src/HOL/Analysis/Measure_Space.thy
 author nipkow Wed Jan 10 15:25:09 2018 +0100 (19 months ago) changeset 67399 eab6ce8368fa parent 66453 cc19f7ca2ed6 child 67673 c8caefb20564 permissions -rw-r--r--
ran isabelle update_op on all sources
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 and their properties\<close>
9 theory Measure_Space
10 imports
11   Measurable "HOL-Library.Extended_Nonnegative_Real"
12 begin
14 subsection "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 Sigma_Algebra}, as it is also used to
59   represent sigma algebras (with an arbitrary emeasure).
60 \<close>
62 subsection "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 \<open>Properties of a premeasure @{term \<mu>}\<close>
96 text \<open>
97   The definitions for @{const positive} and @{const countably_additive} should be here, by they are
98   necessary to define @{typ "'a measure"} in @{theory Sigma_Algebra}.
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: "AS \<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: "AS \<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 UNIV A \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (UNION UNIV A)"
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_ereal[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 \<open>Properties of @{const emeasure}\<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   "FI \<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: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: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: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 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 ae_filter :: "'a measure \<Rightarrow> 'a filter" where
993   "ae_filter M = (INF N: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 \verb+[symmetric]+ 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_strong: "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 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 measurable}-functions\<close>
1390 definition distr :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b measure" where
1391   "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f - A \<inter> space M))"
1393 lemma
1394   shows sets_distr[simp, measurable_cong]: "sets (distr M N f) = sets N"
1395     and space_distr[simp]: "space (distr M N f) = space N"
1396   by (auto simp: distr_def)
1398 lemma
1399   shows measurable_distr_eq1[simp]: "measurable (distr Mf Nf f) Mf' = measurable Nf Mf'"
1400     and measurable_distr_eq2[simp]: "measurable Mg' (distr Mg Ng g) = measurable Mg' Ng"
1401   by (auto simp: measurable_def)
1403 lemma distr_cong:
1404   "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"
1405   using sets_eq_imp_space_eq[of N L] by (simp add: distr_def Int_def cong: rev_conj_cong)
1407 lemma emeasure_distr:
1408   fixes f :: "'a \<Rightarrow> 'b"
1409   assumes f: "f \<in> measurable M N" and A: "A \<in> sets N"
1410   shows "emeasure (distr M N f) A = emeasure M (f - A \<inter> space M)" (is "_ = ?\<mu> A")
1411   unfolding distr_def
1412 proof (rule emeasure_measure_of_sigma)
1413   show "positive (sets N) ?\<mu>"
1414     by (auto simp: positive_def)
1416   show "countably_additive (sets N) ?\<mu>"
1418     fix A :: "nat \<Rightarrow> 'b set" assume "range A \<subseteq> sets N" "disjoint_family A"
1419     then have A: "\<And>i. A i \<in> sets N" "(\<Union>i. A i) \<in> sets N" by auto
1420     then have *: "range (\<lambda>i. f - (A i) \<inter> space M) \<subseteq> sets M"
1421       using f by (auto simp: measurable_def)
1422     moreover have "(\<Union>i. f -  A i \<inter> space M) \<in> sets M"
1423       using * by blast
1424     moreover have **: "disjoint_family (\<lambda>i. f - A i \<inter> space M)"
1425       using \<open>disjoint_family A\<close> by (auto simp: disjoint_family_on_def)
1426     ultimately show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)"
1427       using suminf_emeasure[OF _ **] A f
1428       by (auto simp: comp_def vimage_UN)
1429   qed
1430   show "sigma_algebra (space N) (sets N)" ..
1431 qed fact
1433 lemma emeasure_Collect_distr:
1434   assumes X[measurable]: "X \<in> measurable M N" "Measurable.pred N P"
1435   shows "emeasure (distr M N X) {x\<in>space N. P x} = emeasure M {x\<in>space M. P (X x)}"
1436   by (subst emeasure_distr)
1437      (auto intro!: arg_cong2[where f=emeasure] X(1)[THEN measurable_space])
1439 lemma emeasure_lfp2[consumes 1, case_names cont f measurable]:
1440   assumes "P M"
1441   assumes cont: "sup_continuous F"
1442   assumes f: "\<And>M. P M \<Longrightarrow> f \<in> measurable M' M"
1443   assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)"
1444   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)})"
1445 proof (subst (1 2) emeasure_Collect_distr[symmetric, where X=f])
1446   show "f \<in> measurable M' M"  "f \<in> measurable M' M"
1447     using f[OF \<open>P M\<close>] by auto
1448   { fix i show "Measurable.pred M ((F ^^ i) (\<lambda>x. False))"
1449     using \<open>P M\<close> by (induction i arbitrary: M) (auto intro!: *) }
1450   show "Measurable.pred M (lfp F)"
1451     using \<open>P M\<close> cont * by (rule measurable_lfp_coinduct[of P])
1453   have "emeasure (distr M' M f) {x \<in> space (distr M' M f). lfp F x} =
1454     (SUP i. emeasure (distr M' M f) {x \<in> space (distr M' M f). (F ^^ i) (\<lambda>x. False) x})"
1455     using \<open>P M\<close>
1456   proof (coinduction arbitrary: M rule: emeasure_lfp')
1457     case (measurable A N) then have "\<And>N. P N \<Longrightarrow> Measurable.pred (distr M' N f) A"
1458       by metis
1459     then have "\<And>N. P N \<Longrightarrow> Measurable.pred N A"
1460       by simp
1461     with \<open>P N\<close>[THEN *] show ?case
1462       by auto
1463   qed fact
1464   then show "emeasure (distr M' M f) {x \<in> space M. lfp F x} =
1465     (SUP i. emeasure (distr M' M f) {x \<in> space M. (F ^^ i) (\<lambda>x. False) x})"
1466    by simp
1467 qed
1469 lemma distr_id[simp]: "distr N N (\<lambda>x. x) = N"
1470   by (rule measure_eqI) (auto simp: emeasure_distr)
1472 lemma distr_id2: "sets M = sets N \<Longrightarrow> distr N M (\<lambda>x. x) = N"
1473   by (rule measure_eqI) (auto simp: emeasure_distr)
1475 lemma measure_distr:
1476   "f \<in> measurable M N \<Longrightarrow> S \<in> sets N \<Longrightarrow> measure (distr M N f) S = measure M (f - S \<inter> space M)"
1477   by (simp add: emeasure_distr measure_def)
1479 lemma distr_cong_AE:
1480   assumes 1: "M = K" "sets N = sets L" and
1481     2: "(AE x in M. f x = g x)" and "f \<in> measurable M N" and "g \<in> measurable K L"
1482   shows "distr M N f = distr K L g"
1483 proof (rule measure_eqI)
1484   fix A assume "A \<in> sets (distr M N f)"
1485   with assms show "emeasure (distr M N f) A = emeasure (distr K L g) A"
1486     by (auto simp add: emeasure_distr intro!: emeasure_eq_AE measurable_sets)
1487 qed (insert 1, simp)
1489 lemma AE_distrD:
1490   assumes f: "f \<in> measurable M M'"
1491     and AE: "AE x in distr M M' f. P x"
1492   shows "AE x in M. P (f x)"
1493 proof -
1494   from AE[THEN AE_E] guess N .
