. N i)" .
1158   moreover from N have "(\<Union>i. N i) \<in> null_sets M"
1159     by (intro null_sets_UN) auto
1160   ultimately show "AE x in M. \<forall>i. P i x"
1161     unfolding eventually_ae_filter by auto
1162 qed auto
1164 lemma AE_ball_countable:
1165   assumes [intro]: "countable X"
1166   shows "(AE x in M. \<forall>y\<in>X. P x y) \<longleftrightarrow> (\<forall>y\<in>X. AE x in M. P x y)"
1167 proof
1168   assume "\<forall>y\<in>X. AE x in M. P x y"
1169   from this[unfolded eventually_ae_filter Bex_def, THEN bchoice]
1170   obtain N where N: "\<And>y. y \<in> X \<Longrightarrow> N y \<in> null_sets M" "\<And>y. y \<in> X \<Longrightarrow> {x\<in>space M. \<not> P x y} \<subseteq> N y"
1171     by auto
1172   have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. {x\<in>space M. \<not> P x y})"
1173     by auto
1174   also have "\<dots> \<subseteq> (\<Union>y\<in>X. N y)"
1175     using N by auto
1176   finally have "{x\<in>space M. \<not> (\<forall>y\<in>X. P x y)} \<subseteq> (\<Union>y\<in>X. N y)" .
1177   moreover from N have "(\<Union>y\<in>X. N y) \<in> null_sets M"
1178     by (intro null_sets_UN') auto
1179   ultimately show "AE x in M. \<forall>y\<in>X. P x y"
1180     unfolding eventually_ae_filter by auto
1181 qed auto
1183 lemma AE_ball_countable':
1184   "(\<And>N. N \<in> I \<Longrightarrow> AE x in M. P N x) \<Longrightarrow> countable I \<Longrightarrow> AE x in M. \<forall>N \<in> I. P N x"
1185   unfolding AE_ball_countable by simp
1187 lemma pairwise_alt: "pairwise R S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S-{x}. R x y)"
1188   by (auto simp add: pairwise_def)
1190 lemma AE_pairwise: "countable F \<Longrightarrow> pairwise (\<lambda>A B. AE x in M. R x A B) F \<longleftrightarrow> (AE x in M. pairwise (R x) F)"
1191   unfolding pairwise_alt by (simp add: AE_ball_countable)
1193 lemma AE_discrete_difference:
1194   assumes X: "countable X"
1195   assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"
1196   assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
1197   shows "AE x in M. x \<notin> X"
1198 proof -
1199   have "(\<Union>x\<in>X. {x}) \<in> null_sets M"
1200     using assms by (intro null_sets_UN') auto
1201   from AE_not_in[OF this] show "AE x in M. x \<notin> X"
1202     by auto
1203 qed
1205 lemma AE_finite_all:
1206   assumes f: "finite S" shows "(AE x in M. \<forall>i\<in>S. P i x) \<longleftrightarrow> (\<forall>i\<in>S. AE x in M. P i x)"
1207   using f by induct auto
1209 lemma AE_finite_allI:
1210   assumes "finite S"
1211   shows "(\<And>s. s \<in> S \<Longrightarrow> AE x in M. Q s x) \<Longrightarrow> AE x in M. \<forall>s\<in>S. Q s x"
1212   using AE_finite_all[OF \<open>finite S\<close>] by auto
1214 lemma emeasure_mono_AE:
1215   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B"
1216     and B: "B \<in> sets M"
1217   shows "emeasure M A \<le> emeasure M B"
1218 proof cases
1219   assume A: "A \<in> sets M"
1220   from imp obtain N where N: "{x\<in>space M. \<not> (x \<in> A \<longrightarrow> x \<in> B)} \<subseteq> N" "N \<in> null_sets M"
1221     by (auto simp: eventually_ae_filter)
1222   have "emeasure M A = emeasure M (A - N)"
1223     using N A by (subst emeasure_Diff_null_set) auto
1224   also have "emeasure M (A - N) \<le> emeasure M (B - N)"
1225     using N A B sets.sets_into_space by (auto intro!: emeasure_mono)
1226   also have "emeasure M (B - N) = emeasure M B"
1227     using N B by (subst emeasure_Diff_null_set) auto
1228   finally show ?thesis .
1231 lemma emeasure_eq_AE:
1232   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
1233   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
1234   shows "emeasure M A = emeasure M B"
1235   using assms by (safe intro!: antisym emeasure_mono_AE) auto
1237 lemma emeasure_Collect_eq_AE:
1238   "AE x in M. P x \<longleftrightarrow> Q x \<Longrightarrow> Measurable.pred M Q \<Longrightarrow> Measurable.pred M P \<Longrightarrow>
1239    emeasure M {x\<in>space M. P x} = emeasure M {x\<in>space M. Q x}"
1240    by (intro emeasure_eq_AE) auto
1242 lemma emeasure_eq_0_AE: "AE x in M. \<not> P x \<Longrightarrow> emeasure M {x\<in>space M. P x} = 0"
1243   using AE_iff_measurable[OF _ refl, of M "\<lambda>x. \<not> P x"]
1244   by (cases "{x\<in>space M. P x} \<in> sets M") (simp_all add: emeasure_notin_sets)
1246 lemma emeasure_0_AE:
1247   assumes "emeasure M (space M) = 0"
1248   shows "AE x in M. P x"
1249 using eventually_ae_filter assms by blast
1252   assumes [measurable]: "A \<in> sets M" "B \<in> sets M" "C \<in> sets M"
1253   assumes 1: "AE x in M. x \<in> C \<longleftrightarrow> x \<in> A \<or> x \<in> B"
1254   assumes 2: "AE x in M. \<not> (x \<in> A \<and> x \<in> B)"
1255   shows "emeasure M C = emeasure M A + emeasure M B"
1256 proof -
1257   have "emeasure M C = emeasure M (A \<union> B)"
1258     by (rule emeasure_eq_AE) (insert 1, auto)
1259   also have "\<dots> = emeasure M A + emeasure M (B - A)"
1260     by (subst plus_emeasure) auto
1261   also have "emeasure M (B - A) = emeasure M B"
1262     by (rule emeasure_eq_AE) (insert 2, auto)
1263   finally show ?thesis .
1264 qed
1266 subsection \<open>\<open>\<sigma>\<close>-finite Measures\<close>
1268 locale%important sigma_finite_measure =
1269   fixes M :: "'a measure"
1270   assumes sigma_finite_countable:
1271     "\<exists>A::'a set set. countable A \<and> A \<subseteq> sets M \<and> (\<Union>A) = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"
1273 lemma (in sigma_finite_measure) sigma_finite:
1274   obtains A :: "nat \<Rightarrow> 'a set"
1275   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
1276 proof -
1277   obtain A :: "'a set set" where
1278     [simp]: "countable A" and
1279     A: "A \<subseteq> sets M" "(\<Union>A) = space M" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
1280     using sigma_finite_countable by metis
1281   show thesis
1282   proof cases
1283     assume "A = {}" with \<open>(\<Union>A) = space M\<close> show thesis
1284       by (intro that[of "\<lambda>_. {}"]) auto
1285   next
1286     assume "A \<noteq> {}"
1287     show thesis
1288     proof
1289       show "range (from_nat_into A) \<subseteq> sets M"
1290         using \<open>A \<noteq> {}\<close> A by auto
1291       have "(\<Union>i. from_nat_into A i) = \<Union>A"
1292         using range_from_nat_into[OF \<open>A \<noteq> {}\<close> \<open>countable A\<close>] by auto
1293       with A show "(\<Union>i. from_nat_into A i) = space M"
1294         by auto
1295     qed (intro A from_nat_into \<open>A \<noteq> {}\<close>)
1296   qed
1297 qed
1299 lemma (in sigma_finite_measure) sigma_finite_disjoint:
1300   obtains A :: "nat \<Rightarrow> 'a set"
1301   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "disjoint_family A"
1302 proof -
1303   obtain A :: "nat \<Rightarrow> 'a set" where
1304     range: "range A \<subseteq> sets M" and
1305     space: "(\<Union>i. A i) = space M" and
1306     measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
1307     using sigma_finite by blast
1308   show thesis
1309   proof (rule that[of "disjointed A"])
1310     show "range (disjointed A) \<subseteq> sets M"
1311       by (rule sets.range_disjointed_sets[OF range])
1312     show "(\<Union>i. disjointed A i) = space M"
1313       and "disjoint_family (disjointed A)"
1314       using disjoint_family_disjointed UN_disjointed_eq[of A] space range
1315       by auto
1316     show "emeasure M (disjointed A i) \<noteq> \<infinity>" for i
1317     proof -
1318       have "emeasure M (disjointed A i) \<le> emeasure M (A i)"
1319         using range disjointed_subset[of A i] by (auto intro!: emeasure_mono)
1320       then show ?thesis using measure[of i] by (auto simp: top_unique)
1321     qed
1322   qed
1323 qed
1325 lemma (in sigma_finite_measure) sigma_finite_incseq:
1326   obtains A :: "nat \<Rightarrow> 'a set"
1327   where "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
1328 proof -
1329   obtain F :: "nat \<Rightarrow> 'a set" where
1330     F: "range F \<subseteq> sets M" "(\<Union>i. F i) = space M" "\<And>i. emeasure M (F i) \<noteq> \<infinity>"
1331     using sigma_finite by blast
1332   show thesis
1333   proof (rule that[of "\<lambda>n. \<Union>i\<le>n. F i"])
1334     show "range (\<lambda>n. \<Union>i\<le>n. F i) \<subseteq> sets M"
1335       using F by (force simp: incseq_def)
1336     show "(\<Union>n. \<Union>i\<le>n. F i) = space M"
1337     proof -
1338       from F have "\<And>x. x \<in> space M \<Longrightarrow> \<exists>i. x \<in> F i" by auto
1339       with F show ?thesis by fastforce
1340     qed
1341     show "emeasure M (\<Union>i\<le>n. F i) \<noteq> \<infinity>" for n
1342     proof -
1343       have "emeasure M (\<Union>i\<le>n. F i) \<le> (\<Sum>i\<le>n. emeasure M (F i))"
1344         using F by (auto intro!: emeasure_subadditive_finite)
1345       also have "\<dots> < \<infinity>"
1346         using F by (auto simp: sum_Pinfty less_top)
1347       finally show ?thesis by simp
1348     qed
1349     show "incseq (\<lambda>n. \<Union>i\<le>n. F i)"
1350       by (force simp: incseq_def)
1351   qed
1352 qed
1354 lemma (in sigma_finite_measure) approx_PInf_emeasure_with_finite:
1355   fixes C::real
1356   assumes W_meas: "W \<in> sets M"
1357       and W_inf: "emeasure M W = \<infinity>"
1358   obtains Z where "Z \<in> sets M" "Z \<subseteq> W" "emeasure M Z < \<infinity>" "emeasure M Z > C"
1359 proof -
1360   obtain A :: "nat \<Rightarrow> 'a set"
1361     where A: "range A \<subseteq> sets M" "(\<Union>i. A i) = space M" "\<And>i. emeasure M (A i) \<noteq> \<infinity>" "incseq A"
1362     using sigma_finite_incseq by blast
1363   define B where "B = (\<lambda>i. W \<inter> A i)"
1364   have B_meas: "\<And>i. B i \<in> sets M" using W_meas \<open>range A \<subseteq> sets M\<close> B_def by blast
1365   have b: "\<And>i. B i \<subseteq> W" using B_def by blast
1367   { fix i
1368     have "emeasure M (B i) \<le> emeasure M (A i)"
1369       using A by (intro emeasure_mono) (auto simp: B_def)
1370     also have "emeasure M (A i) < \<infinity>"
1371       using \<open>\<And>i. emeasure M (A i) \<noteq> \<infinity>\<close> by (simp add: less_top)
1372     finally have "emeasure M (B i) < \<infinity>" . }
1373   note c = this
1375   have "W = (\<Union>i. B i)" using B_def \<open>(\<Union>i. A i) = space M\<close> W_meas by auto
1376   moreover have "incseq B" using B_def \<open>incseq A\<close> by (simp add: incseq_def subset_eq)
1377   ultimately have "(\<lambda>i. emeasure M (B i)) \<longlonglongrightarrow> emeasure M W" using W_meas B_meas
1378     by (simp add: B_meas Lim_emeasure_incseq image_subset_iff)
1379   then have "(\<lambda>i. emeasure M (B i)) \<longlonglongrightarrow> \<infinity>" using W_inf by simp
1380   from order_tendstoD(1)[OF this, of C]
1381   obtain i where d: "emeasure M (B i) > C"
1382     by (auto simp: eventually_sequentially)
1383   have "B i \<in> sets M" "B i \<subseteq> W" "emeasure M (B i) < \<infinity>" "emeasure M (B i) > C"
1384     using B_meas b c d by auto
1385   then show ?thesis using that by blast
1386 qed
1388 subsection \<open>Measure space induced by distribution of \<^const>\<open>measurable\<close>-functions\<close>
1390 definition%important distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
1391 "distr M N f =
1392   measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
1394 lemma
1395   shows sets_distr[simp, measurable_cong]: "sets (distr M N f) = sets N"
1396     and space_distr[simp]: "space (distr M N f) = space N"
1397   by (auto simp: distr_def)
1399 lemma
1400   shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"
1401     and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"
1402   by (auto simp: measurable_def)
1404 lemma distr_cong:
1405   "M = K \<Longrightarrow> sets N = sets L \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> distr M N f = distr K L g"
1406   using sets_eq_imp_space_eq[of N L] by (simp add: distr_def Int_def cong: rev_conj_cong)
1408 lemma emeasure_distr:
1409   fixes f :: "'a \<Rightarrow> 'b"
1410   assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"
1411   shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)" (is "_ = ?\<mu> A")
1412   unfolding distr_def
1413 proof (rule emeasure_measure_of_sigma)
1414   show "positive (sets N) ?\<mu>"
1415     by (auto simp: positive_def)
1417   show "countably_additive (sets N) ?\<mu>"
1419     fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"
1420     then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto
1421     then have *: "range (\<lambda>i. f -` (A i) \<inter> space M) \<subseteq> sets M"
1422       using f by (auto simp: measurable_def)
1423     moreover have "(\<Union>i. f -`  A i \<inter> space M) \<in> sets M"
1424       using * by blast
1425     moreover have **: "disjoint_family (\<lambda>i. f -` A i \<inter> space M)"
1426       using \<open>disjoint_family A\<close> by (auto simp: disjoint_family_on_def)
1427     ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"
1428       using suminf_emeasure[OF _ **] A f
1429       by (auto simp: comp_def vimage_UN)
1430   qed
1431   show "sigma_algebra (space N) (sets N)" ..
1432 qed fact
1434 lemma emeasure_Collect_distr:
1435   assumes X[measurable]: "X \<in> measurable M N" "Measurable.pred N P"
1436   shows "emeasure (distr M N X) {x\<in>space N. P x} = emeasure M {x\<in>space M. P (X x)}"
1437   by (subst emeasure_distr)
1438      (auto intro!: arg_cong2[where f=emeasure] X(1)[THEN measurable_space])
1440 lemma emeasure_lfp2[consumes 1, case_names cont f measurable]:
1441   assumes "P M"
1442   assumes cont: "sup_continuous F"
1443   assumes f: "\<And>M. P M \<Longrightarrow> f \<in> measurable M' M"
1444   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
1445   shows "emeasure M' {x\<in>space M'. lfp F (f x)} = (SUP i. emeasure M' {x\<in>space M'. (F ^^ i) (\<lambda>x. False) (f x)})"
1446 proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])
1447   show "f \<in> measurable M' M"  "f \<in> measurable M' M"
1448     using f[OF \<open>P M\<close>] by auto
1449   { fix i show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))"
1450     using \<open>P M\<close> by (induction i arbitrary: M) (auto intro!: *) }
1451   show "Measurable.pred M (lfp F)"
1452     using \<open>P M\<close> cont * by (rule measurable_lfp_coinduct[of P])
1454   have "emeasure (distr M' M f) {x \<in> space (distr M' M f). lfp F x} =
1455     (SUP i. emeasure (distr M' M f) {x \<in> space (distr M' M f). (F ^^ i) (\<lambda>x. False) x})"
1456     using \<open>P M\<close>
1457   proof (coinduction arbitrary: M rule: emeasure_lfp')
1458     case (measurable A N) then have "\<And>N. P N \<Longrightarrow> Measurable.pred (distr M' N f) A"
1459       by metis
1460     then have "\<And>N. P N \<Longrightarrow> Measurable.pred N A"
1461       by simp
1462     with \<open>P N\<close>[THEN *] show ?case
1463       by auto
1464   qed fact
1465   then show "emeasure (distr M' M f) {x \<in> space M. lfp F x} =
1466     (SUP i. emeasure (distr M' M f) {x \<in> space M. (F ^^ i) (\<lambda>x. False) x})"
1467    by simp
1468 qed
1470 lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"
1471   by (rule measure_eqI) (auto simp: emeasure_distr)
1473 lemma distr_id2: "sets M = sets N \<Longrightarrow> distr N M (\<lambda>x. x) = N"
1474   by (rule measure_eqI) (auto simp: emeasure_distr)
1476 lemma measure_distr:
1477   "f \<in> measurable M N \<Longrightarrow> S \<in> sets N \<Longrightarrow> measure (distr M N f) S = measure M (f -` S \<inter> space M)"
1478   by (simp add: emeasure_distr measure_def)
1480 lemma distr_cong_AE:
1481   assumes 1: "M = K" "sets N = sets L" and
1482     2: "(AE x in M. f x = g x)" and "f \<in> measurable M N" and "g \<in> measurable K L"
1483   shows "distr M N f = distr K L g"
1484 proof (rule measure_eqI)
1485   fix A assume "A \<in> sets (distr M N f)"
1486   with assms show "emeasure (distr M N f) A = emeasure (distr K L g) A"
1487     by (auto simp add: emeasure_distr intro!: emeasure_eq_AE measurable_sets)
1488 qed (insert 1, simp)
1490 lemma AE_distrD:
1491   assumes f: "f \<in> measurable M M'"
1492     and AE: "AE x in distr M M' f. P x"
1493   shows "AE x in M. P (f x)"
1494 proof -
1495   from AE[THEN AE_E] guess N .
1496   with f show ?thesis
1497     unfolding eventually_ae_filter
1498     by (intro bexI[of _ "f -` N \<inter> space M"])
1499        (auto simp: emeasure_distr measurable_def)
1500 qed
1502 lemma AE_distr_iff:
1503   assumes f[measurable]: "f \<in> measurable M N" and P[measurable]: "{x \<in> space N. P x} \<in> sets N"
1504   shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))"
1505 proof (subst (1 2) AE_iff_measurable[OF _ refl])
1506   have "f -` {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"
1507     using f[THEN measurable_space] by auto
1508   then show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =
1509     (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"
1511 qed auto
1513 lemma null_sets_distr_iff:
1514   "f \<in> measurable M N \<Longrightarrow> A \<in> null_sets (distr M N f) \<longleftrightarrow> f -` A \<inter> space M \<in> null_sets M \<and> A \<in> sets N"
1515   by (auto simp add: null_sets_def emeasure_distr)
1517 proposition distr_distr:
1518   "g \<in> measurable N L \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> distr (distr M N f) L g = distr M L (g \<circ> f)"
1519   by (auto simp add: emeasure_distr measurable_space
1520            intro!: arg_cong[where f="emeasure M"] measure_eqI)
1522 subsection%unimportant \<open>Real measure values\<close>
1524 lemma ring_of_finite_sets: "ring_of_sets (space M) {A\<in>sets M. emeasure M A \<noteq> top}"
1525 proof (rule ring_of_setsI)
1526   show "a \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow> b \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow>
1527     a \<union> b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b
1528     using emeasure_subadditive[of a M b] by (auto simp: top_unique)
1530   show "a \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow> b \<in> {A \<in> sets M. emeasure M A \<noteq> top} \<Longrightarrow>
1531     a - b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b
1532     using emeasure_mono[of "a - b" a M] by (auto simp: top_unique)
1533 qed (auto dest: sets.sets_into_space)
1535 lemma measure_nonneg[simp]: "0 \<le> measure M A"
1536   unfolding measure_def by auto
1538 lemma measure_nonneg' [simp]: "\<not> measure M A < 0"
1539   using measure_nonneg not_le by blast
1541 lemma zero_less_measure_iff: "0 < measure M A \<longleftrightarrow> measure M A \<noteq> 0"
1542   using measure_nonneg[of M A] by (auto simp add: le_less)
1544 lemma measure_le_0_iff: "measure M X \<le> 0 \<longleftrightarrow> measure M X = 0"
1545   using measure_nonneg[of M X] by linarith
1547 lemma measure_empty[simp]: "measure M {} = 0"
1548   unfolding measure_def by (simp add: zero_ennreal.rep_eq)
1550 lemma emeasure_eq_ennreal_measure:
1551   "emeasure M A \<noteq> top \<Longrightarrow> emeasure M A = ennreal (measure M A)"
1552   by (cases "emeasure M A" rule: ennreal_cases) (auto simp: measure_def)
1554 lemma measure_zero_top: "emeasure M A = top \<Longrightarrow> measure M A = 0"
1555   by (simp add: measure_def enn2ereal_top)
1557 lemma measure_eq_emeasure_eq_ennreal: "0 \<le> x \<Longrightarrow> emeasure M A = ennreal x \<Longrightarrow> measure M A = x"
1558   using emeasure_eq_ennreal_measure[of M A]
1559   by (cases "A \<in> M") (auto simp: measure_notin_sets emeasure_notin_sets)
1561 lemma enn2real_plus:"a < top \<Longrightarrow> b < top \<Longrightarrow> enn2real (a + b) = enn2real a + enn2real b"
1563            del: real_of_ereal_enn2ereal)
1565 lemma measure_eq_AE:
1566   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
1567   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
1568   shows "measure M A = measure M B"
1569   using assms emeasure_eq_AE[OF assms] by (simp add: measure_def)
1571 lemma measure_Union:
1572   "emeasure M A \<noteq> \<infinity> \<Longrightarrow> emeasure M B \<noteq> \<infinity> \<Longrightarrow> A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow>
1573     measure M (A \<union> B) = measure M A + measure M B"
1574   by (simp add: measure_def plus_emeasure[symmetric] enn2real_plus less_top)
1576 lemma disjoint_family_on_insert:
1577   "i \<notin> I \<Longrightarrow> disjoint_family_on A (insert i I) \<longleftrightarrow> A i \<inter> (\<Union>i\<in>I. A i) = {} \<and> disjoint_family_on A I"
1578   by (fastforce simp: disjoint_family_on_def)
1580 lemma measure_finite_Union:
1581   "finite S \<Longrightarrow> A`S \<subseteq> sets M \<Longrightarrow> disjoint_family_on A S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>) \<Longrightarrow>
1582     measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
1583   by (induction S rule: finite_induct)
1584      (auto simp: disjoint_family_on_insert measure_Union sum_emeasure[symmetric] sets.countable_UN'[OF countable_finite])
1586 lemma measure_Diff:
1587   assumes finite: "emeasure M A \<noteq> \<infinity>"
1588   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
1589   shows "measure M (A - B) = measure M A - measure M B"
1590 proof -
1591   have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"
1592     using measurable by (auto intro!: emeasure_mono)
1593   hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"
1594     using measurable finite by (rule_tac measure_Union) (auto simp: top_unique)
1595   thus ?thesis using \<open>B \<subseteq> A\<close> by (auto simp: Un_absorb2)
1596 qed
1598 lemma measure_UNION:
1599   assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
1600   assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
1601   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
1602 proof -
1603   have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))"
1604     unfolding suminf_emeasure[OF measurable, symmetric] by (simp add: summable_sums)
1605   moreover
1606   { fix i
1607     have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"
1608       using measurable by (auto intro!: emeasure_mono)
1609     then have "emeasure M (A i) = ennreal ((measure M (A i)))"
1610       using finite by (intro emeasure_eq_ennreal_measure) (auto simp: top_unique) }
1611   ultimately show ?thesis using finite
1612     by (subst (asm) (2) emeasure_eq_ennreal_measure) simp_all
1613 qed
1616   assumes measurable: "A \<in> sets M" "B \<in> sets M"
1617   and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
1618   shows "measure M (A \<union> B) \<le> measure M A + measure M B"
1619 proof -
1620   have "emeasure M (A \<union> B) \<noteq> \<infinity>"
1621     using emeasure_subadditive[OF measurable] fin by (auto simp: top_unique)
1622   then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
1624     apply simp
1625     apply (subst (asm) (2 3 4) emeasure_eq_ennreal_measure)
1626     apply (auto simp flip: ennreal_plus)
1627     done
1628 qed
1631   assumes A: "finite I" "A`I \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
1632   shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
1633 proof -
1634   { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
1636     also have "\<dots> < \<infinity>"
1637       using fin by (simp add: less_top A)
1638     finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> top" by simp }
1639   note * = this
1640   show ?thesis
1642     unfolding emeasure_eq_ennreal_measure[OF *]
1643     by (simp_all add: sum_nonneg emeasure_eq_ennreal_measure)
1644 qed
1647   assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"
1648   shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
1649 proof -
1650   from fin have **: "\<And>i. emeasure M (A i) \<noteq> top"
1651     using ennreal_suminf_lessD[of "\<lambda>i. emeasure M (A i)"] by (simp add: less_top)
1652   { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"
1654     also have "\<dots> < \<infinity>"
1655       using fin by (simp add: less_top)
1656     finally have "emeasure M (\<Union>i. A i) \<noteq> top" by simp }
1657   then have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"
1658     by (rule emeasure_eq_ennreal_measure[symmetric])
1659   also have "\<dots> \<le> (\<Sum>i. emeasure M (A i))"
1661   also have "\<dots> = ennreal (\<Sum>i. measure M (A i))"
1662     using fin unfolding emeasure_eq_ennreal_measure[OF **]
1663     by (subst suminf_ennreal) (auto simp: **)
1664   finally show ?thesis
1665     apply (rule ennreal_le_iff[THEN iffD1, rotated])
1666     apply (intro suminf_nonneg allI measure_nonneg summable_suminf_not_top)
1667     using fin
1668     apply (simp add: emeasure_eq_ennreal_measure[OF **])
1669     done
1670 qed
1672 lemma measure_Un_null_set: "A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> measure M (A \<union> B) = measure M A"
1673   by (simp add: measure_def emeasure_Un_null_set)
1675 lemma measure_Diff_null_set: "A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> measure M (A - B) = measure M A"
1676   by (simp add: measure_def emeasure_Diff_null_set)
1678 lemma measure_eq_sum_singleton:
1679   "finite S \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M) \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>) \<Longrightarrow>
1680     measure M S = (\<Sum>x\<in>S. measure M {x})"
1681   using emeasure_eq_sum_singleton[of S M]
1682   by (intro measure_eq_emeasure_eq_ennreal) (auto simp: sum_nonneg emeasure_eq_ennreal_measure)
1684 lemma Lim_measure_incseq:
1685   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
1686   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"
1687 proof (rule tendsto_ennrealD)
1688   have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"
1689     using fin by (auto simp: emeasure_eq_ennreal_measure)
1690   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i
1691     using assms emeasure_mono[of "A _" "\<Union>i. A i" M]
1692     by (intro emeasure_eq_ennreal_measure[symmetric]) (auto simp: less_top UN_upper intro: le_less_trans)
1693   ultimately show "(\<lambda>x. ennreal (measure M (A x))) \<longlonglongrightarrow> ennreal (measure M (\<Union>i. A i))"
1694     using A by (auto intro!: Lim_emeasure_incseq)
1695 qed auto
1697 lemma Lim_measure_decseq:
1698   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
1699   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"
1700 proof (rule tendsto_ennrealD)
1701   have "ennreal (measure M (\<Inter>i. A i)) = emeasure M (\<Inter>i. A i)"
1702     using fin[of 0] A emeasure_mono[of "\<Inter>i. A i" "A 0" M]
1703     by (auto intro!: emeasure_eq_ennreal_measure[symmetric] simp: INT_lower less_top intro: le_less_trans)
1704   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i
1705     using A fin[of i] by (intro emeasure_eq_ennreal_measure[symmetric]) auto
1706   ultimately show "(\<lambda>x. ennreal (measure M (A x))) \<longlonglongrightarrow> ennreal (measure M (\<Inter>i. A i))"
1707     using fin A by (auto intro!: Lim_emeasure_decseq)
1708 qed auto
1710 subsection \<open>Set of measurable sets with finite measure\<close>
1712 definition%important fmeasurable :: "'a measure \<Rightarrow> 'a set set" where
1713 "fmeasurable M = {A\<in>sets M. emeasure M A < \<infinity>}"
1715 lemma fmeasurableD[dest, measurable_dest]: "A \<in> fmeasurable M \<Longrightarrow> A \<in> sets M"
1716   by (auto simp: fmeasurable_def)
1718 lemma fmeasurableD2: "A \<in> fmeasurable M \<Longrightarrow> emeasure M A \<noteq> top"
1719   by (auto simp: fmeasurable_def)
1721 lemma fmeasurableI: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> A \<in> fmeasurable M"
1722   by (auto simp: fmeasurable_def)
1724 lemma fmeasurableI_null_sets: "A \<in> null_sets M \<Longrightarrow> A \<in> fmeasurable M"
1725   by (auto simp: fmeasurable_def)
1727 lemma fmeasurableI2: "A \<in> fmeasurable M \<Longrightarrow> B \<subseteq> A \<Longrightarrow> B \<in> sets M \<Longrightarrow> B \<in> fmeasurable M"
1728   using emeasure_mono[of B A M] by (auto simp: fmeasurable_def)
1730 lemma measure_mono_fmeasurable:
1731   "A \<subseteq> B \<Longrightarrow> A \<in> sets M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow> measure M A \<le> measure M B"
1732   by (auto simp: measure_def fmeasurable_def intro!: emeasure_mono enn2real_mono)
1734 lemma emeasure_eq_measure2: "A \<in> fmeasurable M \<Longrightarrow> emeasure M A = measure M A"
1735   by (simp add: emeasure_eq_ennreal_measure fmeasurable_def less_top)
1737 interpretation fmeasurable: ring_of_sets "space M" "fmeasurable M"
1738 proof (rule ring_of_setsI)
1739   show "fmeasurable M \<subseteq> Pow (space M)" "{} \<in> fmeasurable M"
1740     by (auto simp: fmeasurable_def dest: sets.sets_into_space)
1741   fix a b assume *: "a \<in> fmeasurable M" "b \<in> fmeasurable M"
1742   then have "emeasure M (a \<union> b) \<le> emeasure M a + emeasure M b"
1744   also have "\<dots> < top"
1745     using * by (auto simp: fmeasurable_def)
1746   finally show  "a \<union> b \<in> fmeasurable M"
1747     using * by (auto intro: fmeasurableI)
1748   show "a - b \<in> fmeasurable M"
1749     using emeasure_mono[of "a - b" a M] * by (auto simp: fmeasurable_def)
1750 qed
1752 subsection%unimportant\<open>Measurable sets formed by unions and intersections\<close>
1754 lemma fmeasurable_Diff: "A \<in> fmeasurable M \<Longrightarrow> B \<in> sets M \<Longrightarrow> A - B \<in> fmeasurable M"
1755   using fmeasurableI2[of A M "A - B"] by auto
1757 lemma fmeasurable_Int_fmeasurable:
1758    "\<lbrakk>S \<in> fmeasurable M; T \<in> sets M\<rbrakk> \<Longrightarrow> (S \<inter> T) \<in> fmeasurable M"
1759   by (meson fmeasurableD fmeasurableI2 inf_le1 sets.Int)
1761 lemma fmeasurable_UN:
1762   assumes "countable I" "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> A" "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" "A \<in> fmeasurable M"
1763   shows "(\<Union>i\<in>I. F i) \<in> fmeasurable M"
1764 proof (rule fmeasurableI2)
1765   show "A \<in> fmeasurable M" "(\<Union>i\<in>I. F i) \<subseteq> A" using assms by auto
1766   show "(\<Union>i\<in>I. F i) \<in> sets M"
1767     using assms by (intro sets.countable_UN') auto
1768 qed
1770 lemma fmeasurable_INT:
1771   assumes "countable I" "i \<in> I" "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" "F i \<in> fmeasurable M"
1772   shows "(\<Inter>i\<in>I. F i) \<in> fmeasurable M"
1773 proof (rule fmeasurableI2)
1774   show "F i \<in> fmeasurable M" "(\<Inter>i\<in>I. F i) \<subseteq> F i"
1775     using assms by auto
1776   show "(\<Inter>i\<in>I. F i) \<in> sets M"
1777     using assms by (intro sets.countable_INT') auto
1778 qed
1780 lemma measurable_measure_Diff:
1781   assumes "A \<in> fmeasurable M" "B \<in> sets M" "B \<subseteq> A"
1782   shows "measure M (A - B) = measure M A - measure M B"
1783   by (simp add: assms fmeasurableD fmeasurableD2 measure_Diff)
1785 lemma measurable_Un_null_set:
1786   assumes "B \<in> null_sets M"
1787   shows "(A \<union> B \<in> fmeasurable M \<and> A \<in> sets M) \<longleftrightarrow> A \<in> fmeasurable M"
1788   using assms  by (fastforce simp add: fmeasurable.Un fmeasurableI_null_sets intro: fmeasurableI2)
1790 lemma measurable_Diff_null_set:
1791   assumes "B \<in> null_sets M"
1792   shows "(A - B) \<in> fmeasurable M \<and> A \<in> sets M \<longleftrightarrow> A \<in> fmeasurable M"
1793   using assms
1794   by (metis Un_Diff_cancel2 fmeasurable.Diff fmeasurableD fmeasurableI_null_sets measurable_Un_null_set)
1796 lemma fmeasurable_Diff_D:
1797     assumes m: "T - S \<in> fmeasurable M" "S \<in> fmeasurable M" and sub: "S \<subseteq> T"
1798     shows "T \<in> fmeasurable M"
1799 proof -
1800   have "T = S \<union> (T - S)"
1801     using assms by blast
1802   then show ?thesis
1803     by (metis m fmeasurable.Un)
1804 qed
1806 lemma measure_Un2:
1807   "A \<in> fmeasurable M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"
1808   using measure_Union[of M A "B - A"] by (auto simp: fmeasurableD2 fmeasurable.Diff)
1810 lemma measure_Un3:
1811   assumes "A \<in> fmeasurable M" "B \<in> fmeasurable M"
1812   shows "measure M (A \<union> B) = measure M A + measure M B - measure M (A \<inter> B)"
1813 proof -
1814   have "measure M (A \<union> B) = measure M A + measure M (B - A)"
1815     using assms by (rule measure_Un2)
1816   also have "B - A = B - (A \<inter> B)"
1817     by auto
1818   also have "measure M (B - (A \<inter> B)) = measure M B - measure M (A \<inter> B)"
1819     using assms by (intro measure_Diff) (auto simp: fmeasurable_def)
1820   finally show ?thesis
1821     by simp
1822 qed
1824 lemma measure_Un_AE:
1825   "AE x in M. x \<notin> A \<or> x \<notin> B \<Longrightarrow> A \<in> fmeasurable M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow>
1826   measure M (A \<union> B) = measure M A + measure M B"
1827   by (subst measure_Un2) (auto intro!: measure_eq_AE)
1829 lemma measure_UNION_AE:
1830   assumes I: "finite I"
1831   shows "(\<And>i. i \<in> I \<Longrightarrow> F i \<in> fmeasurable M) \<Longrightarrow> pairwise (\<lambda>i j. AE x in M. x \<notin> F i \<or> x \<notin> F j) I \<Longrightarrow>
1832     measure M (\<Union>i\<in>I. F i) = (\<Sum>i\<in>I. measure M (F i))"
1833   unfolding AE_pairwise[OF countable_finite, OF I]
1834   using I
1835 proof (induction I rule: finite_induct)
1836   case (insert x I)
1837   have "measure M (F x \<union> \<Union>(F ` I)) = measure M (F x) + measure M (\<Union>(F ` I))"
1838     by (rule measure_Un_AE) (use insert in \<open>auto simp: pairwise_insert\<close>)
1839     with insert show ?case
1840       by (simp add: pairwise_insert )
1841 qed simp
1843 lemma measure_UNION':
1844   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i \<in> fmeasurable M) \<Longrightarrow> pairwise (\<lambda>i j. disjnt (F i) (F j)) I \<Longrightarrow>
1845     measure M (\<Union>i\<in>I. F i) = (\<Sum>i\<in>I. measure M (F i))"
1846   by (intro measure_UNION_AE) (auto simp: disjnt_def elim!: pairwise_mono intro!: always_eventually)
1848 lemma measure_Union_AE:
1849   "finite F \<Longrightarrow> (\<And>S. S \<in> F \<Longrightarrow> S \<in> fmeasurable M) \<Longrightarrow> pairwise (\<lambda>S T. AE x in M. x \<notin> S \<or> x \<notin> T) F \<Longrightarrow>
1850     measure M (\<Union>F) = (\<Sum>S\<in>F. measure M S)"
1851   using measure_UNION_AE[of F "\<lambda>x. x" M] by simp
1853 lemma measure_Union':
1854   "finite F \<Longrightarrow> (\<And>S. S \<in> F \<Longrightarrow> S \<in> fmeasurable M) \<Longrightarrow> pairwise disjnt F \<Longrightarrow> measure M (\<Union>F) = (\<Sum>S\<in>F. measure M S)"
1855   using measure_UNION'[of F "\<lambda>x. x" M] by simp
1857 lemma measure_Un_le:
1858   assumes "A \<in> sets M" "B \<in> sets M" shows "measure M (A \<union> B) \<le> measure M A + measure M B"
1859 proof cases
1860   assume "A \<in> fmeasurable M \<and> B \<in> fmeasurable M"
1861   with measure_subadditive[of A M B] assms show ?thesis
1862     by (auto simp: fmeasurableD2)
1863 next
1864   assume "\<not> (A \<in> fmeasurable M \<and> B \<in> fmeasurable M)"
1865   then have "A \<union> B \<notin> fmeasurable M"
1866     using fmeasurableI2[of "A \<union> B" M A] fmeasurableI2[of "A \<union> B" M B] assms by auto
1867   with assms show ?thesis
1868     by (auto simp: fmeasurable_def measure_def less_top[symmetric])
1869 qed
1871 lemma measure_UNION_le:
1872   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M) \<Longrightarrow> measure M (\<Union>i\<in>I. F i) \<le> (\<Sum>i\<in>I. measure M (F i))"
1873 proof (induction I rule: finite_induct)
1874   case (insert i I)
1875   then have "measure M (\<Union>i\<in>insert i I. F i) \<le> measure M (F i) + measure M (\<Union>i\<in>I. F i)"
1876     by (auto intro!: measure_Un_le)
1877   also have "measure M (\<Union>i\<in>I. F i) \<le> (\<Sum>i\<in>I. measure M (F i))"
1878     using insert by auto
1879   finally show ?case
1880     using insert by simp
1881 qed simp
1883 lemma measure_Union_le:
1884   "finite F \<Longrightarrow> (\<And>S. S \<in> F \<Longrightarrow> S \<in> sets M) \<Longrightarrow> measure M (\<Union>F) \<le> (\<Sum>S\<in>F. measure M S)"
1885   using measure_UNION_le[of F "\<lambda>x. x" M] by simp
1887 text\<open>Version for indexed union over a countable set\<close>
1888 lemma
1889   assumes "countable I" and I: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M"
1890     and bound: "\<And>I'. I' \<subseteq> I \<Longrightarrow> finite I' \<Longrightarrow> measure M (\<Union>i\<in>I'. A i) \<le> B"
1891   shows fmeasurable_UN_bound: "(\<Union>i\<in>I. A i) \<in> fmeasurable M" (is ?fm)
1892     and measure_UN_bound: "measure M (\<Union>i\<in>I. A i) \<le> B" (is ?m)
1893 proof -
1894   have "B \<ge> 0"
1895     using bound by force
1896   have "?fm \<and> ?m"
1897   proof cases
1898     assume "I = {}"
1899     with \<open>B \<ge> 0\<close> show ?thesis
1900       by simp
1901   next
1902     assume "I \<noteq> {}"
1903     have "(\<Union>i\<in>I. A i) = (\<Union>i. (\<Union>n\<le>i. A (from_nat_into I n)))"
1904       by (subst range_from_nat_into[symmetric, OF \<open>I \<noteq> {}\<close> \<open>countable I\<close>]) auto
1905     then have "emeasure M (\<Union>i\<in>I. A i) = emeasure M (\<Union>i. (\<Union>n\<le>i. A (from_nat_into I n)))" by simp
1906     also have "\<dots> = (SUP i. emeasure M (\<Union>n\<le>i. A (from_nat_into I n)))"
1907       using I \<open>I \<noteq> {}\<close>[THEN from_nat_into] by (intro SUP_emeasure_incseq[symmetric]) (fastforce simp: incseq_Suc_iff)+
1908     also have "\<dots> \<le> B"
1909     proof (intro SUP_least)
1910       fix i :: nat
1911       have "emeasure M (\<Union>n\<le>i. A (from_nat_into I n)) = measure M (\<Union>n\<le>i. A (from_nat_into I n))"
1912         using I \<open>I \<noteq> {}\<close>[THEN from_nat_into] by (intro emeasure_eq_measure2 fmeasurable.finite_UN) auto
1913       also have "\<dots> = measure M (\<Union>n\<in>from_nat_into I ` {..i}. A n)"
1914         by simp
1915       also have "\<dots> \<le> B"
1916         by (intro ennreal_leI bound) (auto intro:  from_nat_into[OF \<open>I \<noteq> {}\<close>])
1917       finally show "emeasure M (\<Union>n\<le>i. A (from_nat_into I n)) \<le> ennreal B" .