1495   with f show ?thesis
1496     unfolding eventually_ae_filter
1497     by (intro bexI[of _ "f - N \<inter> space M"])
1498        (auto simp: emeasure_distr measurable_def)
1499 qed
1501 lemma AE_distr_iff:
1502   assumes f[measurable]: "f \<in> measurable M N" and P[measurable]: "{x \<in> space N. P x} \<in> sets N"
1503   shows "(AE x in distr M N f. P x) \<longleftrightarrow> (AE x in M. P (f x))"
1504 proof (subst (1 2) AE_iff_measurable[OF _ refl])
1505   have "f - {x\<in>space N. \<not> P x} \<inter> space M = {x \<in> space M. \<not> P (f x)}"
1506     using f[THEN measurable_space] by auto
1507   then show "(emeasure (distr M N f) {x \<in> space (distr M N f). \<not> P x} = 0) =
1508     (emeasure M {x \<in> space M. \<not> P (f x)} = 0)"
1510 qed auto
1512 lemma null_sets_distr_iff:
1513   "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"
1514   by (auto simp add: null_sets_def emeasure_distr)
1516 lemma distr_distr:
1517   "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)"
1518   by (auto simp add: emeasure_distr measurable_space
1519            intro!: arg_cong[where f="emeasure M"] measure_eqI)
1521 subsection \<open>Real measure values\<close>
1523 lemma ring_of_finite_sets: "ring_of_sets (space M) {A\<in>sets M. emeasure M A \<noteq> top}"
1524 proof (rule ring_of_setsI)
1525   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>
1526     a \<union> b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b
1527     using emeasure_subadditive[of a M b] by (auto simp: top_unique)
1529   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>
1530     a - b \<in> {A \<in> sets M. emeasure M A \<noteq> top}" for a b
1531     using emeasure_mono[of "a - b" a M] by (auto simp: Diff_subset top_unique)
1532 qed (auto dest: sets.sets_into_space)
1534 lemma measure_nonneg[simp]: "0 \<le> measure M A"
1535   unfolding measure_def by auto
1537 lemma zero_less_measure_iff: "0 < measure M A \<longleftrightarrow> measure M A \<noteq> 0"
1538   using measure_nonneg[of M A] by (auto simp add: le_less)
1540 lemma measure_le_0_iff: "measure M X \<le> 0 \<longleftrightarrow> measure M X = 0"
1541   using measure_nonneg[of M X] by linarith
1543 lemma measure_empty[simp]: "measure M {} = 0"
1544   unfolding measure_def by (simp add: zero_ennreal.rep_eq)
1546 lemma emeasure_eq_ennreal_measure:
1547   "emeasure M A \<noteq> top \<Longrightarrow> emeasure M A = ennreal (measure M A)"
1548   by (cases "emeasure M A" rule: ennreal_cases) (auto simp: measure_def)
1550 lemma measure_zero_top: "emeasure M A = top \<Longrightarrow> measure M A = 0"
1551   by (simp add: measure_def enn2ereal_top)
1553 lemma measure_eq_emeasure_eq_ennreal: "0 \<le> x \<Longrightarrow> emeasure M A = ennreal x \<Longrightarrow> measure M A = x"
1554   using emeasure_eq_ennreal_measure[of M A]
1555   by (cases "A \<in> M") (auto simp: measure_notin_sets emeasure_notin_sets)
1557 lemma enn2real_plus:"a < top \<Longrightarrow> b < top \<Longrightarrow> enn2real (a + b) = enn2real a + enn2real b"
1559            del: real_of_ereal_enn2ereal)
1561 lemma measure_eq_AE:
1562   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
1563   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
1564   shows "measure M A = measure M B"
1565   using assms emeasure_eq_AE[OF assms] by (simp add: measure_def)
1567 lemma measure_Union:
1568   "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>
1569     measure M (A \<union> B) = measure M A + measure M B"
1570   by (simp add: measure_def plus_emeasure[symmetric] enn2real_plus less_top)
1572 lemma disjoint_family_on_insert:
1573   "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"
1574   by (fastforce simp: disjoint_family_on_def)
1576 lemma measure_finite_Union:
1577   "finite S \<Longrightarrow> AS \<subseteq> sets M \<Longrightarrow> disjoint_family_on A S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>) \<Longrightarrow>
1578     measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
1579   by (induction S rule: finite_induct)
1580      (auto simp: disjoint_family_on_insert measure_Union sum_emeasure[symmetric] sets.countable_UN'[OF countable_finite])
1582 lemma measure_Diff:
1583   assumes finite: "emeasure M A \<noteq> \<infinity>"
1584   and measurable: "A \<in> sets M" "B \<in> sets M" "B \<subseteq> A"
1585   shows "measure M (A - B) = measure M A - measure M B"
1586 proof -
1587   have "emeasure M (A - B) \<le> emeasure M A" "emeasure M B \<le> emeasure M A"
1588     using measurable by (auto intro!: emeasure_mono)
1589   hence "measure M ((A - B) \<union> B) = measure M (A - B) + measure M B"
1590     using measurable finite by (rule_tac measure_Union) (auto simp: top_unique)
1591   thus ?thesis using \<open>B \<subseteq> A\<close> by (auto simp: Un_absorb2)
1592 qed
1594 lemma measure_UNION:
1595   assumes measurable: "range A \<subseteq> sets M" "disjoint_family A"
1596   assumes finite: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
1597   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
1598 proof -
1599   have "(\<lambda>i. emeasure M (A i)) sums (emeasure M (\<Union>i. A i))"
1600     unfolding suminf_emeasure[OF measurable, symmetric] by (simp add: summable_sums)
1601   moreover
1602   { fix i
1603     have "emeasure M (A i) \<le> emeasure M (\<Union>i. A i)"
1604       using measurable by (auto intro!: emeasure_mono)
1605     then have "emeasure M (A i) = ennreal ((measure M (A i)))"
1606       using finite by (intro emeasure_eq_ennreal_measure) (auto simp: top_unique) }
1607   ultimately show ?thesis using finite
1608     by (subst (asm) (2) emeasure_eq_ennreal_measure) simp_all
1609 qed
1612   assumes measurable: "A \<in> sets M" "B \<in> sets M"
1613   and fin: "emeasure M A \<noteq> \<infinity>" "emeasure M B \<noteq> \<infinity>"
1614   shows "measure M (A \<union> B) \<le> measure M A + measure M B"
1615 proof -
1616   have "emeasure M (A \<union> B) \<noteq> \<infinity>"
1617     using emeasure_subadditive[OF measurable] fin by (auto simp: top_unique)
1618   then show "(measure M (A \<union> B)) \<le> (measure M A) + (measure M B)"
1620     apply simp
1621     apply (subst (asm) (2 3 4) emeasure_eq_ennreal_measure)
1622     apply (auto simp add: ennreal_plus[symmetric] simp del: ennreal_plus)
1623     done
1624 qed
1627   assumes A: "finite I" "AI \<subseteq> sets M" and fin: "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>"
1628   shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
1629 proof -
1630   { have "emeasure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. emeasure M (A i))"
1632     also have "\<dots> < \<infinity>"
1633       using fin by (simp add: less_top A)
1634     finally have "emeasure M (\<Union>i\<in>I. A i) \<noteq> top" by simp }
1635   note * = this
1636   show ?thesis
1638     unfolding emeasure_eq_ennreal_measure[OF *]
1639     by (simp_all add: sum_nonneg emeasure_eq_ennreal_measure)
1640 qed
1643   assumes A: "range A \<subseteq> sets M" and fin: "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"
1644   shows "measure M (\<Union>i. A i) \<le> (\<Sum>i. measure M (A i))"
1645 proof -
1646   from fin have **: "\<And>i. emeasure M (A i) \<noteq> top"
1647     using ennreal_suminf_lessD[of "\<lambda>i. emeasure M (A i)"] by (simp add: less_top)
1648   { have "emeasure M (\<Union>i. A i) \<le> (\<Sum>i. emeasure M (A i))"
1650     also have "\<dots> < \<infinity>"
1651       using fin by (simp add: less_top)
1652     finally have "emeasure M (\<Union>i. A i) \<noteq> top" by simp }
1653   then have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"
1654     by (rule emeasure_eq_ennreal_measure[symmetric])
1655   also have "\<dots> \<le> (\<Sum>i. emeasure M (A i))"
1657   also have "\<dots> = ennreal (\<Sum>i. measure M (A i))"
1658     using fin unfolding emeasure_eq_ennreal_measure[OF **]
1659     by (subst suminf_ennreal) (auto simp: **)
1660   finally show ?thesis
1661     apply (rule ennreal_le_iff[THEN iffD1, rotated])
1662     apply (intro suminf_nonneg allI measure_nonneg summable_suminf_not_top)
1663     using fin
1664     apply (simp add: emeasure_eq_ennreal_measure[OF **])
1665     done
1666 qed
1668 lemma measure_Un_null_set: "A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> measure M (A \<union> B) = measure M A"
1669   by (simp add: measure_def emeasure_Un_null_set)
1671 lemma measure_Diff_null_set: "A \<in> sets M \<Longrightarrow> B \<in> null_sets M \<Longrightarrow> measure M (A - B) = measure M A"
1672   by (simp add: measure_def emeasure_Diff_null_set)
1674 lemma measure_eq_sum_singleton:
1675   "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>
1676     measure M S = (\<Sum>x\<in>S. measure M {x})"
1677   using emeasure_eq_sum_singleton[of S M]
1678   by (intro measure_eq_emeasure_eq_ennreal) (auto simp: sum_nonneg emeasure_eq_ennreal_measure)
1680 lemma Lim_measure_incseq:
1681   assumes A: "range A \<subseteq> sets M" "incseq A" and fin: "emeasure M (\<Union>i. A i) \<noteq> \<infinity>"
1682   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"
1683 proof (rule tendsto_ennrealD)
1684   have "ennreal (measure M (\<Union>i. A i)) = emeasure M (\<Union>i. A i)"
1685     using fin by (auto simp: emeasure_eq_ennreal_measure)
1686   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i
1687     using assms emeasure_mono[of "A _" "\<Union>i. A i" M]
1688     by (intro emeasure_eq_ennreal_measure[symmetric]) (auto simp: less_top UN_upper intro: le_less_trans)