1918     qed
1919     finally have *: "emeasure M (\<Union>i\<in>I. A i) \<le> B" .
1920     then have ?fm
1921       using I \<open>countable I\<close> by (intro fmeasurableI conjI) (auto simp: less_top[symmetric] top_unique)
1922     with * \<open>0\<le>B\<close> show ?thesis
1924   qed
1925   then show ?fm ?m by auto
1926 qed
1928 text\<open>Version for big union of a countable set\<close>
1929 lemma
1930   assumes "countable \<D>"
1931     and meas: "\<And>D. D \<in> \<D> \<Longrightarrow> D \<in> fmeasurable M"
1932     and bound:  "\<And>\<E>. \<lbrakk>\<E> \<subseteq> \<D>; finite \<E>\<rbrakk> \<Longrightarrow> measure M (\<Union>\<E>) \<le> B"
1933  shows fmeasurable_Union_bound: "\<Union>\<D> \<in> fmeasurable M"  (is ?fm)
1934     and measure_Union_bound: "measure M (\<Union>\<D>) \<le> B"     (is ?m)
1935 proof -
1936   have "B \<ge> 0"
1937     using bound by force
1938   have "?fm \<and> ?m"
1939   proof (cases "\<D> = {}")
1940     case True
1941     with \<open>B \<ge> 0\<close> show ?thesis
1942       by auto
1943   next
1944     case False
1945     then obtain D :: "nat \<Rightarrow> 'a set" where D: "\<D> = range D"
1946       using \<open>countable \<D>\<close> uncountable_def by force
1947       have 1: "\<And>i. D i \<in> fmeasurable M"
1948         by (simp add: D meas)
1949       have 2: "\<And>I'. finite I' \<Longrightarrow> measure M (\<Union>x\<in>I'. D x) \<le> B"
1950         by (simp add: D bound image_subset_iff)
1951       show ?thesis
1952         unfolding D
1953         by (intro conjI fmeasurable_UN_bound [OF _ 1 2] measure_UN_bound [OF _ 1 2]) auto
1954     qed
1955     then show ?fm ?m by auto
1956 qed
1958 text\<open>Version for indexed union over the type of naturals\<close>
1959 lemma
1960   fixes S :: "nat \<Rightarrow> 'a set"
1961   assumes S: "\<And>i. S i \<in> fmeasurable M" and B: "\<And>n. measure M (\<Union>i\<le>n. S i) \<le> B"
1962   shows fmeasurable_countable_Union: "(\<Union>i. S i) \<in> fmeasurable M"
1963     and measure_countable_Union_le: "measure M (\<Union>i. S i) \<le> B"
1964 proof -
1965   have mB: "measure M (\<Union>i\<in>I. S i) \<le> B" if "finite I" for I
1966   proof -
1967     have "(\<Union>i\<in>I. S i) \<subseteq> (\<Union>i\<le>Max I. S i)"
1968       using Max_ge that by force
1969     then have "measure M (\<Union>i\<in>I. S i) \<le> measure M (\<Union>i \<le> Max I. S i)"
1970       by (rule measure_mono_fmeasurable) (use S in \<open>blast+\<close>)
1971     then show ?thesis
1972       using B order_trans by blast
1973   qed
1974   show "(\<Union>i. S i) \<in> fmeasurable M"
1975     by (auto intro: fmeasurable_UN_bound [OF _ S mB])
1976   show "measure M (\<Union>n. S n) \<le> B"
1977     by (auto intro: measure_UN_bound [OF _ S mB])
1978 qed
1980 lemma measure_diff_le_measure_setdiff:
1981   assumes "S \<in> fmeasurable M" "T \<in> fmeasurable M"
1982   shows "measure M S - measure M T \<le> measure M (S - T)"
1983 proof -
1984   have "measure M S \<le> measure M ((S - T) \<union> T)"
1985     by (simp add: assms fmeasurable.Un fmeasurableD measure_mono_fmeasurable)
1986   also have "\<dots> \<le> measure M (S - T) + measure M T"
1987     using assms by (blast intro: measure_Un_le)
1988   finally show ?thesis
1990 qed
1992 lemma suminf_exist_split2:
1993   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
1994   assumes "summable f"
1995   shows "(\<lambda>n. (\<Sum>k. f(k+n))) \<longlonglongrightarrow> 0"
1996 by (subst lim_sequentially, auto simp add: dist_norm suminf_exist_split[OF _ assms])
1998 lemma emeasure_union_summable:
1999   assumes [measurable]: "\<And>n. A n \<in> sets M"
2000     and "\<And>n. emeasure M (A n) < \<infinity>" "summable (\<lambda>n. measure M (A n))"
2001   shows "emeasure M (\<Union>n. A n) < \<infinity>" "emeasure M (\<Union>n. A n) \<le> (\<Sum>n. measure M (A n))"
2002 proof -
2003   define B where "B = (\<lambda>N. (\<Union>n\<in>{..<N}. A n))"
2004   have [measurable]: "B N \<in> sets M" for N unfolding B_def by auto
2005   have "(\<lambda>N. emeasure M (B N)) \<longlonglongrightarrow> emeasure M (\<Union>N. B N)"
2006     apply (rule Lim_emeasure_incseq) unfolding B_def by (auto simp add: SUP_subset_mono incseq_def)
2007   moreover have "emeasure M (B N) \<le> ennreal (\<Sum>n. measure M (A n))" for N
2008   proof -
2009     have *: "(\<Sum>n\<in>{..<N}. measure M (A n)) \<le> (\<Sum>n. measure M (A n))"
2010       using assms(3) measure_nonneg sum_le_suminf by blast
2012     have "emeasure M (B N) \<le> (\<Sum>n\<in>{..<N}. emeasure M (A n))"
2013       unfolding B_def by (rule emeasure_subadditive_finite, auto)
2014     also have "... = (\<Sum>n\<in>{..<N}. ennreal(measure M (A n)))"
2015       using assms(2) by (simp add: emeasure_eq_ennreal_measure less_top)
2016     also have "... = ennreal (\<Sum>n\<in>{..<N}. measure M (A n))"
2017       by auto
2018     also have "... \<le> ennreal (\<Sum>n. measure M (A n))"
2019       using * by (auto simp: ennreal_leI)
2020     finally show ?thesis by simp
2021   qed
2022   ultimately have "emeasure M (\<Union>N. B N) \<le> ennreal (\<Sum>n. measure M (A n))"
2024   then show "emeasure M (\<Union>n. A n) \<le> (\<Sum>n. measure M (A n))"
2025     unfolding B_def by (metis UN_UN_flatten UN_lessThan_UNIV)
2026   then show "emeasure M (\<Union>n. A n) < \<infinity>"
2027     by (auto simp: less_top[symmetric] top_unique)
2028 qed
2030 lemma borel_cantelli_limsup1:
2031   assumes [measurable]: "\<And>n. A n \<in> sets M"
2032     and "\<And>n. emeasure M (A n) < \<infinity>" "summable (\<lambda>n. measure M (A n))"
2033   shows "limsup A \<in> null_sets M"
2034 proof -
2035   have "emeasure M (limsup A) \<le> 0"
2036   proof (rule LIMSEQ_le_const)
2037     have "(\<lambda>n. (\<Sum>k. measure M (A (k+n)))) \<longlonglongrightarrow> 0" by (rule suminf_exist_split2[OF assms(3)])
2038     then show "(\<lambda>n. ennreal (\<Sum>k. measure M (A (k+n)))) \<longlonglongrightarrow> 0"
2039       unfolding ennreal_0[symmetric] by (intro tendsto_ennrealI)
2040     have "emeasure M (limsup A) \<le> (\<Sum>k. measure M (A (k+n)))" for n
2041     proof -
2042       have I: "(\<Union>k\<in>{n..}. A k) = (\<Union>k. A (k+n))" by (auto, metis le_add_diff_inverse2, fastforce)
2043       have "emeasure M (limsup A) \<le> emeasure M (\<Union>k\<in>{n..}. A k)"
2044         by (rule emeasure_mono, auto simp add: limsup_INF_SUP)
2045       also have "... = emeasure M (\<Union>k. A (k+n))"
2046         using I by auto
2047       also have "... \<le> (\<Sum>k. measure M (A (k+n)))"
2048         apply (rule emeasure_union_summable)
2049         using assms summable_ignore_initial_segment[OF assms(3), of n] by auto
2050       finally show ?thesis by simp
2051     qed
2052     then show "\<exists>N. \<forall>n\<ge>N. emeasure M (limsup A) \<le> (\<Sum>k. measure M (A (k+n)))"
2053       by auto
2054   qed
2055   then show ?thesis using assms(1) measurable_limsup by auto
2056 qed
2058 lemma borel_cantelli_AE1:
2059   assumes [measurable]: "\<And>n. A n \<in> sets M"
2060     and "\<And>n. emeasure M (A n) < \<infinity>" "summable (\<lambda>n. measure M (A n))"
2061   shows "AE x in M. eventually (\<lambda>n. x \<in> space M - A n) sequentially"
2062 proof -
2063   have "AE x in M. x \<notin> limsup A"
2064     using borel_cantelli_limsup1[OF assms] unfolding eventually_ae_filter by auto
2065   moreover
2066   {
2067     fix x assume "x \<notin> limsup A"
2068     then obtain N where "x \<notin> (\<Union>n\<in>{N..}. A n)" unfolding limsup_INF_SUP by blast
2069     then have "eventually (\<lambda>n. x \<notin> A n) sequentially" using eventually_sequentially by auto
2070   }
2071   ultimately show ?thesis by auto
2072 qed
2074 subsection \<open>Measure spaces with \<^term>\<open>emeasure M (space M) < \<infinity>\<close>\<close>
2076 locale%important finite_measure = sigma_finite_measure M for M +
2077   assumes finite_emeasure_space: "emeasure M (space M) \<noteq> top"
2079 lemma finite_measureI[Pure.intro!]:
2080   "emeasure M (space M) \<noteq> \<infinity> \<Longrightarrow> finite_measure M"
2081   proof qed (auto intro!: exI[of _ "{space M}"])
2083 lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> top"
2084   using finite_emeasure_space emeasure_space[of M A] by (auto simp: top_unique)
2086 lemma (in finite_measure) fmeasurable_eq_sets: "fmeasurable M = sets M"
2087   by (auto simp: fmeasurable_def less_top[symmetric])
2089 lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ennreal (measure M A)"
2090   by (intro emeasure_eq_ennreal_measure) simp
2092 lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ennreal r"
2093   using emeasure_finite[of A] by (cases "emeasure M A" rule: ennreal_cases) auto
2095 lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"
2096   using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)
2098 lemma (in finite_measure) finite_measure_Diff:
2099   assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
2100   shows "measure M (A - B) = measure M A - measure M B"
2101   using measure_Diff[OF _ assms] by simp
2103 lemma (in finite_measure) finite_measure_Union:
2104   assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
2105   shows "measure M (A \<union> B) = measure M A + measure M B"
2106   using measure_Union[OF _ _ assms] by simp
2108 lemma (in finite_measure) finite_measure_finite_Union:
2109   assumes measurable: "finite S" "A`S \<subseteq> sets M" "disjoint_family_on A S"
2110   shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
2111   using measure_finite_Union[OF assms] by simp
2113 lemma (in finite_measure) finite_measure_UNION:
2114   assumes A: "range A \<subseteq> sets M" "disjoint_family A"
2115   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
2116   using measure_UNION[OF A] by simp
2118 lemma (in finite_measure) finite_measure_mono:
2119   assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"
2120   using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)
2123   assumes m: "A \<in> sets M" "B \<in> sets M"
2124   shows "measure M (A \<union> B) \<le> measure M A + measure M B"
2125   using measure_subadditive[OF m] by simp
2128   assumes "finite I" "A`I \<subseteq> sets M" shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
2129   using measure_subadditive_finite[OF assms] by simp
2132   "range A \<subseteq> sets M \<Longrightarrow> summable (\<lambda>i. measure M (A i)) \<Longrightarrow> measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
2136 lemma (in finite_measure) finite_measure_eq_sum_singleton:
2137   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
2138   shows "measure M S = (\<Sum>x\<in>S. measure M {x})"
2139   using measure_eq_sum_singleton[OF assms] by simp
2141 lemma (in finite_measure) finite_Lim_measure_incseq:
2142   assumes A: "range A \<subseteq> sets M" "incseq A"
2143   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"
2144   using Lim_measure_incseq[OF A] by simp
2146 lemma (in finite_measure) finite_Lim_measure_decseq:
2147   assumes A: "range A \<subseteq> sets M" "decseq A"
2148   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"
2149   using Lim_measure_decseq[OF A] by simp
2151 lemma (in finite_measure) finite_measure_compl:
2152   assumes S: "S \<in> sets M"
2153   shows "measure M (space M - S) = measure M (space M) - measure M S"
2154   using measure_Diff[OF _ sets.top S sets.sets_into_space] S by simp
2156 lemma (in finite_measure) finite_measure_mono_AE:
2157   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"
2158   shows "measure M A \<le> measure M B"
2159   using assms emeasure_mono_AE[OF imp B]
2162 lemma (in finite_measure) finite_measure_eq_AE:
2163   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
2164   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
2165   shows "measure M A = measure M B"
2166   using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)
2168 lemma (in finite_measure) measure_increasing: "increasing M (measure M)"
2169   by (auto intro!: finite_measure_mono simp: increasing_def)
2171 lemma (in finite_measure) measure_zero_union:
2172   assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"
2173   shows "measure M (s \<union> t) = measure M s"
2174 using assms
2175 proof -
2176   have "measure M (s \<union> t) \<le> measure M s"
2177     using finite_measure_subadditive[of s t] assms by auto
2178   moreover have "measure M (s \<union> t) \<ge> measure M s"
2179     using assms by (blast intro: finite_measure_mono)
2180   ultimately show ?thesis by simp
2181 qed
2183 lemma (in finite_measure) measure_eq_compl:
2184   assumes "s \<in> sets M" "t \<in> sets M"
2185   assumes "measure M (space M - s) = measure M (space M - t)"
2186   shows "measure M s = measure M t"
2187   using assms finite_measure_compl by auto
2189 lemma (in finite_measure) measure_eq_bigunion_image:
2190   assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M"
2191   assumes "disjoint_family f" "disjoint_family g"
2192   assumes "\<And> n :: nat. measure M (f n) = measure M (g n)"
2193   shows "measure M (\<Union>i. f i) = measure M (\<Union>i. g i)"
2194 using assms
2195 proof -
2196   have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union>i. f i))"
2197     by (rule finite_measure_UNION[OF assms(1,3)])
2198   have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union>i. g i))"
2199     by (rule finite_measure_UNION[OF assms(2,4)])
2200   show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
2201 qed
2203 lemma (in finite_measure) measure_countably_zero:
2204   assumes "range c \<subseteq> sets M"
2205   assumes "\<And> i. measure M (c i) = 0"
2206   shows "measure M (\<Union>i :: nat. c i) = 0"
2207 proof (rule antisym)
2208   show "measure M (\<Union>i :: nat. c i) \<le> 0"
2210 qed simp
2212 lemma (in finite_measure) measure_space_inter:
2213   assumes events:"s \<in> sets M" "t \<in> sets M"
2214   assumes "measure M t = measure M (space M)"
2215   shows "measure M (s \<inter> t) = measure M s"
2216 proof -
2217   have "measure M ((space M - s) \<union> (space M - t)) = measure M (space M - s)"
2218     using events assms finite_measure_compl[of "t"] by (auto intro!: measure_zero_union)
2219   also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"
2220     by blast
2221   finally show "measure M (s \<inter> t) = measure M s"
2222     using events by (auto intro!: measure_eq_compl[of "s \<inter> t" s])
2223 qed
2225 lemma (in finite_measure) measure_equiprobable_finite_unions:
2226   assumes s: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> sets M"
2227   assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}"
2228   shows "measure M s = real (card s) * measure M {SOME x. x \<in> s}"
2229 proof cases
2230   assume "s \<noteq> {}"
2231   then have "\<exists> x. x \<in> s" by blast
2232   from someI_ex[OF this] assms
2233   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast
2234   have "measure M s = (\<Sum> x \<in> s. measure M {x})"
2235     using finite_measure_eq_sum_singleton[OF s] by simp
2236   also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto
2237   also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"
2238     using sum_constant assms by simp
2239   finally show ?thesis by simp
2240 qed simp
2242 lemma (in finite_measure) measure_real_sum_image_fn:
2243   assumes "e \<in> sets M"
2244   assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> sets M"
2245   assumes "finite s"
2246   assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
2247   assumes upper: "space M \<subseteq> (\<Union>i \<in> s. f i)"
2248   shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
2249 proof -
2250   have "e \<subseteq> (\<Union>i\<in>s. f i)"
2251     using \<open>e \<in> sets M\<close> sets.sets_into_space upper by blast
2252   then have e: "e = (\<Union>i \<in> s. e \<inter> f i)"
2253     by auto
2254   hence "measure M e = measure M (\<Union>i \<in> s. e \<inter> f i)" by simp
2255   also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
2256   proof (rule finite_measure_finite_Union)
2257     show "finite s" by fact
2258     show "(\<lambda>i. e \<inter> f i)`s \<subseteq> sets M" using assms(2) by auto
2259     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
2260       using disjoint by (auto simp: disjoint_family_on_def)
2261   qed
2262   finally show ?thesis .
2263 qed
2265 lemma (in finite_measure) measure_exclude:
2266   assumes "A \<in> sets M" "B \<in> sets M"
2267   assumes "measure M A = measure M (space M)" "A \<inter> B = {}"
2268   shows "measure M B = 0"
2269   using measure_space_inter[of B A] assms by (auto simp: ac_simps)
2270 lemma (in finite_measure) finite_measure_distr:
2271   assumes f: "f \<in> measurable M M'"
2272   shows "finite_measure (distr M M' f)"
2273 proof (rule finite_measureI)
2274   have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
2275   with f show "emeasure (distr M M' f) (space (distr M M' f)) \<noteq> \<infinity>" by (auto simp: emeasure_distr)
2276 qed
2278 lemma emeasure_gfp[consumes 1, case_names cont measurable]:
2279   assumes sets[simp]: "\<And>s. sets (M s) = sets N"
2280   assumes "\<And>s. finite_measure (M s)"
2281   assumes cont: "inf_continuous F" "inf_continuous f"
2282   assumes meas: "\<And>P. Measurable.pred N P \<Longrightarrow> Measurable.pred N (F P)"
2283   assumes iter: "\<And>P s. Measurable.pred N P \<Longrightarrow> emeasure (M s) {x\<in>space N. F P x} = f (\<lambda>s. emeasure (M s) {x\<in>space N. P x}) s"
2284   assumes bound: "\<And>P. f P \<le> f (\<lambda>s. emeasure (M s) (space (M s)))"
2285   shows "emeasure (M s) {x\<in>space N. gfp F x} = gfp f s"
2286 proof (subst gfp_transfer_bounded[where \<alpha>="\<lambda>F s. emeasure (M s) {x\<in>space N. F x}" and g=f and f=F and
2287     P="Measurable.pred N", symmetric])
2288   interpret finite_measure "M s" for s by fact
2289   fix C assume "decseq C" "\<And>i. Measurable.pred N (C i)"
2290   then show "(\<lambda>s. emeasure (M s) {x \<in> space N. (INF i. C i) x}) = (INF i. (\<lambda>s. emeasure (M s) {x \<in> space N. C i x}))"
2291     unfolding INF_apply[abs_def]
2292     by (subst INF_emeasure_decseq) (auto simp: antimono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])
2293 next
2294   show "f x \<le> (\<lambda>s. emeasure (M s) {x \<in> space N. F top x})" for x
2295     using bound[of x] sets_eq_imp_space_eq[OF sets] by (simp add: iter)
2296 qed (auto simp add: iter le_fun_def INF_apply[abs_def] intro!: meas cont)
2298 subsection%unimportant \<open>Counting space\<close>
2300 lemma strict_monoI_Suc:
2301   assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f"
2302   unfolding strict_mono_def
2303 proof safe
2304   fix n m :: nat assume "n < m" then show "f n < f m"
2305     by (induct m) (auto simp: less_Suc_eq intro: less_trans ord)
2306 qed
2308 lemma emeasure_count_space:
2309   assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else \<infinity>)"
2310     (is "_ = ?M X")
2311   unfolding count_space_def
2312 proof (rule emeasure_measure_of_sigma)
2313   show "X \<in> Pow A" using \<open>X \<subseteq> A\<close> by auto
2314   show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)
2315   show positive: "positive (Pow A) ?M"
2316     by (auto simp: positive_def)
2318     by (auto simp: card_Un_disjoint additive_def)
2320   interpret ring_of_sets A "Pow A"
2321     by (rule ring_of_setsI) auto
2322   show "countably_additive (Pow A) ?M"
2324   proof safe
2325     fix F :: "nat \<Rightarrow> 'a set" assume "incseq F"
2326     show "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"
2327     proof cases
2328       assume "\<exists>i. \<forall>j\<ge>i. F i = F j"
2329       then guess i .. note i = this
2330       { fix j from i \<open>incseq F\<close> have "F j \<subseteq> F i"
2331           by (cases "i \<le> j") (auto simp: incseq_def) }
2332       then have eq: "(\<Union>i. F i) = F i"
2333         by auto
2334       with i show ?thesis
2335         by (auto intro!: Lim_transform_eventually[OF _ tendsto_const] eventually_sequentiallyI[where c=i])
2336     next
2337       assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)"
2338       then obtain f where f: "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis
2339       then have "\<And>i. F i \<subseteq> F (f i)" using \<open>incseq F\<close> by (auto simp: incseq_def)
2340       with f have *: "\<And>i. F i \<subset> F (f i)" by auto
2342       have "incseq (\<lambda>i. ?M (F i))"
2343         using \<open>incseq F\<close> unfolding incseq_def by (auto simp: card_mono dest: finite_subset)
2344       then have "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> (SUP n. ?M (F n))"
2345         by (rule LIMSEQ_SUP)
2347       moreover have "(SUP n. ?M (F n)) = top"
2348       proof (rule ennreal_SUP_eq_top)
2349         fix n :: nat show "\<exists>k::nat\<in>UNIV. of_nat n \<le> ?M (F k)"
2350         proof (induct n)
2351           case (Suc n)
2352           then guess k .. note k = this
2353           moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))"
2354             using \<open>F k \<subset> F (f k)\<close> by (simp add: psubset_card_mono)
2355           moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)"
2356             using \<open>k \<le> f k\<close> \<open>incseq F\<close> by (auto simp: incseq_def dest: finite_subset)
2357           ultimately show ?case
2358             by (auto intro!: exI[of _ "f k"] simp del: of_nat_Suc)
2359         qed auto
2360       qed
2362       moreover
2363       have "inj (\<lambda>n. F ((f ^^ n) 0))"
2364         by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *)
2365       then have 1: "infinite (range (\<lambda>i. F ((f ^^ i) 0)))"
2366         by (rule range_inj_infinite)
2367       have "infinite (Pow (\<Union>i. F i))"
2368         by (rule infinite_super[OF _ 1]) auto
2369       then have "infinite (\<Union>i. F i)"
2370         by auto
2371       ultimately show ?thesis by (simp only:) simp
2373     qed
2374   qed
2375 qed
2377 lemma distr_bij_count_space:
2378   assumes f: "bij_betw f A B"
2379   shows "distr (count_space A) (count_space B) f = count_space B"
2380 proof (rule measure_eqI)
2381   have f': "f \<in> measurable (count_space A) (count_space B)"
2382     using f unfolding Pi_def bij_betw_def by auto
2383   fix X assume "X \<in> sets (distr (count_space A) (count_space B) f)"
2384   then have X: "X \<in> sets (count_space B)" by auto
2385   moreover from X have "f -` X \<inter> A = the_inv_into A f ` X"
2386     using f by (auto simp: bij_betw_def subset_image_iff image_iff the_inv_into_f_f intro: the_inv_into_f_f[symmetric])
2387   moreover have "inj_on (the_inv_into A f) B"
2388     using X f by (auto simp: bij_betw_def inj_on_the_inv_into)
2389   with X have "inj_on (the_inv_into A f) X"
2390     by (auto intro: subset_inj_on)
2391   ultimately show "emeasure (distr (count_space A) (count_space B) f) X = emeasure (count_space B) X"
2392     using f unfolding emeasure_distr[OF f' X]
2393     by (subst (1 2) emeasure_count_space) (auto simp: card_image dest: finite_imageD)
2394 qed simp
2396 lemma emeasure_count_space_finite[simp]:
2397   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = of_nat (card X)"
2398   using emeasure_count_space[of X A] by simp
2400 lemma emeasure_count_space_infinite[simp]:
2401   "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"
2402   using emeasure_count_space[of X A] by simp
2404 lemma measure_count_space: "measure (count_space A) X = (if X \<subseteq> A then of_nat (card X) else 0)"
2405   by (cases "finite X") (auto simp: measure_notin_sets ennreal_of_nat_eq_real_of_nat
2406                                     measure_zero_top measure_eq_emeasure_eq_ennreal)
2408 lemma emeasure_count_space_eq_0:
2409   "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"
2410 proof cases
2411   assume X: "X \<subseteq> A"
2412   then show ?thesis
2413   proof (intro iffI impI)
2414     assume "emeasure (count_space A) X = 0"
2415     with X show "X = {}"
2416       by (subst (asm) emeasure_count_space) (auto split: if_split_asm)
2417   qed simp
2420 lemma null_sets_count_space: "null_sets (count_space A) = { {} }"
2421   unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)
2423 lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
2424   unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)
2426 lemma sigma_finite_measure_count_space_countable:
2427   assumes A: "countable A"
2428   shows "sigma_finite_measure (count_space A)"
2429   proof qed (insert A, auto intro!: exI[of _ "(\<lambda>a. {a}) ` A"])
2431 lemma sigma_finite_measure_count_space:
2432   fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)"
2433   by (rule sigma_finite_measure_count_space_countable) auto
2435 lemma finite_measure_count_space:
2436   assumes [simp]: "finite A"
2437   shows "finite_measure (count_space A)"
2438   by rule simp
2440 lemma sigma_finite_measure_count_space_finite:
2441   assumes A: "finite A" shows "sigma_finite_measure (count_space A)"
2442 proof -
2443   interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)
2444   show "sigma_finite_measure (count_space A)" ..
2445 qed
2447 subsection%unimportant \<open>Measure restricted to space\<close>
2449 lemma emeasure_restrict_space:
2450   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"
2451   shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"
2452 proof (cases "A \<in> sets M")
2453   case True
2454   show ?thesis
2455   proof (rule emeasure_measure_of[OF restrict_space_def])
2456     show "(\<inter>) \<Omega> ` sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"
2457       using \<open>A \<subseteq> \<Omega>\<close> \<open>A \<in> sets M\<close> sets.space_closed by (auto simp: sets_restrict_space)
2458     show "positive (sets (restrict_space M \<Omega>)) (emeasure M)"
2459       by (auto simp: positive_def)
2460     show "countably_additive (sets (restrict_space M \<Omega>)) (emeasure M)"
2462       fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> sets (restrict_space M \<Omega>)" "disjoint_family A"
2463       with assms have "\<And>i. A i \<in> sets M" "\<And>i. A i \<subseteq> space M" "disjoint_family A"
2464         by (fastforce simp: sets_restrict_space_iff[OF assms(1)] image_subset_iff
2465                       dest: sets.sets_into_space)+
2466       then show "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
2467         by (subst suminf_emeasure) (auto simp: disjoint_family_subset)
2468     qed
2469   qed
2470 next
2471   case False
2472   with assms have "A \<notin> sets (restrict_space M \<Omega>)"
2474   with False show ?thesis
2476 qed
2478 lemma measure_restrict_space:
2479   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"
2480   shows "measure (restrict_space M \<Omega>) A = measure M A"
2481   using emeasure_restrict_space[OF assms] by (simp add: measure_def)
2483 lemma AE_restrict_space_iff:
2484   assumes "\<Omega> \<inter> space M \<in> sets M"
2485   shows "(AE x in restrict_space M \<Omega>. P x) \<longleftrightarrow> (AE x in M. x \<in> \<Omega> \<longrightarrow> P x)"
2486 proof -
2487   have ex_cong: "\<And>P Q f. (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> (\<And>x. Q x \<Longrightarrow> P (f x)) \<Longrightarrow> (\<exists>x. P x) \<longleftrightarrow> (\<exists>x. Q x)"
2488     by auto
2489   { fix X assume X: "X \<in> sets M" "emeasure M X = 0"
2490     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) \<le> emeasure M X"
2491       by (intro emeasure_mono) auto
2492     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) = 0"
2493       using X by (auto intro!: antisym) }
2494   with assms show ?thesis
2495     unfolding eventually_ae_filter
2496     by (auto simp add: space_restrict_space null_sets_def sets_restrict_space_iff
2497                        emeasure_restrict_space cong: conj_cong
2498              intro!: ex_cong[where f="\<lambda>X. (\<Omega> \<inter> space M) \<inter> X"])
2499 qed
2501 lemma restrict_restrict_space:
2502   assumes "A \<inter> space M \<in> sets M" "B \<inter> space M \<in> sets M"
2503   shows "restrict_space (restrict_space M A) B = restrict_space M (A \<inter> B)" (is "?l = ?r")
2504 proof (rule measure_eqI[symmetric])
2505   show "sets ?r = sets ?l"
2506     unfolding sets_restrict_space image_comp by (intro image_cong) auto
2507 next
2508   fix X assume "X \<in> sets (restrict_space M (A \<inter> B))"
2509   then obtain Y where "Y \<in> sets M" "X = Y \<inter> A \<inter> B"
2510     by (auto simp: sets_restrict_space)
2511   with assms sets.Int[OF assms] show "emeasure ?r X = emeasure ?l X"
2512     by (subst (1 2) emeasure_restrict_space)
2513        (auto simp: space_restrict_space sets_restrict_space_iff emeasure_restrict_space ac_simps)
2514 qed
2516 lemma restrict_count_space: "restrict_space (count_space B) A = count_space (A \<inter> B)"
2517 proof (rule measure_eqI)
2518   show "sets (restrict_space (count_space B) A) = sets (count_space (A \<inter> B))"
2519     by (subst sets_restrict_space) auto
2520   moreover fix X assume "X \<in> sets (restrict_space (count_space B) A)"
2521   ultimately have "X \<subseteq> A \<inter> B" by auto
2522   then show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space (A \<inter> B)) X"
2523     by (cases "finite X") (auto simp add: emeasure_restrict_space)
2524 qed
2526 lemma sigma_finite_measure_restrict_space:
2527   assumes "sigma_finite_measure M"
2528   and A: "A \<in> sets M"
2529   shows "sigma_finite_measure (restrict_space M A)"
2530 proof -
2531   interpret sigma_finite_measure M by fact
2532   from sigma_finite_countable obtain C
2533     where C: "countable C" "C \<subseteq> sets M" "(\<Union>C) = space M" "\<forall>a\<in>C. emeasure M a \<noteq> \<infinity>"
2534     by blast
2535   let ?C = "(\<inter>) A ` C"
2536   from C have "countable ?C" "?C \<subseteq> sets (restrict_space M A)" "(\<Union>?C) = space (restrict_space M A)"
2537     by(auto simp add: sets_restrict_space space_restrict_space)
2538   moreover {
2539     fix a
2540     assume "a \<in> ?C"
2541     then obtain a' where "a = A \<inter> a'" "a' \<in> C" ..