1689   ultimately show "(\<lambda>x. ennreal (Sigma_Algebra.measure M (A x))) \<longlonglongrightarrow> ennreal (Sigma_Algebra.measure M (\<Union>i. A i))"
1690     using A by (auto intro!: Lim_emeasure_incseq)
1691 qed auto
1693 lemma Lim_measure_decseq:
1694   assumes A: "range A \<subseteq> sets M" "decseq A" and fin: "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
1695   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"
1696 proof (rule tendsto_ennrealD)
1697   have "ennreal (measure M (\<Inter>i. A i)) = emeasure M (\<Inter>i. A i)"
1698     using fin[of 0] A emeasure_mono[of "\<Inter>i. A i" "A 0" M]
1699     by (auto intro!: emeasure_eq_ennreal_measure[symmetric] simp: INT_lower less_top intro: le_less_trans)
1700   moreover have "ennreal (measure M (A i)) = emeasure M (A i)" for i
1701     using A fin[of i] by (intro emeasure_eq_ennreal_measure[symmetric]) auto
1702   ultimately show "(\<lambda>x. ennreal (Sigma_Algebra.measure M (A x))) \<longlonglongrightarrow> ennreal (Sigma_Algebra.measure M (\<Inter>i. A i))"
1703     using fin A by (auto intro!: Lim_emeasure_decseq)
1704 qed auto
1706 subsection \<open>Set of measurable sets with finite measure\<close>
1708 definition fmeasurable :: "'a measure \<Rightarrow> 'a set set"
1709 where
1710   "fmeasurable M = {A\<in>sets M. emeasure M A < \<infinity>}"
1712 lemma fmeasurableD[dest, measurable_dest]: "A \<in> fmeasurable M \<Longrightarrow> A \<in> sets M"
1713   by (auto simp: fmeasurable_def)
1715 lemma fmeasurableD2: "A \<in> fmeasurable M \<Longrightarrow> emeasure M A \<noteq> top"
1716   by (auto simp: fmeasurable_def)
1718 lemma fmeasurableI: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> A \<in> fmeasurable M"
1719   by (auto simp: fmeasurable_def)
1721 lemma fmeasurableI_null_sets: "A \<in> null_sets M \<Longrightarrow> A \<in> fmeasurable M"
1722   by (auto simp: fmeasurable_def)
1724 lemma fmeasurableI2: "A \<in> fmeasurable M \<Longrightarrow> B \<subseteq> A \<Longrightarrow> B \<in> sets M \<Longrightarrow> B \<in> fmeasurable M"
1725   using emeasure_mono[of B A M] by (auto simp: fmeasurable_def)
1727 lemma measure_mono_fmeasurable:
1728   "A \<subseteq> B \<Longrightarrow> A \<in> sets M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow> measure M A \<le> measure M B"
1729   by (auto simp: measure_def fmeasurable_def intro!: emeasure_mono enn2real_mono)
1731 lemma emeasure_eq_measure2: "A \<in> fmeasurable M \<Longrightarrow> emeasure M A = measure M A"
1732   by (simp add: emeasure_eq_ennreal_measure fmeasurable_def less_top)
1734 interpretation fmeasurable: ring_of_sets "space M" "fmeasurable M"
1735 proof (rule ring_of_setsI)
1736   show "fmeasurable M \<subseteq> Pow (space M)" "{} \<in> fmeasurable M"
1737     by (auto simp: fmeasurable_def dest: sets.sets_into_space)
1738   fix a b assume *: "a \<in> fmeasurable M" "b \<in> fmeasurable M"
1739   then have "emeasure M (a \<union> b) \<le> emeasure M a + emeasure M b"
1741   also have "\<dots> < top"
1742     using * by (auto simp: fmeasurable_def)
1743   finally show  "a \<union> b \<in> fmeasurable M"
1744     using * by (auto intro: fmeasurableI)
1745   show "a - b \<in> fmeasurable M"
1746     using emeasure_mono[of "a - b" a M] * by (auto simp: fmeasurable_def Diff_subset)
1747 qed
1749 lemma fmeasurable_Diff: "A \<in> fmeasurable M \<Longrightarrow> B \<in> sets M \<Longrightarrow> A - B \<in> fmeasurable M"
1750   using fmeasurableI2[of A M "A - B"] by auto
1752 lemma fmeasurable_UN:
1753   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"
1754   shows "(\<Union>i\<in>I. F i) \<in> fmeasurable M"
1755 proof (rule fmeasurableI2)
1756   show "A \<in> fmeasurable M" "(\<Union>i\<in>I. F i) \<subseteq> A" using assms by auto
1757   show "(\<Union>i\<in>I. F i) \<in> sets M"
1758     using assms by (intro sets.countable_UN') auto
1759 qed
1761 lemma fmeasurable_INT:
1762   assumes "countable I" "i \<in> I" "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" "F i \<in> fmeasurable M"
1763   shows "(\<Inter>i\<in>I. F i) \<in> fmeasurable M"
1764 proof (rule fmeasurableI2)
1765   show "F i \<in> fmeasurable M" "(\<Inter>i\<in>I. F i) \<subseteq> F i"
1766     using assms by auto
1767   show "(\<Inter>i\<in>I. F i) \<in> sets M"
1768     using assms by (intro sets.countable_INT') auto
1769 qed
1771 lemma measure_Un2:
1772   "A \<in> fmeasurable M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"
1773   using measure_Union[of M A "B - A"] by (auto simp: fmeasurableD2 fmeasurable.Diff)
1775 lemma measure_Un3:
1776   assumes "A \<in> fmeasurable M" "B \<in> fmeasurable M"
1777   shows "measure M (A \<union> B) = measure M A + measure M B - measure M (A \<inter> B)"
1778 proof -
1779   have "measure M (A \<union> B) = measure M A + measure M (B - A)"
1780     using assms by (rule measure_Un2)