2542     then have "emeasure (restrict_space M A) a \<le> emeasure M a'"
2543       using A C by(auto simp add: emeasure_restrict_space intro: emeasure_mono)
2544     also have "\<dots> < \<infinity>" using C(4)[rule_format, of a'] \<open>a' \<in> C\<close> by (simp add: less_top)
2545     finally have "emeasure (restrict_space M A) a \<noteq> \<infinity>" by simp }
2546   ultimately show ?thesis
2547     by unfold_locales (rule exI conjI|assumption|blast)+
2548 qed
2550 lemma finite_measure_restrict_space:
2551   assumes "finite_measure M"
2552   and A: "A \<in> sets M"
2553   shows "finite_measure (restrict_space M A)"
2554 proof -
2555   interpret finite_measure M by fact
2556   show ?thesis
2557     by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space)
2558 qed
2560 lemma restrict_distr:
2561   assumes [measurable]: "f \<in> measurable M N"
2562   assumes [simp]: "\<Omega> \<inter> space N \<in> sets N" and restrict: "f \<in> space M \<rightarrow> \<Omega>"
2563   shows "restrict_space (distr M N f) \<Omega> = distr M (restrict_space N \<Omega>) f"
2564   (is "?l = ?r")
2565 proof (rule measure_eqI)
2566   fix A assume "A \<in> sets (restrict_space (distr M N f) \<Omega>)"
2567   with restrict show "emeasure ?l A = emeasure ?r A"
2568     by (subst emeasure_distr)
2569        (auto simp: sets_restrict_space_iff emeasure_restrict_space emeasure_distr
2570              intro!: measurable_restrict_space2)
2573 lemma measure_eqI_restrict_generator:
2574   assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
2575   assumes sets_eq: "sets M = sets N" and \<Omega>: "\<Omega> \<in> sets M"
2576   assumes "sets (restrict_space M \<Omega>) = sigma_sets \<Omega> E"
2577   assumes "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E"
2578   assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>"
2579   assumes A: "countable A" "A \<noteq> {}" "A \<subseteq> E" "\<Union>A = \<Omega>" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
2580   shows "M = N"
2581 proof (rule measure_eqI)
2582   fix X assume X: "X \<in> sets M"
2583   then have "emeasure M X = emeasure (restrict_space M \<Omega>) (X \<inter> \<Omega>)"
2584     using ae \<Omega> by (auto simp add: emeasure_restrict_space intro!: emeasure_eq_AE)
2585   also have "restrict_space M \<Omega> = restrict_space N \<Omega>"
2586   proof (rule measure_eqI_generator_eq)
2587     fix X assume "X \<in> E"
2588     then show "emeasure (restrict_space M \<Omega>) X = emeasure (restrict_space N \<Omega>) X"
2589       using E \<Omega> by (subst (1 2) emeasure_restrict_space) (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq])
2590   next
2591     show "range (from_nat_into A) \<subseteq> E" "(\<Union>i. from_nat_into A i) = \<Omega>"
2592       using A by (auto cong del: SUP_cong_simp)
2593   next
2594     fix i
2595     have "emeasure (restrict_space M \<Omega>) (from_nat_into A i) = emeasure M (from_nat_into A i)"
2596       using A \<Omega> by (subst emeasure_restrict_space)
2597                    (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq] intro: from_nat_into)
2598     with A show "emeasure (restrict_space M \<Omega>) (from_nat_into A i) \<noteq> \<infinity>"
2599       by (auto intro: from_nat_into)
2600   qed fact+
2601   also have "emeasure (restrict_space N \<Omega>) (X \<inter> \<Omega>) = emeasure N X"
2602     using X ae \<Omega> by (auto simp add: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)
2603   finally show "emeasure M X = emeasure N X" .
2604 qed fact
2606 subsection%unimportant \<open>Null measure\<close>
2608 definition null_measure :: "'a measure \<Rightarrow> 'a measure" where
2609 "null_measure M = sigma (space M) (sets M)"
2611 lemma space_null_measure[simp]: "space (null_measure M) = space M"
2614 lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M"
2617 lemma emeasure_null_measure[simp]: "emeasure (null_measure M) X = 0"
2618   by (cases "X \<in> sets M", rule emeasure_measure_of)
2619      (auto simp: positive_def countably_additive_def emeasure_notin_sets null_measure_def
2620            dest: sets.sets_into_space)
2622 lemma measure_null_measure[simp]: "measure (null_measure M) X = 0"
2623   by (intro measure_eq_emeasure_eq_ennreal) auto
2625 lemma null_measure_idem [simp]: "null_measure (null_measure M) = null_measure M"
2626   by(rule measure_eqI) simp_all
2628 subsection \<open>Scaling a measure\<close>
2630 definition%important scale_measure :: "ennreal \<Rightarrow> 'a measure \<Rightarrow> 'a measure" where
2631 "scale_measure r M = measure_of (space M) (sets M) (\<lambda>A. r * emeasure M A)"
2633 lemma space_scale_measure: "space (scale_measure r M) = space M"
2636 lemma sets_scale_measure [simp, measurable_cong]: "sets (scale_measure r M) = sets M"
2639 lemma emeasure_scale_measure [simp]:
2640   "emeasure (scale_measure r M) A = r * emeasure M A"
2641   (is "_ = ?\<mu> A")
2642 proof(cases "A \<in> sets M")
2643   case True
2644   show ?thesis unfolding scale_measure_def
2645   proof(rule emeasure_measure_of_sigma)
2646     show "sigma_algebra (space M) (sets M)" ..
2647     show "positive (sets M) ?\<mu>" by (simp add: positive_def)
2648     show "countably_additive (sets M) ?\<mu>"
2650       fix A :: "nat \<Rightarrow> _"  assume *: "range A \<subseteq> sets M" "disjoint_family A"
2651       have "(\<Sum>i. ?\<mu> (A i)) = r * (\<Sum>i. emeasure M (A i))"
2652         by simp
2653       also have "\<dots> = ?\<mu> (\<Union>i. A i)" using * by(simp add: suminf_emeasure)
2654       finally show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)" .
2655     qed
2656   qed(fact True)
2659 lemma scale_measure_1 [simp]: "scale_measure 1 M = M"
2660   by(rule measure_eqI) simp_all
2662 lemma scale_measure_0[simp]: "scale_measure 0 M = null_measure M"
2663   by(rule measure_eqI) simp_all
2665 lemma measure_scale_measure [simp]: "0 \<le> r \<Longrightarrow> measure (scale_measure r M) A = r * measure M A"
2666   using emeasure_scale_measure[of r M A]
2667     emeasure_eq_ennreal_measure[of M A]
2668     measure_eq_emeasure_eq_ennreal[of _ "scale_measure r M" A]
2669   by (cases "emeasure (scale_measure r M) A = top")
2670      (auto simp del: emeasure_scale_measure
2671            simp: ennreal_top_eq_mult_iff ennreal_mult_eq_top_iff measure_zero_top ennreal_mult[symmetric])
2673 lemma scale_scale_measure [simp]:
2674   "scale_measure r (scale_measure r' M) = scale_measure (r * r') M"
2675   by (rule measure_eqI) (simp_all add: max_def mult.assoc)
2677 lemma scale_null_measure [simp]: "scale_measure r (null_measure M) = null_measure M"
2678   by (rule measure_eqI) simp_all
2681 subsection \<open>Complete lattice structure on measures\<close>
2683 lemma (in finite_measure) finite_measure_Diff':
2684   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A - B) = measure M A - measure M (A \<inter> B)"
2685   using finite_measure_Diff[of A "A \<inter> B"] by (auto simp: Diff_Int)
2687 lemma (in finite_measure) finite_measure_Union':
2688   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"
2689   using finite_measure_Union[of A "B - A"] by auto
2691 lemma finite_unsigned_Hahn_decomposition:
2692   assumes "finite_measure M" "finite_measure N" and [simp]: "sets N = sets M"
2693   shows "\<exists>Y\<in>sets M. (\<forall>X\<in>sets M. X \<subseteq> Y \<longrightarrow> N X \<le> M X) \<and> (\<forall>X\<in>sets M. X \<inter> Y = {} \<longrightarrow> M X \<le> N X)"
2694 proof -
2695   interpret M: finite_measure M by fact
2696   interpret N: finite_measure N by fact
2698   define d where "d X = measure M X - measure N X" for X
2700   have [intro]: "bdd_above (d`sets M)"
2701     using sets.sets_into_space[of _ M]
2702     by (intro bdd_aboveI[where M="measure M (space M)"])
2703        (auto simp: d_def field_simps subset_eq intro!: add_increasing M.finite_measure_mono)
2705   define \<gamma> where "\<gamma> = (SUP X\<in>sets M. d X)"
2706   have le_\<gamma>[intro]: "X \<in> sets M \<Longrightarrow> d X \<le> \<gamma>" for X
2707     by (auto simp: \<gamma>_def intro!: cSUP_upper)
2709   have "\<exists>f. \<forall>n. f n \<in> sets M \<and> d (f n) > \<gamma> - 1 / 2^n"
2710   proof (intro choice_iff[THEN iffD1] allI)
2711     fix n
2712     have "\<exists>X\<in>sets M. \<gamma> - 1 / 2^n < d X"
2713       unfolding \<gamma>_def by (intro less_cSUP_iff[THEN iffD1]) auto
2714     then show "\<exists>y. y \<in> sets M \<and> \<gamma> - 1 / 2 ^ n < d y"
2715       by auto
2716   qed
2717   then obtain E where [measurable]: "E n \<in> sets M" and E: "d (E n) > \<gamma> - 1 / 2^n" for n
2718     by auto
2720   define F where "F m n = (if m \<le> n then \<Inter>i\<in>{m..n}. E i else space M)" for m n
2722   have [measurable]: "m \<le> n \<Longrightarrow> F m n \<in> sets M" for m n
2723     by (auto simp: F_def)
2725   have 1: "\<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)" if "m \<le> n" for m n
2726     using that
2727   proof (induct rule: dec_induct)
2728     case base with E[of m] show ?case
2729       by (simp add: F_def field_simps)
2730   next
2731     case (step i)
2732     have F_Suc: "F m (Suc i) = F m i \<inter> E (Suc i)"
2733       using \<open>m \<le> i\<close> by (auto simp: F_def le_Suc_eq)
2735     have "\<gamma> + (\<gamma> - 2 / 2^m + 1 / 2 ^ Suc i) \<le> (\<gamma> - 1 / 2^Suc i) + (\<gamma> - 2 / 2^m + 1 / 2^i)"
2737     also have "\<dots> \<le> d (E (Suc i)) + d (F m i)"
2738       using E[of "Suc i"] by (intro add_mono step) auto
2739     also have "\<dots> = d (E (Suc i)) + d (F m i - E (Suc i)) + d (F m (Suc i))"
2740       using \<open>m \<le> i\<close> by (simp add: d_def field_simps F_Suc M.finite_measure_Diff' N.finite_measure_Diff')
2741     also have "\<dots> = d (E (Suc i) \<union> F m i) + d (F m (Suc i))"
2742       using \<open>m \<le> i\<close> by (simp add: d_def field_simps M.finite_measure_Union' N.finite_measure_Union')
2743     also have "\<dots> \<le> \<gamma> + d (F m (Suc i))"
2744       using \<open>m \<le> i\<close> by auto
2745     finally show ?case
2746       by auto
2747   qed
2749   define F' where "F' m = (\<Inter>i\<in>{m..}. E i)" for m
2750   have F'_eq: "F' m = (\<Inter>i. F m (i + m))" for m
2751     by (fastforce simp: le_iff_add[of m] F'_def F_def)
2753   have [measurable]: "F' m \<in> sets M" for m
2754     by (auto simp: F'_def)
2756   have \<gamma>_le: "\<gamma> - 0 \<le> d (\<Union>m. F' m)"
2757   proof (rule LIMSEQ_le)
2758     show "(\<lambda>n. \<gamma> - 2 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 0"
2759       by (intro tendsto_intros LIMSEQ_divide_realpow_zero) auto
2760     have "incseq F'"
2761       by (auto simp: incseq_def F'_def)
2762     then show "(\<lambda>m. d (F' m)) \<longlonglongrightarrow> d (\<Union>m. F' m)"
2763       unfolding d_def
2764       by (intro tendsto_diff M.finite_Lim_measure_incseq N.finite_Lim_measure_incseq) auto
2766     have "\<gamma> - 2 / 2 ^ m + 0 \<le> d (F' m)" for m
2767     proof (rule LIMSEQ_le)
2768       have *: "decseq (\<lambda>n. F m (n + m))"
2769         by (auto simp: decseq_def F_def)
2770       show "(\<lambda>n. d (F m n)) \<longlonglongrightarrow> d (F' m)"
2771         unfolding d_def F'_eq
2772         by (rule LIMSEQ_offset[where k=m])
2773            (auto intro!: tendsto_diff M.finite_Lim_measure_decseq N.finite_Lim_measure_decseq *)
2774       show "(\<lambda>n. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 2 / 2 ^ m + 0"
2775         by (intro tendsto_add LIMSEQ_divide_realpow_zero tendsto_const) auto
2776       show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)"
2777         using 1[of m] by (intro exI[of _ m]) auto
2778     qed
2779     then show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ n \<le> d (F' n)"
2780       by auto
2781   qed
2783   show ?thesis
2784   proof (safe intro!: bexI[of _ "\<Union>m. F' m"])
2785     fix X assume [measurable]: "X \<in> sets M" and X: "X \<subseteq> (\<Union>m. F' m)"
2786     have "d (\<Union>m. F' m) - d X = d ((\<Union>m. F' m) - X)"
2787       using X by (auto simp: d_def M.finite_measure_Diff N.finite_measure_Diff)
2788     also have "\<dots> \<le> \<gamma>"
2789       by auto
2790     finally have "0 \<le> d X"
2791       using \<gamma>_le by auto
2792     then show "emeasure N X \<le> emeasure M X"
2793       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)
2794   next
2795     fix X assume [measurable]: "X \<in> sets M" and X: "X \<inter> (\<Union>m. F' m) = {}"
2796     then have "d (\<Union>m. F' m) + d X = d (X \<union> (\<Union>m. F' m))"
2797       by (auto simp: d_def M.finite_measure_Union N.finite_measure_Union)
2798     also have "\<dots> \<le> \<gamma>"
2799       by auto
2800     finally have "d X \<le> 0"
2801       using \<gamma>_le by auto
2802     then show "emeasure M X \<le> emeasure N X"
2803       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)
2804   qed auto
2805 qed
2807 proposition unsigned_Hahn_decomposition:
2808   assumes [simp]: "sets N = sets M" and [measurable]: "A \<in> sets M"
2809     and [simp]: "emeasure M A \<noteq> top" "emeasure N A \<noteq> top"
2810   shows "\<exists>Y\<in>sets M. Y \<subseteq> A \<and> (\<forall>X\<in>sets M. X \<subseteq> Y \<longrightarrow> N X \<le> M X) \<and> (\<forall>X\<in>sets M. X \<subseteq> A \<longrightarrow> X \<inter> Y = {} \<longrightarrow> M X \<le> N X)"
2811 proof -
2812   have "\<exists>Y\<in>sets (restrict_space M A).
2813     (\<forall>X\<in>sets (restrict_space M A). X \<subseteq> Y \<longrightarrow> (restrict_space N A) X \<le> (restrict_space M A) X) \<and>
2814     (\<forall>X\<in>sets (restrict_space M A). X \<inter> Y = {} \<longrightarrow> (restrict_space M A) X \<le> (restrict_space N A) X)"
2815   proof (rule finite_unsigned_Hahn_decomposition)
2816     show "finite_measure (restrict_space M A)" "finite_measure (restrict_space N A)"
2817       by (auto simp: space_restrict_space emeasure_restrict_space less_top intro!: finite_measureI)
2819   then guess Y ..
2820   then show ?thesis
2821     apply (intro bexI[of _ Y] conjI ballI conjI)
2822     apply (simp_all add: sets_restrict_space emeasure_restrict_space)
2823     apply safe
2824     subgoal for X Z
2825       by (erule ballE[of _ _ X]) (auto simp add: Int_absorb1)
2826     subgoal for X Z
2827       by (erule ballE[of _ _ X])  (auto simp add: Int_absorb1 ac_simps)
2828     apply auto
2829     done
2830 qed
2832 text%important \<open>
2833   Define a lexicographical order on \<^type>\<open>measure\<close>, in the order space, sets and measure. The parts
2834   of the lexicographical order are point-wise ordered.