1781   also have "B - A = B - (A \<inter> B)"
1782     by auto
1783   also have "measure M (B - (A \<inter> B)) = measure M B - measure M (A \<inter> B)"
1784     using assms by (intro measure_Diff) (auto simp: fmeasurable_def)
1785   finally show ?thesis
1786     by simp
1787 qed
1789 lemma measure_Un_AE:
1790   "AE x in M. x \<notin> A \<or> x \<notin> B \<Longrightarrow> A \<in> fmeasurable M \<Longrightarrow> B \<in> fmeasurable M \<Longrightarrow>
1791   measure M (A \<union> B) = measure M A + measure M B"
1792   by (subst measure_Un2) (auto intro!: measure_eq_AE)
1794 lemma measure_UNION_AE:
1795   assumes I: "finite I"
1796   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>
1797     measure M (\<Union>i\<in>I. F i) = (\<Sum>i\<in>I. measure M (F i))"
1798   unfolding AE_pairwise[OF countable_finite, OF I]
1799   using I
1800   apply (induction I rule: finite_induct)
1801    apply simp
1803   apply (subst measure_Un_AE)
1804   apply auto
1805   done
1807 lemma measure_UNION':
1808   "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>
1809     measure M (\<Union>i\<in>I. F i) = (\<Sum>i\<in>I. measure M (F i))"
1810   by (intro measure_UNION_AE) (auto simp: disjnt_def elim!: pairwise_mono intro!: always_eventually)
1812 lemma measure_Union_AE:
1813   "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>
1814     measure M (\<Union>F) = (\<Sum>S\<in>F. measure M S)"
1815   using measure_UNION_AE[of F "\<lambda>x. x" M] by simp
1817 lemma measure_Union':
1818   "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)"
1819   using measure_UNION'[of F "\<lambda>x. x" M] by simp
1821 lemma measure_Un_le:
1822   assumes "A \<in> sets M" "B \<in> sets M" shows "measure M (A \<union> B) \<le> measure M A + measure M B"
1823 proof cases
1824   assume "A \<in> fmeasurable M \<and> B \<in> fmeasurable M"
1825   with measure_subadditive[of A M B] assms show ?thesis
1826     by (auto simp: fmeasurableD2)
1827 next
1828   assume "\<not> (A \<in> fmeasurable M \<and> B \<in> fmeasurable M)"
1829   then have "A \<union> B \<notin> fmeasurable M"
1830     using fmeasurableI2[of "A \<union> B" M A] fmeasurableI2[of "A \<union> B" M B] assms by auto
1831   with assms show ?thesis
1832     by (auto simp: fmeasurable_def measure_def less_top[symmetric])
1833 qed
1835 lemma measure_UNION_le:
1836   "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))"
1837 proof (induction I rule: finite_induct)
1838   case (insert i I)
1839   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)"
1840     by (auto intro!: measure_Un_le)
1841   also have "measure M (\<Union>i\<in>I. F i) \<le> (\<Sum>i\<in>I. measure M (F i))"
1842     using insert by auto
1843   finally show ?case
1844     using insert by simp
1845 qed simp
1847 lemma measure_Union_le:
1848   "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)"
1849   using measure_UNION_le[of F "\<lambda>x. x" M] by simp
1851 lemma
1852   assumes "countable I" and I: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M"
1853     and bound: "\<And>I'. I' \<subseteq> I \<Longrightarrow> finite I' \<Longrightarrow> measure M (\<Union>i\<in>I'. A i) \<le> B" and "0 \<le> B"
1854   shows fmeasurable_UN_bound: "(\<Union>i\<in>I. A i) \<in> fmeasurable M" (is ?fm)
1855     and measure_UN_bound: "measure M (\<Union>i\<in>I. A i) \<le> B" (is ?m)
1856 proof -
1857   have "?fm \<and> ?m"
1858   proof cases
1859     assume "I = {}" with \<open>0 \<le> B\<close> show ?thesis by simp
1860   next
1861     assume "I \<noteq> {}"
1862     have "(\<Union>i\<in>I. A i) = (\<Union>i. (\<Union>n\<le>i. A (from_nat_into I n)))"
1863       by (subst range_from_nat_into[symmetric, OF \<open>I \<noteq> {}\<close> \<open>countable I\<close>]) auto
1864     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
1865     also have "\<dots> = (SUP i. emeasure M (\<Union>n\<le>i. A (from_nat_into I n)))"
1866       using I \<open>I \<noteq> {}\<close>[THEN from_nat_into] by (intro SUP_emeasure_incseq[symmetric]) (fastforce simp: incseq_Suc_iff)+
1867     also have "\<dots> \<le> B"
1868     proof (intro SUP_least)
1869       fix i :: nat
1870       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))"
1871         using I \<open>I \<noteq> {}\<close>[THEN from_nat_into] by (intro emeasure_eq_measure2 fmeasurable.finite_UN) auto
1872       also have "\<dots> = measure M (\<Union>n\<in>from_nat_into I  {..i}. A n)"
1873         by simp
1874       also have "\<dots> \<le> B"
1875         by (intro ennreal_leI bound) (auto intro:  from_nat_into[OF \<open>I \<noteq> {}\<close>])
1876       finally show "emeasure M (\<Union>n\<le>i. A (from_nat_into I n)) \<le> ennreal B" .
1877     qed
1878     finally have *: "emeasure M (\<Union>i\<in>I. A i) \<le> B" .
1879     then have ?fm
1880       using I \<open>countable I\<close> by (intro fmeasurableI conjI) (auto simp: less_top[symmetric] top_unique)
1881     with * \<open>0\<le>B\<close> show ?thesis
1883   qed
1884   then show ?fm ?m by auto
1885 qed
1887 lemma suminf_exist_split2:
1888   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
1889   assumes "summable f"
1890   shows "(\<lambda>n. (\<Sum>k. f(k+n))) \<longlonglongrightarrow> 0"
1891 by (subst lim_sequentially, auto simp add: dist_norm suminf_exist_split[OF _ assms])
1893 lemma emeasure_union_summable:
1894   assumes [measurable]: "\<And>n. A n \<in> sets M"
1895     and "\<And>n. emeasure M (A n) < \<infinity>" "summable (\<lambda>n. measure M (A n))"
1896   shows "emeasure M (\<Union>n. A n) < \<infinity>" "emeasure M (\<Union>n. A n) \<le> (\<Sum>n. measure M (A n))"
1897 proof -
1898   define B where "B = (\<lambda>N. (\<Union>n\<in>{..<N}. A n))"
1899   have [measurable]: "B N \<in> sets M" for N unfolding B_def by auto
1900   have "(\<lambda>N. emeasure M (B N)) \<longlonglongrightarrow> emeasure M (\<Union>N. B N)"
1901     apply (rule Lim_emeasure_incseq) unfolding B_def by (auto simp add: SUP_subset_mono incseq_def)
1902   moreover have "emeasure M (B N) \<le> ennreal (\<Sum>n. measure M (A n))" for N
1903   proof -
1904     have *: "(\<Sum>n\<in>{..<N}. measure M (A n)) \<le> (\<Sum>n. measure M (A n))"
1905       using assms(3) measure_nonneg sum_le_suminf by blast
1907     have "emeasure M (B N) \<le> (\<Sum>n\<in>{..<N}. emeasure M (A n))"
1908       unfolding B_def by (rule emeasure_subadditive_finite, auto)
1909     also have "... = (\<Sum>n\<in>{..<N}. ennreal(measure M (A n)))"
1910       using assms(2) by (simp add: emeasure_eq_ennreal_measure less_top)
1911     also have "... = ennreal (\<Sum>n\<in>{..<N}. measure M (A n))"
1912       by auto
1913     also have "... \<le> ennreal (\<Sum>n. measure M (A n))"
1914       using * by (auto simp: ennreal_leI)
1915     finally show ?thesis by simp
1916   qed
1917   ultimately have "emeasure M (\<Union>N. B N) \<le> ennreal (\<Sum>n. measure M (A n))"
1919   then show "emeasure M (\<Union>n. A n) \<le> (\<Sum>n. measure M (A n))"
1920     unfolding B_def by (metis UN_UN_flatten UN_lessThan_UNIV)
1921   then show "emeasure M (\<Union>n. A n) < \<infinity>"
1922     by (auto simp: less_top[symmetric] top_unique)
1923 qed
1925 lemma borel_cantelli_limsup1:
1926   assumes [measurable]: "\<And>n. A n \<in> sets M"
1927     and "\<And>n. emeasure M (A n) < \<infinity>" "summable (\<lambda>n. measure M (A n))"
1928   shows "limsup A \<in> null_sets M"
1929 proof -
1930   have "emeasure M (limsup A) \<le> 0"
1931   proof (rule LIMSEQ_le_const)
1932     have "(\<lambda>n. (\<Sum>k. measure M (A (k+n)))) \<longlonglongrightarrow> 0" by (rule suminf_exist_split2[OF assms(3)])
1933     then show "(\<lambda>n. ennreal (\<Sum>k. measure M (A (k+n)))) \<longlonglongrightarrow> 0"
1934       unfolding ennreal_0[symmetric] by (intro tendsto_ennrealI)
1935     have "emeasure M (limsup A) \<le> (\<Sum>k. measure M (A (k+n)))" for n
1936     proof -
1937       have I: "(\<Union>k\<in>{n..}. A k) = (\<Union>k. A (k+n))" by (auto, metis le_add_diff_inverse2, fastforce)
1938       have "emeasure M (limsup A) \<le> emeasure M (\<Union>k\<in>{n..}. A k)"
1939         by (rule emeasure_mono, auto simp add: limsup_INF_SUP)
1940       also have "... = emeasure M (\<Union>k. A (k+n))"
1941         using I by auto
1942       also have "... \<le> (\<Sum>k. measure M (A (k+n)))"
1943         apply (rule emeasure_union_summable)
1944         using assms summable_ignore_initial_segment[OF assms(3), of n] by auto
1945       finally show ?thesis by simp
1946     qed
1947     then show "\<exists>N. \<forall>n\<ge>N. emeasure M (limsup A) \<le> (\<Sum>k. measure M (A (k+n)))"
1948       by auto
1949   qed
1950   then show ?thesis using assms(1) measurable_limsup by auto
1951 qed
1953 lemma borel_cantelli_AE1:
1954   assumes [measurable]: "\<And>n. A n \<in> sets M"
1955     and "\<And>n. emeasure M (A n) < \<infinity>" "summable (\<lambda>n. measure M (A n))"
1956   shows "AE x in M. eventually (\<lambda>n. x \<in> space M - A n) sequentially"
1957 proof -
1958   have "AE x in M. x \<notin> limsup A"
1959     using borel_cantelli_limsup1[OF assms] unfolding eventually_ae_filter by auto
1960   moreover
1961   {
1962     fix x assume "x \<notin> limsup A"
1963     then obtain N where "x \<notin> (\<Union>n\<in>{N..}. A n)" unfolding limsup_INF_SUP by blast
1964     then have "eventually (\<lambda>n. x \<notin> A n) sequentially" using eventually_sequentially by auto
1965   }
1966   ultimately show ?thesis by auto
1967 qed
1969 subsection \<open>Measure spaces with @{term "emeasure M (space M) < \<infinity>"}\<close>
1971 locale finite_measure = sigma_finite_measure M for M +
1972   assumes finite_emeasure_space: "emeasure M (space M) \<noteq> top"
1974 lemma finite_measureI[Pure.intro!]:
1975   "emeasure M (space M) \<noteq> \<infinity> \<Longrightarrow> finite_measure M"
1976   proof qed (auto intro!: exI[of _ "{space M}"])
1978 lemma (in finite_measure) emeasure_finite[simp, intro]: "emeasure M A \<noteq> top"
1979   using finite_emeasure_space emeasure_space[of M A] by (auto simp: top_unique)
1981 lemma (in finite_measure) fmeasurable_eq_sets: "fmeasurable M = sets M"
1982   by (auto simp: fmeasurable_def less_top[symmetric])
1984 lemma (in finite_measure) emeasure_eq_measure: "emeasure M A = ennreal (measure M A)"
1985   by (intro emeasure_eq_ennreal_measure) simp