2835 \<close>
2837 instantiation measure :: (type) order_bot
2838 begin
2840 inductive less_eq_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where
2841   "space M \<subset> space N \<Longrightarrow> less_eq_measure M N"
2842 | "space M = space N \<Longrightarrow> sets M \<subset> sets N \<Longrightarrow> less_eq_measure M N"
2843 | "space M = space N \<Longrightarrow> sets M = sets N \<Longrightarrow> emeasure M \<le> emeasure N \<Longrightarrow> less_eq_measure M N"
2845 lemma le_measure_iff:
2846   "M \<le> N \<longleftrightarrow> (if space M = space N then
2847     if sets M = sets N then emeasure M \<le> emeasure N else sets M \<subseteq> sets N else space M \<subseteq> space N)"
2848   by (auto elim: less_eq_measure.cases intro: less_eq_measure.intros)
2850 definition%important less_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where
2851   "less_measure M N \<longleftrightarrow> (M \<le> N \<and> \<not> N \<le> M)"
2853 definition%important bot_measure :: "'a measure" where
2854   "bot_measure = sigma {} {}"
2856 lemma
2857   shows space_bot[simp]: "space bot = {}"
2858     and sets_bot[simp]: "sets bot = {{}}"
2859     and emeasure_bot[simp]: "emeasure bot X = 0"
2860   by (auto simp: bot_measure_def sigma_sets_empty_eq emeasure_sigma)
2862 instance
2863 proof standard
2864   show "bot \<le> a" for a :: "'a measure"
2865     by (simp add: le_measure_iff bot_measure_def sigma_sets_empty_eq emeasure_sigma le_fun_def)
2866 qed (auto simp: le_measure_iff less_measure_def split: if_split_asm intro: measure_eqI)
2868 end
2870 proposition le_measure: "sets M = sets N \<Longrightarrow> M \<le> N \<longleftrightarrow> (\<forall>A\<in>sets M. emeasure M A \<le> emeasure N A)"
2871   apply -
2872   apply (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq)
2873   subgoal for X
2874     by (cases "X \<in> sets M") (auto simp: emeasure_notin_sets)
2875   done
2877 definition%important sup_measure' :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure" where
2878 "sup_measure' A B =
2879   measure_of (space A) (sets A)
2880     (\<lambda>X. SUP Y\<in>sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"
2882 lemma assumes [simp]: "sets B = sets A"
2883   shows space_sup_measure'[simp]: "space (sup_measure' A B) = space A"
2884     and sets_sup_measure'[simp]: "sets (sup_measure' A B) = sets A"
2885   using sets_eq_imp_space_eq[OF assms] by (simp_all add: sup_measure'_def)
2887 lemma emeasure_sup_measure':
2888   assumes sets_eq[simp]: "sets B = sets A" and [simp, intro]: "X \<in> sets A"
2889   shows "emeasure (sup_measure' A B) X = (SUP Y\<in>sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"
2890     (is "_ = ?S X")
2891 proof -
2892   note sets_eq_imp_space_eq[OF sets_eq, simp]
2893   show ?thesis
2894     using sup_measure'_def
2895   proof (rule emeasure_measure_of)
2896     let ?d = "\<lambda>X Y. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)"
2897     show "countably_additive (sets (sup_measure' A B)) (\<lambda>X. SUP Y \<in> sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"
2899       case (1 X)
2900       then have [measurable]: "\<And>i. X i \<in> sets A" and "disjoint_family X"
2901         by auto
2902       have disjoint: "disjoint_family (\<lambda>i. X i \<inter> Y)" "disjoint_family (\<lambda>i. X i - Y)" for Y
2903         by (auto intro: disjoint_family_on_bisimulation [OF \<open>disjoint_family X\<close>, simplified])
2904       have "(\<Sum>i. ?S (X i)) = (SUP Y\<in>sets A. \<Sum>i. ?d (X i) Y)"
2905       proof (rule ennreal_suminf_SUP_eq_directed)
2906         fix J :: "nat set" and a b assume "finite J" and [measurable]: "a \<in> sets A" "b \<in> sets A"
2907         have "\<exists>c\<in>sets A. c \<subseteq> X i \<and> (\<forall>a\<in>sets A. ?d (X i) a \<le> ?d (X i) c)" for i
2908         proof cases
2909           assume "emeasure A (X i) = top \<or> emeasure B (X i) = top"
2910           then show ?thesis
2911           proof
2912             assume "emeasure A (X i) = top" then show ?thesis
2913               by (intro bexI[of _ "X i"]) auto
2914           next
2915             assume "emeasure B (X i) = top" then show ?thesis
2916               by (intro bexI[of _ "{}"]) auto
2917           qed
2918         next
2919           assume finite: "\<not> (emeasure A (X i) = top \<or> emeasure B (X i) = top)"
2920           then have "\<exists>Y\<in>sets A. Y \<subseteq> X i \<and> (\<forall>C\<in>sets A. C \<subseteq> Y \<longrightarrow> B C \<le> A C) \<and> (\<forall>C\<in>sets A. C \<subseteq> X i \<longrightarrow> C \<inter> Y = {} \<longrightarrow> A C \<le> B C)"
2921             using unsigned_Hahn_decomposition[of B A "X i"] by simp
2922           then obtain Y where [measurable]: "Y \<in> sets A" and [simp]: "Y \<subseteq> X i"
2923             and B_le_A: "\<And>C. C \<in> sets A \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> B C \<le> A C"
2924             and A_le_B: "\<And>C. C \<in> sets A \<Longrightarrow> C \<subseteq> X i \<Longrightarrow> C \<inter> Y = {} \<Longrightarrow> A C \<le> B C"
2925             by auto
2927           show ?thesis
2928           proof (intro bexI[of _ Y] ballI conjI)
2929             fix a assume [measurable]: "a \<in> sets A"
2930             have *: "(X i \<inter> a \<inter> Y \<union> (X i \<inter> a - Y)) = X i \<inter> a" "(X i - a) \<inter> Y \<union> (X i - a - Y) = X i \<inter> - a"
2931               for a Y by auto
2932             then have "?d (X i) a =
2933               (A (X i \<inter> a \<inter> Y) + A (X i \<inter> a \<inter> - Y)) + (B (X i \<inter> - a \<inter> Y) + B (X i \<inter> - a \<inter> - Y))"
2934               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric])
2935             also have "\<dots> \<le> (A (X i \<inter> a \<inter> Y) + B (X i \<inter> a \<inter> - Y)) + (A (X i \<inter> - a \<inter> Y) + B (X i \<inter> - a \<inter> - Y))"
2936               by (intro add_mono order_refl B_le_A A_le_B) (auto simp: Diff_eq[symmetric])
2937             also have "\<dots> \<le> (A (X i \<inter> Y \<inter> a) + A (X i \<inter> Y \<inter> - a)) + (B (X i \<inter> - Y \<inter> a) + B (X i \<inter> - Y \<inter> - a))"
2939             also have "\<dots> \<le> A (X i \<inter> Y) + B (X i \<inter> - Y)"
2940               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric] *)
2941             finally show "?d (X i) a \<le> ?d (X i) Y" .
2942           qed auto
2943         qed
2944         then obtain C where [measurable]: "C i \<in> sets A" and "C i \<subseteq> X i"
2945           and C: "\<And>a. a \<in> sets A \<Longrightarrow> ?d (X i) a \<le> ?d (X i) (C i)" for i
2946           by metis
2947         have *: "X i \<inter> (\<Union>i. C i) = X i \<inter> C i" for i
2948         proof safe
2949           fix x j assume "x \<in> X i" "x \<in> C j"
2950           moreover have "i = j \<or> X i \<inter> X j = {}"
2951             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)
2952           ultimately show "x \<in> C i"
2953             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto
2954         qed auto
2955         have **: "X i \<inter> - (\<Union>i. C i) = X i \<inter> - C i" for i
2956         proof safe
2957           fix x j assume "x \<in> X i" "x \<notin> C i" "x \<in> C j"
2958           moreover have "i = j \<or> X i \<inter> X j = {}"
2959             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)
2960           ultimately show False
2961             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto
2962         qed auto
2963         show "\<exists>c\<in>sets A. \<forall>i\<in>J. ?d (X i) a \<le> ?d (X i) c \<and> ?d (X i) b \<le> ?d (X i) c"
2964           apply (intro bexI[of _ "\<Union>i. C i"])
2965           unfolding * **
2966           apply (intro C ballI conjI)
2967           apply auto
2968           done
2969       qed
2970       also have "\<dots> = ?S (\<Union>i. X i)"
2971         apply (simp only: UN_extend_simps(4))
2972         apply (rule arg_cong [of _ _ Sup])
2973         apply (rule image_cong)
2974          apply (fact refl)
2975         using disjoint
2976         apply (auto simp add: suminf_add [symmetric] Diff_eq [symmetric] image_subset_iff suminf_emeasure simp del: UN_simps)
2977         done
2978       finally show "(\<Sum>i. ?S (X i)) = ?S (\<Union>i. X i)" .
2979     qed
2980   qed (auto dest: sets.sets_into_space simp: positive_def intro!: SUP_const)
2981 qed
2983 lemma le_emeasure_sup_measure'1:
2984   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure A X \<le> emeasure (sup_measure' A B) X"
2985   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "X"] assms)
2987 lemma le_emeasure_sup_measure'2:
2988   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure B X \<le> emeasure (sup_measure' A B) X"
2989   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "{}"] assms)
2991 lemma emeasure_sup_measure'_le2:
2992   assumes [simp]: "sets B = sets C" "sets A = sets C" and [measurable]: "X \<in> sets C"
2993   assumes A: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure A Y \<le> emeasure C Y"
2994   assumes B: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure B Y \<le> emeasure C Y"
2995   shows "emeasure (sup_measure' A B) X \<le> emeasure C X"
2996 proof (subst emeasure_sup_measure')
2997   show "(SUP Y\<in>sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)) \<le> emeasure C X"
2998     unfolding \<open>sets A = sets C\<close>
2999   proof (intro SUP_least)
3000     fix Y assume [measurable]: "Y \<in> sets C"
3001     have [simp]: "X \<inter> Y \<union> (X - Y) = X"
3002       by auto
3003     have "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C (X \<inter> Y) + emeasure C (X \<inter> - Y)"
3004       by (intro add_mono A B) (auto simp: Diff_eq[symmetric])
3005     also have "\<dots> = emeasure C X"
3006       by (subst plus_emeasure) (auto simp: Diff_eq[symmetric])
3007     finally show "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C X" .
3008   qed
3009 qed simp_all
3011 definition%important sup_lexord :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b::order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" where
3012 "sup_lexord A B k s c =
3013   (if k A = k B then c else
3014    if \<not> k A \<le> k B \<and> \<not> k B \<le> k A then s else
3015    if k B \<le> k A then A else B)"
3017 lemma sup_lexord:
3018   "(k A < k B \<Longrightarrow> P B) \<Longrightarrow> (k B < k A \<Longrightarrow> P A) \<Longrightarrow> (k A = k B \<Longrightarrow> P c) \<Longrightarrow>
3019     (\<not> k B \<le> k A \<Longrightarrow> \<not> k A \<le> k B \<Longrightarrow> P s) \<Longrightarrow> P (sup_lexord A B k s c)"
3020   by (auto simp: sup_lexord_def)
3022 lemmas le_sup_lexord = sup_lexord[where P="\<lambda>a. c \<le> a" for c]
3024 lemma sup_lexord1: "k A = k B \<Longrightarrow> sup_lexord A B k s c = c"
3027 lemma sup_lexord_commute: "sup_lexord A B k s c = sup_lexord B A k s c"
3028   by (auto simp: sup_lexord_def)
3030 lemma sigma_sets_le_sets_iff: "(sigma_sets (space x) \<A> \<subseteq> sets x) = (\<A> \<subseteq> sets x)"
3031   using sets.sigma_sets_subset[of \<A> x] by auto
3033 lemma sigma_le_iff: "\<A> \<subseteq> Pow \<Omega> \<Longrightarrow> sigma \<Omega> \<A> \<le> x \<longleftrightarrow> (\<Omega> \<subseteq> space x \<and> (space x = \<Omega> \<longrightarrow> \<A> \<subseteq> sets x))"
3034   by (cases "\<Omega> = space x")
3035      (simp_all add: eq_commute[of _ "sets x"] le_measure_iff emeasure_sigma le_fun_def
3036                     sigma_sets_superset_generator sigma_sets_le_sets_iff)
3038 instantiation measure :: (type) semilattice_sup
3039 begin
3041 definition%important sup_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure" where
3042   "sup_measure A B =
3043     sup_lexord A B space (sigma (space A \<union> space B) {})
3044       (sup_lexord A B sets (sigma (space A) (sets A \<union> sets B)) (sup_measure' A B))"
3046 instance
3047 proof
3048   fix x y z :: "'a measure"
3049   show "x \<le> sup x y"
3050     unfolding sup_measure_def
3051   proof (intro le_sup_lexord)
3052     assume "space x = space y"
3053     then have *: "sets x \<union> sets y \<subseteq> Pow (space x)"
3054       using sets.space_closed by auto
3055     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"
3056     then have "sets x \<subset> sets x \<union> sets y"
3057       by auto
3058     also have "\<dots> \<le> sigma (space x) (sets x \<union> sets y)"
3059       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)
3060     finally show "x \<le> sigma (space x) (sets x \<union> sets y)"
3061       by (simp add: space_measure_of[OF *] less_eq_measure.intros(2))
3062   next
3063     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"
3064     then show "x \<le> sigma (space x \<union> space y) {}"
3065       by (intro less_eq_measure.intros) auto
3066   next
3067     assume "sets x = sets y" then show "x \<le> sup_measure' x y"
3068       by (simp add: le_measure le_emeasure_sup_measure'1)
3069   qed (auto intro: less_eq_measure.intros)
3070   show "y \<le> sup x y"
3071     unfolding sup_measure_def
3072   proof (intro le_sup_lexord)
3073     assume **: "space x = space y"
3074     then have *: "sets x \<union> sets y \<subseteq> Pow (space y)"
3075       using sets.space_closed by auto
3076     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"
3077     then have "sets y \<subset> sets x \<union> sets y"
3078       by auto
3079     also have "\<dots> \<le> sigma (space y) (sets x \<union> sets y)"
3080       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)
3081     finally show "y \<le> sigma (space x) (sets x \<union> sets y)"
3082       by (simp add: ** space_measure_of[OF *] less_eq_measure.intros(2))
3083   next
3084     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"
3085     then show "y \<le> sigma (space x \<union> space y) {}"
3086       by (intro less_eq_measure.intros) auto
3087   next
3088     assume "sets x = sets y" then show "y \<le> sup_measure' x y"
3089       by (simp add: le_measure le_emeasure_sup_measure'2)
3090   qed (auto intro: less_eq_measure.intros)
3091   show "x \<le> y \<Longrightarrow> z \<le> y \<Longrightarrow> sup x z \<le> y"
3092     unfolding sup_measure_def
3093   proof (intro sup_lexord[where P="\<lambda>x. x \<le> y"])
3094     assume "x \<le> y" "z \<le> y" and [simp]: "space x = space z" "sets x = sets z"
3095     from \<open>x \<le> y\<close> show "sup_measure' x z \<le> y"
3096     proof cases
3097       case 1 then show ?thesis
3098         by (intro less_eq_measure.intros(1)) simp
3099     next
3100       case 2 then show ?thesis
3101         by (intro less_eq_measure.intros(2)) simp_all
3102     next
3103       case 3 with \<open>z \<le> y\<close> \<open>x \<le> y\<close> show ?thesis
3104         by (auto simp add: le_measure intro!: emeasure_sup_measure'_le2)
3105     qed
3106   next
3107     assume **: "x \<le> y" "z \<le> y" "space x = space z" "\<not> sets z \<subseteq> sets x" "\<not> sets x \<subseteq> sets z"
3108     then have *: "sets x \<union> sets z \<subseteq> Pow (space x)"
3109       using sets.space_closed by auto
3110     show "sigma (space x) (sets x \<union> sets z) \<le> y"
3111       unfolding sigma_le_iff[OF *] using ** by (auto simp: le_measure_iff split: if_split_asm)
3112   next
3113     assume "x \<le> y" "z \<le> y" "\<not> space z \<subseteq> space x" "\<not> space x \<subseteq> space z"
3114     then have "space x \<subseteq> space y" "space z \<subseteq> space y"
3115       by (auto simp: le_measure_iff split: if_split_asm)
3116     then show "sigma (space x \<union> space z) {} \<le> y"
3118   qed
3119 qed
3121 end
3123 lemma space_empty_eq_bot: "space a = {} \<longleftrightarrow> a = bot"
3124   using space_empty[of a] by (auto intro!: measure_eqI)
3126 lemma sets_eq_iff_bounded: "A \<le> B \<Longrightarrow> B \<le> C \<Longrightarrow> sets A = sets C \<Longrightarrow> sets B = sets A"
3127   by (auto dest: sets_eq_imp_space_eq simp add: le_measure_iff split: if_split_asm)
3129 lemma sets_sup: "sets A = sets M \<Longrightarrow> sets B = sets M \<Longrightarrow> sets (sup A B) = sets M"
3130   by (auto simp add: sup_measure_def sup_lexord_def dest: sets_eq_imp_space_eq)
3132 lemma le_measureD1: "A \<le> B \<Longrightarrow> space A \<le> space B"
3133   by (auto simp: le_measure_iff split: if_split_asm)
3135 lemma le_measureD2: "A \<le> B \<Longrightarrow> space A = space B \<Longrightarrow> sets A \<le> sets B"
3136   by (auto simp: le_measure_iff split: if_split_asm)
3138 lemma le_measureD3: "A \<le> B \<Longrightarrow> sets A = sets B \<Longrightarrow> emeasure A X \<le> emeasure B X"
3139   by (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq split: if_split_asm)
3141 lemma UN_space_closed: "\<Union>(sets ` S) \<subseteq> Pow (\<Union>(space ` S))"
3142   using sets.space_closed by auto
3144 definition%important
3145   Sup_lexord :: "('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> 'a"
3146 where
3147   "Sup_lexord k c s A =
3148   (let U = (SUP a\<in>A. k a)
3149    in if \<exists>a\<in>A. k a = U then c {a\<in>A. k a = U} else s A)"
3151 lemma Sup_lexord:
3152   "(\<And>a S. a \<in> A \<Longrightarrow> k a = (SUP a\<in>A. k a) \<Longrightarrow> S = {a'\<in>A. k a' = k a} \<Longrightarrow> P (c S)) \<Longrightarrow> ((\<And>a. a \<in> A \<Longrightarrow> k a \<noteq> (SUP a\<in>A. k a)) \<Longrightarrow> P (s A)) \<Longrightarrow>
3153     P (Sup_lexord k c s A)"
3154   by (auto simp: Sup_lexord_def Let_def)
3156 lemma Sup_lexord1:
3157   assumes A: "A \<noteq> {}" "(\<And>a. a \<in> A \<Longrightarrow> k a = (\<Union>a\<in>A. k a))" "P (c A)"
3158   shows "P (Sup_lexord k c s A)"
3159   unfolding Sup_lexord_def Let_def
3160 proof (clarsimp, safe)
3161   show "\<forall>a\<in>A. k a \<noteq> (\<Union>x\<in>A. k x) \<Longrightarrow> P (s A)"
3162     by (metis assms(1,2) ex_in_conv)
3163 next
3164   fix a assume "a \<in> A" "k a = (\<Union>x\<in>A. k x)"
3165   then have "{a \<in> A. k a = (\<Union>x\<in>A. k x)} = {a \<in> A. k a = k a}"
3166     by (metis A(2)[symmetric])
3167   then show "P (c {a \<in> A. k a = (\<Union>x\<in>A. k x)})"
3169 qed
3171 instantiation measure :: (type) complete_lattice
3172 begin
3174 interpretation sup_measure: comm_monoid_set sup "bot :: 'a measure"
3175   by standard (auto intro!: antisym)
3177 lemma sup_measure_F_mono':
3178   "finite J \<Longrightarrow> finite I \<Longrightarrow> sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"
3179 proof (induction J rule: finite_induct)
3180   case empty then show ?case
3181     by simp
3182 next
3183   case (insert i J)
3184   show ?case
3185   proof cases
3186     assume "i \<in> I" with insert show ?thesis
3187       by (auto simp: insert_absorb)
3188   next
3189     assume "i \<notin> I"
3190     have "sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"
3191       by (intro insert)
3192     also have "\<dots> \<le> sup_measure.F id (insert i (I \<union> J))"
3193       using insert \<open>i \<notin> I\<close> by (subst sup_measure.insert) auto
3194     finally show ?thesis
3195       by auto
3196   qed
3197 qed
3199 lemma sup_measure_F_mono: "finite I \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sup_measure.F id J \<le> sup_measure.F id I"
3200   using sup_measure_F_mono'[of I J] by (auto simp: finite_subset Un_absorb1)
3202 lemma sets_sup_measure_F:
3203   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> sets i = sets M) \<Longrightarrow> sets (sup_measure.F id I) = sets M"
3204   by (induction I rule: finite_ne_induct) (simp_all add: sets_sup)
3206 definition%important Sup_measure' :: "'a measure set \<Rightarrow> 'a measure" where
3207 "Sup_measure' M =
3208   measure_of (\<Union>a\<in>M. space a) (\<Union>a\<in>M. sets a)
3209     (\<lambda>X. (SUP P\<in>{P. finite P \<and> P \<subseteq> M }. sup_measure.F id P X))"
3211 lemma space_Sup_measure'2: "space (Sup_measure' M) = (\<Union>m\<in>M. space m)"
3212   unfolding Sup_measure'_def by (intro space_measure_of[OF UN_space_closed])
3214 lemma sets_Sup_measure'2: "sets (Sup_measure' M) = sigma_sets (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)"
3215   unfolding Sup_measure'_def by (intro sets_measure_of[OF UN_space_closed])
3217 lemma sets_Sup_measure':
3218   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"
3219   shows "sets (Sup_measure' M) = sets A"
3220   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close> by (simp add: Sup_measure'_def)
3222 lemma space_Sup_measure':
3223   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"
3224   shows "space (Sup_measure' M) = space A"
3225   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close>
3226   by (simp add: Sup_measure'_def )
3228 lemma emeasure_Sup_measure':
3229   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "X \<in> sets A" "M \<noteq> {}"
3230   shows "emeasure (Sup_measure' M) X = (SUP P\<in>{P. finite P \<and> P \<subseteq> M}. sup_measure.F id P X)"
3231     (is "_ = ?S X")
3232   using Sup_measure'_def
3233 proof (rule emeasure_measure_of)
3234   note sets_eq[THEN sets_eq_imp_space_eq, simp]
3235   have *: "sets (Sup_measure' M) = sets A" "space (Sup_measure' M) = space A"
3236     using \<open>M \<noteq> {}\<close> by (simp_all add: Sup_measure'_def)
3237   let ?\<mu> = "sup_measure.F id"
3238   show "countably_additive (sets (Sup_measure' M)) ?S"
3240     case (1 F)
3241     then have **: "range F \<subseteq> sets A"
3242       by (auto simp: *)
3243     show "(\<Sum>i. ?S (F i)) = ?S (\<Union>i. F i)"
3244     proof (subst ennreal_suminf_SUP_eq_directed)
3245       fix i j and N :: "nat set" assume ij: "i \<in> {P. finite P \<and> P \<subseteq> M}" "j \<in> {P. finite P \<and> P \<subseteq> M}"
3246       have "(i \<noteq> {} \<longrightarrow> sets (?\<mu> i) = sets A) \<and> (j \<noteq> {} \<longrightarrow> sets (?\<mu> j) = sets A) \<and>
3247         (i \<noteq> {} \<or> j \<noteq> {} \<longrightarrow> sets (?\<mu> (i \<union> j)) = sets A)"
3248         using ij by (intro impI sets_sup_measure_F conjI) auto
3249       then have "?\<mu> j (F n) \<le> ?\<mu> (i \<union> j) (F n) \<and> ?\<mu> i (F n) \<le> ?\<mu> (i \<union> j) (F n)" for n
3250         using ij
3251         by (cases "i = {}"; cases "j = {}")
3252            (auto intro!: le_measureD3 sup_measure_F_mono simp: sets_sup_measure_F
3253                  simp del: id_apply)
3254       with ij show "\<exists>k\<in>{P. finite P \<and> P \<subseteq> M}. \<forall>n\<in>N. ?\<mu> i (F n) \<le> ?\<mu> k (F n) \<and> ?\<mu> j (F n) \<le> ?\<mu> k (F n)"
3255         by (safe intro!: bexI[of _ "i \<union> j"]) auto
3256     next
3257       show "(SUP P \<in> {P. finite P \<and> P \<subseteq> M}. \<Sum>n. ?\<mu> P (F n)) = (SUP P \<in> {P. finite P \<and> P \<subseteq> M}. ?\<mu> P (\<Union>(F ` UNIV)))"
3258       proof (intro arg_cong [of _ _ Sup] image_cong refl)
3259         fix i assume i: "i \<in> {P. finite P \<and> P \<subseteq> M}"
3260         show "(\<Sum>n. ?\<mu> i (F n)) = ?\<mu> i (\<Union>(F ` UNIV))"
3261         proof cases
3262           assume "i \<noteq> {}" with i ** show ?thesis
3263             apply (intro suminf_emeasure \<open>disjoint_family F\<close>)
3264             apply (subst sets_sup_measure_F[OF _ _ sets_eq])
3265             apply auto
3266             done
3267         qed simp
3268       qed
3269     qed
3270   qed
3271   show "positive (sets (Sup_measure' M)) ?S"
3272     by (auto simp: positive_def bot_ennreal[symmetric])
3273   show "X \<in> sets (Sup_measure' M)"
3274     using assms * by auto
3275 qed (rule UN_space_closed)
3277 definition%important Sup_measure :: "'a measure set \<Rightarrow> 'a measure" where
3278 "Sup_measure =
3279   Sup_lexord space
3280     (Sup_lexord sets Sup_measure'
3281       (\<lambda>U. sigma (\<Union>u\<in>U. space u) (\<Union>u\<in>U. sets u)))
3282     (\<lambda>U. sigma (\<Union>u\<in>U. space u) {})"
3284 definition%important Inf_measure :: "'a measure set \<Rightarrow> 'a measure" where
3285   "Inf_measure A = Sup {x. \<forall>a\<in>A. x \<le> a}"
3287 definition%important inf_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure" where
3288   "inf_measure a b = Inf {a, b}"
3290 definition%important top_measure :: "'a measure" where
3291   "top_measure = Inf {}"
3293 instance
3294 proof
3295   note UN_space_closed [simp]
3296   show upper: "x \<le> Sup A" if x: "x \<in> A" for x :: "'a measure" and A
3297     unfolding Sup_measure_def
3298   proof (intro Sup_lexord[where P="\<lambda>y. x \<le> y"])
3299     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"
3300     from this[OF \<open>x \<in> A\<close>] \<open>x \<in> A\<close> show "x \<le> sigma (\<Union>a\<in>A. space a) {}"
3301       by (intro less_eq_measure.intros) auto
3302   next
3303     fix a S assume "a \<in> A" and a: "space a = (\<Union>a\<in>A. space a)" and S: "S = {a' \<in> A. space a' = space a}"
3304       and neq: "\<And>aa. aa \<in> S \<Longrightarrow> sets aa \<noteq> (\<Union>a\<in>S. sets a)"
3305     have sp_a: "space a = (\<Union>(space ` S))"
3306       using \<open>a\<in>A\<close> by (auto simp: S)
3307     show "x \<le> sigma (\<Union>(space ` S)) (\<Union>(sets ` S))"
3308     proof cases
3309       assume [simp]: "space x = space a"
3310       have "sets x \<subset> (\<Union>a\<in>S. sets a)"
3311         using \<open>x\<in>A\<close> neq[of x] by (auto simp: S)
3312       also have "\<dots> \<subseteq> sigma_sets (\<Union>x\<in>S. space x) (\<Union>x\<in>S. sets x)"
3313         by (rule sigma_sets_superset_generator)
3314       finally show ?thesis
3315         by (intro less_eq_measure.intros(2)) (simp_all add: sp_a)
3316     next
3317       assume "space x \<noteq> space a"
3318       moreover have "space x \<le> space a"
3319         unfolding a using \<open>x\<in>A\<close> by auto
3320       ultimately show ?thesis
3321         by (intro less_eq_measure.intros) (simp add: less_le sp_a)
3322     qed
3323   next
3324     fix a b S S' assume "a \<in> A" and a: "space a = (\<Union>a\<in>A. space a)" and S: "S = {a' \<in> A. space a' = space a}"
3325       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"
3326     then have "S' \<noteq> {}" "space b = space a"
3327       by auto
3328     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"
3329       by (auto simp: S')
3330     note sets_eq[THEN sets_eq_imp_space_eq, simp]
3331     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"
3332       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)
3333     show "x \<le> Sup_measure' S'"
3334     proof cases
3335       assume "x \<in> S"
3336       with \<open>b \<in> S\<close> have "space x = space b"
3338       show ?thesis
3339       proof cases
3340         assume "x \<in> S'"
3341         show "x \<le> Sup_measure' S'"
3342         proof (intro le_measure[THEN iffD2] ballI)
3343           show "sets x = sets (Sup_measure' S')"
3344             using \<open>x\<in>S'\<close> * by (simp add: S')
3345           fix X assume "X \<in> sets x"
3346           show "emeasure x X \<le> emeasure (Sup_measure' S') X"
3347           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets x\<close>])
3348             show "emeasure x X \<le> (SUP P \<in> {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X)"
3349               using \<open>x\<in>S'\<close> by (intro SUP_upper2[where i="{x}"]) auto
3350           qed (insert \<open>x\<in>S'\<close> S', auto)
3351         qed
3352       next
3353         assume "x \<notin> S'"
3354         then have "sets x \<noteq> sets b"
3355           using \<open>x\<in>S\<close> by (auto simp: S')
3356         moreover have "sets x \<le> sets b"
3357           using \<open>x\<in>S\<close> unfolding b by auto
3358         ultimately show ?thesis
3359           using * \<open>x \<in> S\<close>
3360           by (intro less_eq_measure.intros(2))
3361              (simp_all add: * \<open>space x = space b\<close> less_le)
3362       qed
3363     next
3364       assume "x \<notin> S"
3365       with \<open>x\<in>A\<close> \<open>x \<notin> S\<close> \<open>space b = space a\<close> show ?thesis
3366         by (intro less_eq_measure.intros)
3367            (simp_all add: * less_le a SUP_upper S)
3368     qed
3369   qed
3370   show least: "Sup A \<le> x" if x: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x" for x :: "'a measure" and A
3371     unfolding Sup_measure_def
3372   proof (intro Sup_lexord[where P="\<lambda>y. y \<le> x"])
3373     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"
3374     show "sigma (\<Union>(space ` A)) {} \<le> x"
3375       using x[THEN le_measureD1] by (subst sigma_le_iff) auto
3376   next
3377     fix a S assume "a \<in> A" "space a = (\<Union>a\<in>A. space a)" and S: "S = {a' \<in> A. space a' = space a}"
3378       "\<And>a. a \<in> S \<Longrightarrow> sets a \<noteq> (\<Union>a\<in>S. sets a)"
3379     have "\<Union>(space ` S) \<subseteq> space x"
3380       using S le_measureD1[OF x] by auto
3381     moreover
3382     have "\<Union>(space ` S) = space a"
3383       using \<open>a\<in>A\<close> S by auto
3384     then have "space x = \<Union>(space ` S) \<Longrightarrow> \<Union>(sets ` S) \<subseteq> sets x"
3385       using \<open>a \<in> A\<close> le_measureD2[OF x] by (auto simp: S)
3386     ultimately show "sigma (\<Union>(space ` S)) (\<Union>(sets ` S)) \<le> x"
3387       by (subst sigma_le_iff) simp_all
3388   next
3389     fix a b S S' assume "a \<in> A" and a: "space a = (\<Union>a\<in>A. space a)" and S: "S = {a' \<in> A. space a' = space a}"
3390       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"
3391     then have "S' \<noteq> {}" "space b = space a"
3392       by auto
3393     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"
3394       by (auto simp: S')
3395     note sets_eq[THEN sets_eq_imp_space_eq, simp]
3396     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"
3397       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)
3398     show "Sup_measure' S' \<le> x"
3399     proof cases
3400       assume "space x = space a"
3401       show ?thesis
3402       proof cases
3403         assume **: "sets x = sets b"
3404         show ?thesis
3405         proof (intro le_measure[THEN iffD2] ballI)
3406           show ***: "sets (Sup_measure' S') = sets x"
3407             by (simp add: * **)
3408           fix X assume "X \<in> sets (Sup_measure' S')"
3409           show "emeasure (Sup_measure' S') X \<le> emeasure x X"
3410             unfolding ***
3411           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets (Sup_measure' S')\<close>])
3412             show "(SUP P \<in> {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X) \<le> emeasure x X"
3413             proof (safe intro!: SUP_least)
3414               fix P assume P: "finite P" "P \<subseteq> S'"
3415               show "emeasure (sup_measure.F id P) X \<le> emeasure x X"
3416               proof cases
3417                 assume "P = {}" then show ?thesis
3418                   by auto
3419               next
3420                 assume "P \<noteq> {}"
3421                 from P have "finite P" "P \<subseteq> A"
3422                   unfolding S' S by (simp_all add: subset_eq)
3423                 then have "sup_measure.F id P \<le> x"
3424                   by (induction P) (auto simp: x)
3425                 moreover have "sets (sup_measure.F id P) = sets x"
3426                   using \<open>finite P\<close> \<open>P \<noteq> {}\<close> \<open>P \<subseteq> S'\<close> \<open>sets x = sets b\<close>
3427                   by (intro sets_sup_measure_F) (auto simp: S')
3428                 ultimately show "emeasure (sup_measure.F id P) X \<le> emeasure x X"
3429                   by (rule le_measureD3)
3430               qed
3431             qed
3432             show "m \<in> S' \<Longrightarrow> sets m = sets (Sup_measure' S')" for m
3433               unfolding * by (simp add: S')
3434           qed fact
3435         qed
3436       next
3437         assume "sets x \<noteq> sets b"
3438         moreover have "sets b \<le> sets x"
3439           unfolding b S using x[THEN le_measureD2] \<open>space x = space a\<close> by auto
3440         ultimately show "Sup_measure' S' \<le> x"
3441           using \<open>space x = space a\<close> \<open>b \<in> S\<close>
3442           by (intro less_eq_measure.intros(2)) (simp_all add: * S)
3443       qed
3444     next
3445       assume "space x \<noteq> space a"
3446       then have "space a < space x"
3447         using le_measureD1[OF x[OF \<open>a\<in>A\<close>]] by auto
3448       then show "Sup_measure' S' \<le> x"
3449         by (intro less_eq_measure.intros) (simp add: * \<open>space b = space a\<close>)
3450     qed
3451   qed
3452   show "Sup {} = (bot::'a measure)" "Inf {} = (top::'a measure)"
3453     by (auto intro!: antisym least simp: top_measure_def)
3454   show lower: "x \<in> A \<Longrightarrow> Inf A \<le> x" for x :: "'a measure" and A
3455     unfolding Inf_measure_def by (intro least) auto
3456   show greatest: "(\<And>z. z \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> x \<le> Inf A" for x :: "'a measure" and A
3457     unfolding Inf_measure_def by (intro upper) auto
3458   show "inf x y \<le> x" "inf x y \<le> y" "x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> inf y z" for x y z :: "'a measure"
3459     by (auto simp: inf_measure_def intro!: lower greatest)
3460 qed
3462 end
3464 lemma sets_SUP:
3465   assumes "\<And>x. x \<in> I \<Longrightarrow> sets (M x) = sets N"
3466   shows "I \<noteq> {} \<Longrightarrow> sets (SUP i\<in>I. M i) = sets N"
3467   unfolding Sup_measure_def
3468   using assms assms[THEN sets_eq_imp_space_eq]
3469     sets_Sup_measure'[where A=N and M="M`I"]
3470   by (intro Sup_lexord1[where P="\<lambda>x. sets x = sets N"]) auto
3472 lemma emeasure_SUP:
3473   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N" "I \<noteq> {}"
3474   shows "emeasure (SUP i\<in>I. M i) X = (SUP J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emeasure (SUP i\<in>J. M i) X)"
3475 proof -
3476   interpret sup_measure: comm_monoid_set sup "bot :: 'b measure"
3477     by standard (auto intro!: antisym)
3478   have eq: "finite J \<Longrightarrow> sup_measure.F id J = (SUP i\<in>J. i)" for J :: "'b measure set"
3479     by (induction J rule: finite_induct) auto
3480   have 1: "J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sets (SUP x\<in>J. M x) = sets N" for J
3481     by (intro sets_SUP sets) (auto )
3482   from \<open>I \<noteq> {}\<close> obtain i where "i\<in>I" by auto
3483   have "Sup_measure' (M`I) X = (SUP P\<in>{P. finite P \<and> P \<subseteq> M`I}. sup_measure.F id P X)"
3484     using sets by (intro emeasure_Sup_measure') auto
3485   also have "Sup_measure' (M`I) = (SUP i\<in>I. M i)"
3486     unfolding Sup_measure_def using \<open>I \<noteq> {}\<close> sets sets(1)[THEN sets_eq_imp_space_eq]
3487     by (intro Sup_lexord1[where P="\<lambda>x. _ = x"]) auto
3488   also have "(SUP P\<in>{P. finite P \<and> P \<subseteq> M`I}. sup_measure.F id P X) =
3489     (SUP J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. (SUP i\<in>J. M i) X)"
3490   proof (intro SUP_eq)
3491     fix J assume "J \<in> {P. finite P \<and> P \<subseteq> M`I}"
3492     then obtain J' where J': "J' \<subseteq> I" "finite J'" and J: "J = M`J'" and "finite J"
3493       using finite_subset_image[of J M I] by auto
3494     show "\<exists>j\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. sup_measure.F id J X \<le> (SUP i\<in>j. M i) X"
3495     proof cases
3496       assume "J' = {}" with \<open>i \<in> I\<close> show ?thesis
3497         by (auto simp add: J)
3498     next
3499       assume "J' \<noteq> {}" with J J' show ?thesis
3500         by (intro bexI[of _ "J'"]) (auto simp add: eq simp del: id_apply)
3501     qed
3502   next
3503     fix J assume J: "J \<in> {P. P \<noteq> {} \<and> finite P \<and> P \<subseteq> I}"
3504     show "\<exists>J'\<in>{J. finite J \<and> J \<subseteq> M`I}. (SUP i\<in>J. M i) X \<le> sup_measure.F id J' X"
3505       using J by (intro bexI[of _ "M`J"]) (auto simp add: eq simp del: id_apply)
3506   qed
3507   finally show ?thesis .