1987 lemma (in finite_measure) emeasure_real: "\<exists>r. 0 \<le> r \<and> emeasure M A = ennreal r"
1988   using emeasure_finite[of A] by (cases "emeasure M A" rule: ennreal_cases) auto
1990 lemma (in finite_measure) bounded_measure: "measure M A \<le> measure M (space M)"
1991   using emeasure_space[of M A] emeasure_real[of A] emeasure_real[of "space M"] by (auto simp: measure_def)
1993 lemma (in finite_measure) finite_measure_Diff:
1994   assumes sets: "A \<in> sets M" "B \<in> sets M" and "B \<subseteq> A"
1995   shows "measure M (A - B) = measure M A - measure M B"
1996   using measure_Diff[OF _ assms] by simp
1998 lemma (in finite_measure) finite_measure_Union:
1999   assumes sets: "A \<in> sets M" "B \<in> sets M" and "A \<inter> B = {}"
2000   shows "measure M (A \<union> B) = measure M A + measure M B"
2001   using measure_Union[OF _ _ assms] by simp
2003 lemma (in finite_measure) finite_measure_finite_Union:
2004   assumes measurable: "finite S" "AS \<subseteq> sets M" "disjoint_family_on A S"
2005   shows "measure M (\<Union>i\<in>S. A i) = (\<Sum>i\<in>S. measure M (A i))"
2006   using measure_finite_Union[OF assms] by simp
2008 lemma (in finite_measure) finite_measure_UNION:
2009   assumes A: "range A \<subseteq> sets M" "disjoint_family A"
2010   shows "(\<lambda>i. measure M (A i)) sums (measure M (\<Union>i. A i))"
2011   using measure_UNION[OF A] by simp
2013 lemma (in finite_measure) finite_measure_mono:
2014   assumes "A \<subseteq> B" "B \<in> sets M" shows "measure M A \<le> measure M B"
2015   using emeasure_mono[OF assms] emeasure_real[of A] emeasure_real[of B] by (auto simp: measure_def)
2018   assumes m: "A \<in> sets M" "B \<in> sets M"
2019   shows "measure M (A \<union> B) \<le> measure M A + measure M B"
2020   using measure_subadditive[OF m] by simp
2023   assumes "finite I" "AI \<subseteq> sets M" shows "measure M (\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. measure M (A i))"
2024   using measure_subadditive_finite[OF assms] by simp
2027   "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))"
2031 lemma (in finite_measure) finite_measure_eq_sum_singleton:
2032   assumes "finite S" and *: "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
2033   shows "measure M S = (\<Sum>x\<in>S. measure M {x})"
2034   using measure_eq_sum_singleton[OF assms] by simp
2036 lemma (in finite_measure) finite_Lim_measure_incseq:
2037   assumes A: "range A \<subseteq> sets M" "incseq A"
2038   shows "(\<lambda>i. measure M (A i)) \<longlonglongrightarrow> measure M (\<Union>i. A i)"
2039   using Lim_measure_incseq[OF A] by simp
2041 lemma (in finite_measure) finite_Lim_measure_decseq:
2042   assumes A: "range A \<subseteq> sets M" "decseq A"
2043   shows "(\<lambda>n. measure M (A n)) \<longlonglongrightarrow> measure M (\<Inter>i. A i)"
2044   using Lim_measure_decseq[OF A] by simp
2046 lemma (in finite_measure) finite_measure_compl:
2047   assumes S: "S \<in> sets M"
2048   shows "measure M (space M - S) = measure M (space M) - measure M S"
2049   using measure_Diff[OF _ sets.top S sets.sets_into_space] S by simp
2051 lemma (in finite_measure) finite_measure_mono_AE:
2052   assumes imp: "AE x in M. x \<in> A \<longrightarrow> x \<in> B" and B: "B \<in> sets M"
2053   shows "measure M A \<le> measure M B"
2054   using assms emeasure_mono_AE[OF imp B]
2057 lemma (in finite_measure) finite_measure_eq_AE:
2058   assumes iff: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
2059   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
2060   shows "measure M A = measure M B"
2061   using assms emeasure_eq_AE[OF assms] by (simp add: emeasure_eq_measure)
2063 lemma (in finite_measure) measure_increasing: "increasing M (measure M)"
2064   by (auto intro!: finite_measure_mono simp: increasing_def)
2066 lemma (in finite_measure) measure_zero_union:
2067   assumes "s \<in> sets M" "t \<in> sets M" "measure M t = 0"
2068   shows "measure M (s \<union> t) = measure M s"
2069 using assms
2070 proof -
2071   have "measure M (s \<union> t) \<le> measure M s"
2072     using finite_measure_subadditive[of s t] assms by auto
2073   moreover have "measure M (s \<union> t) \<ge> measure M s"
2074     using assms by (blast intro: finite_measure_mono)
2075   ultimately show ?thesis by simp
2076 qed
2078 lemma (in finite_measure) measure_eq_compl:
2079   assumes "s \<in> sets M" "t \<in> sets M"
2080   assumes "measure M (space M - s) = measure M (space M - t)"
2081   shows "measure M s = measure M t"
2082   using assms finite_measure_compl by auto
2084 lemma (in finite_measure) measure_eq_bigunion_image:
2085   assumes "range f \<subseteq> sets M" "range g \<subseteq> sets M"
2086   assumes "disjoint_family f" "disjoint_family g"
2087   assumes "\<And> n :: nat. measure M (f n) = measure M (g n)"
2088   shows "measure M (\<Union>i. f i) = measure M (\<Union>i. g i)"
2089 using assms
2090 proof -
2091   have a: "(\<lambda> i. measure M (f i)) sums (measure M (\<Union>i. f i))"
2092     by (rule finite_measure_UNION[OF assms(1,3)])
2093   have b: "(\<lambda> i. measure M (g i)) sums (measure M (\<Union>i. g i))"
2094     by (rule finite_measure_UNION[OF assms(2,4)])
2095   show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
2096 qed
2098 lemma (in finite_measure) measure_countably_zero:
2099   assumes "range c \<subseteq> sets M"
2100   assumes "\<And> i. measure M (c i) = 0"
2101   shows "measure M (\<Union>i :: nat. c i) = 0"
2102 proof (rule antisym)
2103   show "measure M (\<Union>i :: nat. c i) \<le> 0"