3508 qed
3510 lemma emeasure_SUP_chain:
3511   assumes sets: "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N"
3512   assumes ch: "Complete_Partial_Order.chain (\<le>) (M ` A)" and "A \<noteq> {}"
3513   shows "emeasure (SUP i\<in>A. M i) X = (SUP i\<in>A. emeasure (M i) X)"
3514 proof (subst emeasure_SUP[OF sets \<open>A \<noteq> {}\<close>])
3515   show "(SUP J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (Sup (M ` J)) X) = (SUP i\<in>A. emeasure (M i) X)"
3516   proof (rule SUP_eq)
3517     fix J assume "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}"
3518     then have J: "Complete_Partial_Order.chain (\<le>) (M ` J)" "finite J" "J \<noteq> {}" and "J \<subseteq> A"
3519       using ch[THEN chain_subset, of "M`J"] by auto
3520     with in_chain_finite[OF J(1)] obtain j where "j \<in> J" "(SUP j\<in>J. M j) = M j"
3521       by auto
3522     with \<open>J \<subseteq> A\<close> show "\<exists>j\<in>A. emeasure (Sup (M ` J)) X \<le> emeasure (M j) X"
3523       by auto
3524   next
3525     fix j assume "j\<in>A" then show "\<exists>i\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (M j) X \<le> emeasure (Sup (M ` i)) X"
3526       by (intro bexI[of _ "{j}"]) auto
3527   qed
3528 qed
3530 subsubsection%unimportant \<open>Supremum of a set of \<open>\<sigma>\<close>-algebras\<close>
3532 lemma space_Sup_eq_UN: "space (Sup M) = (\<Union>x\<in>M. space x)"
3533   unfolding Sup_measure_def
3534   apply (intro Sup_lexord[where P="\<lambda>x. space x = _"])
3535   apply (subst space_Sup_measure'2)
3536   apply auto []
3537   apply (subst space_measure_of[OF UN_space_closed])
3538   apply auto
3539   done
3541 lemma sets_Sup_eq:
3542   assumes *: "\<And>m. m \<in> M \<Longrightarrow> space m = X" and "M \<noteq> {}"
3543   shows "sets (Sup M) = sigma_sets X (\<Union>x\<in>M. sets x)"
3544   unfolding Sup_measure_def
3545   apply (rule Sup_lexord1)
3546   apply fact
3548   apply (rule Sup_lexord)
3549   subgoal premises that for a S
3550     unfolding that(3) that(2)[symmetric]
3551     using that(1)
3552     apply (subst sets_Sup_measure'2)
3553     apply (intro arg_cong2[where f=sigma_sets])
3554     apply (auto simp: *)
3555     done
3556   apply (subst sets_measure_of[OF UN_space_closed])
3558   done
3560 lemma in_sets_Sup: "(\<And>m. m \<in> M \<Longrightarrow> space m = X) \<Longrightarrow> m \<in> M \<Longrightarrow> A \<in> sets m \<Longrightarrow> A \<in> sets (Sup M)"
3561   by (subst sets_Sup_eq[where X=X]) auto
3563 lemma Sup_lexord_rel:
3564   assumes "\<And>i. i \<in> I \<Longrightarrow> k (A i) = k (B i)"
3565     "R (c (A ` {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))})) (c (B ` {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))}))"
3566     "R (s (A`I)) (s (B`I))"
3567   shows "R (Sup_lexord k c s (A`I)) (Sup_lexord k c s (B`I))"
3568 proof -
3569   have "A ` {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))} =  {a \<in> A ` I. k a = (SUP x\<in>I. k (B x))}"
3570     using assms(1) by auto
3571   moreover have "B ` {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))} =  {a \<in> B ` I. k a = (SUP x\<in>I. k (B x))}"
3572     by auto
3573   ultimately show ?thesis
3574     using assms by (auto simp: Sup_lexord_def Let_def image_comp)
3575 qed
3577 lemma sets_SUP_cong:
3578   assumes eq: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (SUP i\<in>I. M i) = sets (SUP i\<in>I. N i)"
3579   unfolding Sup_measure_def
3580   using eq eq[THEN sets_eq_imp_space_eq]
3581   apply (intro Sup_lexord_rel[where R="\<lambda>x y. sets x = sets y"])
3582   apply simp
3583   apply simp
3585   apply (intro arg_cong2[where f="\<lambda>x y. sets (sigma x y)"])
3586   apply auto
3587   done
3589 lemma sets_Sup_in_sets:
3590   assumes "M \<noteq> {}"
3591   assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N"
3592   assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N"
3593   shows "sets (Sup M) \<subseteq> sets N"
3594 proof -
3595   have *: "\<Union>(space ` M) = space N"
3596     using assms by auto
3597   show ?thesis
3598     unfolding * using assms by (subst sets_Sup_eq[of M "space N"]) (auto intro!: sets.sigma_sets_subset)
3599 qed
3601 lemma measurable_Sup1:
3602   assumes m: "m \<in> M" and f: "f \<in> measurable m N"
3603     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"
3604   shows "f \<in> measurable (Sup M) N"
3605 proof -
3606   have "space (Sup M) = space m"
3607     using m by (auto simp add: space_Sup_eq_UN dest: const_space)
3608   then show ?thesis
3609     using m f unfolding measurable_def by (auto intro: in_sets_Sup[OF const_space])
3610 qed
3612 lemma measurable_Sup2:
3613   assumes M: "M \<noteq> {}"
3614   assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m"
3615     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"
3616   shows "f \<in> measurable N (Sup M)"
3617 proof -
3618   from M obtain m where "m \<in> M" by auto
3619   have space_eq: "\<And>n. n \<in> M \<Longrightarrow> space n = space m"
3620     by (intro const_space \<open>m \<in> M\<close>)
3621   have "f \<in> measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m))"
3622   proof (rule measurable_measure_of)
3623     show "f \<in> space N \<rightarrow> \<Union>(space ` M)"
3624       using measurable_space[OF f] M by auto
3625   qed (auto intro: measurable_sets f dest: sets.sets_into_space)
3626   also have "measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)) = measurable N (Sup M)"
3627     apply (intro measurable_cong_sets refl)
3628     apply (subst sets_Sup_eq[OF space_eq M])
3629     apply simp
3630     apply (subst sets_measure_of[OF UN_space_closed])
3631     apply (simp add: space_eq M)
3632     done
3633   finally show ?thesis .
3634 qed
3636 lemma measurable_SUP2:
3637   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f \<in> measurable N (M i)) \<Longrightarrow>
3638     (\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> space (M i) = space (M j)) \<Longrightarrow> f \<in> measurable N (SUP i\<in>I. M i)"
3639   by (auto intro!: measurable_Sup2)
3641 lemma sets_Sup_sigma:
3642   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"
3643   shows "sets (SUP m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"
3644 proof -
3645   { fix a m assume "a \<in> sigma_sets \<Omega> m" "m \<in> M"
3646     then have "a \<in> sigma_sets \<Omega> (\<Union>M)"
3647      by induction (auto intro: sigma_sets.intros(2-)) }
3648   then show "sets (SUP m\<in>M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"
3649     apply (subst sets_Sup_eq[where X="\<Omega>"])
3650     apply (auto simp add: M) []
3651     apply auto []
3652     apply (simp add: space_measure_of_conv M Union_least)
3653     apply (rule sigma_sets_eqI)
3654     apply auto
3655     done
3656 qed
3658 lemma Sup_sigma:
3659   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"
3660   shows "(SUP m\<in>M. sigma \<Omega> m) = (sigma \<Omega> (\<Union>M))"
3661 proof (intro antisym SUP_least)
3662   have *: "\<Union>M \<subseteq> Pow \<Omega>"
3663     using M by auto
3664   show "sigma \<Omega> (\<Union>M) \<le> (SUP m\<in>M. sigma \<Omega> m)"
3665   proof (intro less_eq_measure.intros(3))
3666     show "space (sigma \<Omega> (\<Union>M)) = space (SUP m\<in>M. sigma \<Omega> m)"
3667       "sets (sigma \<Omega> (\<Union>M)) = sets (SUP m\<in>M. sigma \<Omega> m)"
3668       using sets_Sup_sigma[OF assms] sets_Sup_sigma[OF assms, THEN sets_eq_imp_space_eq]
3669       by auto
3670   qed (simp add: emeasure_sigma le_fun_def)
3671   fix m assume "m \<in> M" then show "sigma \<Omega> m \<le> sigma \<Omega> (\<Union>M)"
3672     by (subst sigma_le_iff) (auto simp add: M *)
3673 qed
3675 lemma SUP_sigma_sigma:
3676   "M \<noteq> {} \<Longrightarrow> (\<And>m. m \<in> M \<Longrightarrow> f m \<subseteq> Pow \<Omega>) \<Longrightarrow> (SUP m\<in>M. sigma \<Omega> (f m)) = sigma \<Omega> (\<Union>m\<in>M. f m)"
3677   using Sup_sigma[of "f`M" \<Omega>] by (auto simp: image_comp)
3679 lemma sets_vimage_Sup_eq:
3680   assumes *: "M \<noteq> {}" "f \<in> X \<rightarrow> Y" "\<And>m. m \<in> M \<Longrightarrow> space m = Y"
3681   shows "sets (vimage_algebra X f (Sup M)) = sets (SUP m \<in> M. vimage_algebra X f m)"
3682   (is "?IS = ?SI")
3683 proof
3684   show "?IS \<subseteq> ?SI"
3685     apply (intro sets_image_in_sets measurable_Sup2)
3686     apply (simp add: space_Sup_eq_UN *)
3688     apply (intro measurable_Sup1)
3689     apply (rule imageI)
3690     apply assumption
3691     apply (rule measurable_vimage_algebra1)
3692     apply (auto simp: *)
3693     done
3694   show "?SI \<subseteq> ?IS"
3695     apply (intro sets_Sup_in_sets)
3696     apply (auto simp: *) []
3697     apply (auto simp: *) []
3698     apply (elim imageE)
3699     apply simp
3700     apply (rule sets_image_in_sets)
3701     apply simp
3703     apply (simp add: * space_Sup_eq_UN sets_vimage_algebra2)
3704     apply (auto intro: in_sets_Sup[OF *(3)])
3705     done
3706 qed
3708 lemma restrict_space_eq_vimage_algebra':
3709   "sets (restrict_space M \<Omega>) = sets (vimage_algebra (\<Omega> \<inter> space M) (\<lambda>x. x) M)"
3710 proof -
3711   have *: "{A \<inter> (\<Omega> \<inter> space M) |A. A \<in> sets M} = {A \<inter> \<Omega> |A. A \<in> sets M}"
3712     using sets.sets_into_space[of _ M] by blast
3714   show ?thesis
3715     unfolding restrict_space_def
3716     by (subst sets_measure_of)
3717        (auto simp add: image_subset_iff sets_vimage_algebra * dest: sets.sets_into_space intro!: arg_cong2[where f=sigma_sets])
3718 qed
3720 lemma sigma_le_sets:
3721   assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"
3722 proof
3723   have "X \<in> sigma_sets X A" "A \<subseteq> sigma_sets X A"
3724     by (auto intro: sigma_sets_top)
3725   moreover assume "sets (sigma X A) \<subseteq> sets N"
3726   ultimately show "X \<in> sets N \<and> A \<subseteq> sets N"
3727     by auto
3728 next
3729   assume *: "X \<in> sets N \<and> A \<subseteq> sets N"
3730   { fix Y assume "Y \<in> sigma_sets X A" from this * have "Y \<in> sets N"
3731       by induction auto }
3732   then show "sets (sigma X A) \<subseteq> sets N"
3733     by auto
3734 qed
3736 lemma measurable_iff_sets:
3737   "f \<in> measurable M N \<longleftrightarrow> (f \<in> space M \<rightarrow> space N \<and> sets (vimage_algebra (space M) f N) \<subseteq> sets M)"
3738 proof -
3739   have *: "{f -` A \<inter> space M |A. A \<in> sets N} \<subseteq> Pow (space M)"
3740     by auto
3741   show ?thesis
3742     unfolding measurable_def
3743     by (auto simp add: vimage_algebra_def sigma_le_sets[OF *])
3744 qed
3746 lemma sets_vimage_algebra_space: "X \<in> sets (vimage_algebra X f M)"
3747   using sets.top[of "vimage_algebra X f M"] by simp
3749 lemma measurable_mono:
3750   assumes N: "sets N' \<le> sets N" "space N = space N'"
3751   assumes M: "sets M \<le> sets M'" "space M = space M'"
3752   shows "measurable M N \<subseteq> measurable M' N'"
3753   unfolding measurable_def
3754 proof safe
3755   fix f A assume "f \<in> space M \<rightarrow> space N" "A \<in> sets N'"
3756   moreover assume "\<forall>y\<in>sets N. f -` y \<inter> space M \<in> sets M" note this[THEN bspec, of A]
3757   ultimately show "f -` A \<inter> space M' \<in> sets M'"
3758     using assms by auto
3759 qed (insert N M, auto)
3761 lemma measurable_Sup_measurable:
3762   assumes f: "f \<in> space N \<rightarrow> A"
3763   shows "f \<in> measurable N (Sup {M. space M = A \<and> f \<in> measurable N M})"
3764 proof (rule measurable_Sup2)
3765   show "{M. space M = A \<and> f \<in> measurable N M} \<noteq> {}"
3766     using f unfolding ex_in_conv[symmetric]
3767     by (intro exI[of _ "sigma A {}"]) (auto intro!: measurable_measure_of)
3768 qed auto
3770 lemma (in sigma_algebra) sigma_sets_subset':
3771   assumes a: "a \<subseteq> M" "\<Omega>' \<in> M"
3772   shows "sigma_sets \<Omega>' a \<subseteq> M"
3773 proof
3774   show "x \<in> M" if x: "x \<in> sigma_sets \<Omega>' a" for x
3775     using x by (induct rule: sigma_sets.induct) (insert a, auto)
3776 qed
3778 lemma in_sets_SUP: "i \<in> I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> space (M i) = Y) \<Longrightarrow> X \<in> sets (M i) \<Longrightarrow> X \<in> sets (SUP i\<in>I. M i)"
3779   by (intro in_sets_Sup[where X=Y]) auto
3781 lemma measurable_SUP1:
3782   "i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<And>m n. m \<in> I \<Longrightarrow> n \<in> I \<Longrightarrow> space (M m) = space (M n)) \<Longrightarrow>
3783     f \<in> measurable (SUP i\<in>I. M i) N"
3784   by (auto intro: measurable_Sup1)
3786 lemma sets_image_in_sets':
3787   assumes X: "X \<in> sets N"
3788   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> f -` A \<inter> X \<in> sets N"
3789   shows "sets (vimage_algebra X f M) \<subseteq> sets N"
3790   unfolding sets_vimage_algebra
3791   by (rule sets.sigma_sets_subset') (auto intro!: measurable_sets X f)
3793 lemma mono_vimage_algebra:
3794   "sets M \<le> sets N \<Longrightarrow> sets (vimage_algebra X f M) \<subseteq> sets (vimage_algebra X f N)"
3795   using sets.top[of "sigma X {f -` A \<inter> X |A. A \<in> sets N}"]
3796   unfolding vimage_algebra_def
3797   apply (subst (asm) space_measure_of)
3798   apply auto []
3799   apply (subst sigma_le_sets)
3800   apply auto
3801   done
3803 lemma mono_restrict_space: "sets M \<le> sets N \<Longrightarrow> sets (restrict_space M X) \<subseteq> sets (restrict_space N X)"
3804   unfolding sets_restrict_space by (rule image_mono)
3806 lemma sets_eq_bot: "sets M = {{}} \<longleftrightarrow> M = bot"
3807   apply safe
3808   apply (intro measure_eqI)
3809   apply auto
3810   done
3812 lemma sets_eq_bot2: "{{}} = sets M \<longleftrightarrow> M = bot"
3813   using sets_eq_bot[of M] by blast
3816 lemma (in finite_measure) countable_support:
3817   "countable {x. measure M {x} \<noteq> 0}"
3818 proof cases
3819   assume "measure M (space M) = 0"
3820   with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
3821     by auto
3822   then show ?thesis
3823     by simp
3824 next
3825   let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
3826   assume "?M \<noteq> 0"
3827   then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
3828     using reals_Archimedean[of "?m x / ?M" for x]
3829     by (auto simp: field_simps not_le[symmetric] divide_le_0_iff measure_le_0_iff)
3830   have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
3831   proof (rule ccontr)
3832     fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
3833     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
3834       by (metis infinite_arbitrarily_large)
3835     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
3836       by auto
3837     { fix x assume "x \<in> X"
3838       from \<open>?M \<noteq> 0\<close> *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
3839       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
3840     note singleton_sets = this
3841     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
3842       using \<open>?M \<noteq> 0\<close>
3843       by (simp add: \<open>card X = Suc (Suc n)\<close> field_simps less_le)
3844     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
3845       by (rule sum_mono) fact
3846     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
3847       using singleton_sets \<open>finite X\<close>
3848       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
3849     finally have "?M < measure M (\<Union>x\<in>X. {x})" .
3850     moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
3851       using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
3852     ultimately show False by simp
3853   qed
3854   show ?thesis
3855     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
3856 qed
3858 end