2105 qed simp
2107 lemma (in finite_measure) measure_space_inter:
2108   assumes events:"s \<in> sets M" "t \<in> sets M"
2109   assumes "measure M t = measure M (space M)"
2110   shows "measure M (s \<inter> t) = measure M s"
2111 proof -
2112   have "measure M ((space M - s) \<union> (space M - t)) = measure M (space M - s)"
2113     using events assms finite_measure_compl[of "t"] by (auto intro!: measure_zero_union)
2114   also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"
2115     by blast
2116   finally show "measure M (s \<inter> t) = measure M s"
2117     using events by (auto intro!: measure_eq_compl[of "s \<inter> t" s])
2118 qed
2120 lemma (in finite_measure) measure_equiprobable_finite_unions:
2121   assumes s: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> sets M"
2122   assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> measure M {x} = measure M {y}"
2123   shows "measure M s = real (card s) * measure M {SOME x. x \<in> s}"
2124 proof cases
2125   assume "s \<noteq> {}"
2126   then have "\<exists> x. x \<in> s" by blast
2127   from someI_ex[OF this] assms
2128   have prob_some: "\<And> x. x \<in> s \<Longrightarrow> measure M {x} = measure M {SOME y. y \<in> s}" by blast
2129   have "measure M s = (\<Sum> x \<in> s. measure M {x})"
2130     using finite_measure_eq_sum_singleton[OF s] by simp
2131   also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto
2132   also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"
2133     using sum_constant assms by simp
2134   finally show ?thesis by simp
2135 qed simp
2137 lemma (in finite_measure) measure_real_sum_image_fn:
2138   assumes "e \<in> sets M"
2139   assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> sets M"
2140   assumes "finite s"
2141   assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
2142   assumes upper: "space M \<subseteq> (\<Union>i \<in> s. f i)"
2143   shows "measure M e = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
2144 proof -
2145   have "e \<subseteq> (\<Union>i\<in>s. f i)"
2146     using \<open>e \<in> sets M\<close> sets.sets_into_space upper by blast
2147   then have e: "e = (\<Union>i \<in> s. e \<inter> f i)"
2148     by auto
2149   hence "measure M e = measure M (\<Union>i \<in> s. e \<inter> f i)" by simp
2150   also have "\<dots> = (\<Sum> x \<in> s. measure M (e \<inter> f x))"
2151   proof (rule finite_measure_finite_Union)
2152     show "finite s" by fact
2153     show "(\<lambda>i. e \<inter> f i)s \<subseteq> sets M" using assms(2) by auto
2154     show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
2155       using disjoint by (auto simp: disjoint_family_on_def)
2156   qed
2157   finally show ?thesis .
2158 qed
2160 lemma (in finite_measure) measure_exclude:
2161   assumes "A \<in> sets M" "B \<in> sets M"
2162   assumes "measure M A = measure M (space M)" "A \<inter> B = {}"
2163   shows "measure M B = 0"
2164   using measure_space_inter[of B A] assms by (auto simp: ac_simps)
2165 lemma (in finite_measure) finite_measure_distr:
2166   assumes f: "f \<in> measurable M M'"
2167   shows "finite_measure (distr M M' f)"
2168 proof (rule finite_measureI)
2169   have "f - space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
2170   with f show "emeasure (distr M M' f) (space (distr M M' f)) \<noteq> \<infinity>" by (auto simp: emeasure_distr)
2171 qed
2173 lemma emeasure_gfp[consumes 1, case_names cont measurable]:
2174   assumes sets[simp]: "\<And>s. sets (M s) = sets N"
2175   assumes "\<And>s. finite_measure (M s)"
2176   assumes cont: "inf_continuous F" "inf_continuous f"
2177   assumes meas: "\<And>P. Measurable.pred N P \<Longrightarrow> Measurable.pred N (F P)"
2178   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"
2179   assumes bound: "\<And>P. f P \<le> f (\<lambda>s. emeasure (M s) (space (M s)))"
2180   shows "emeasure (M s) {x\<in>space N. gfp F x} = gfp f s"
2181 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
2182     P="Measurable.pred N", symmetric])
2183   interpret finite_measure "M s" for s by fact
2184   fix C assume "decseq C" "\<And>i. Measurable.pred N (C i)"
2185   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}))"
2186     unfolding INF_apply[abs_def]
2187     by (subst INF_emeasure_decseq) (auto simp: antimono_def fun_eq_iff intro!: arg_cong2[where f=emeasure])
2188 next
2189   show "f x \<le> (\<lambda>s. emeasure (M s) {x \<in> space N. F top x})" for x
2190     using bound[of x] sets_eq_imp_space_eq[OF sets] by (simp add: iter)
2191 qed (auto simp add: iter le_fun_def INF_apply[abs_def] intro!: meas cont)
2193 subsection \<open>Counting space\<close>
2195 lemma strict_monoI_Suc:
2196   assumes ord [simp]: "(\<And>n. f n < f (Suc n))" shows "strict_mono f"
2197   unfolding strict_mono_def
2198 proof safe
2199   fix n m :: nat assume "n < m" then show "f n < f m"
2200     by (induct m) (auto simp: less_Suc_eq intro: less_trans ord)
2201 qed
2203 lemma emeasure_count_space:
2204   assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else \<infinity>)"
2205     (is "_ = ?M X")
2206   unfolding count_space_def
2207 proof (rule emeasure_measure_of_sigma)
2208   show "X \<in> Pow A" using \<open>X \<subseteq> A\<close> by auto
2209   show "sigma_algebra A (Pow A)" by (rule sigma_algebra_Pow)
2210   show positive: "positive (Pow A) ?M"
2211     by (auto simp: positive_def)
2213     by (auto simp: card_Un_disjoint additive_def)
2215   interpret ring_of_sets A "Pow A"
2216     by (rule ring_of_setsI) auto
2217   show "countably_additive (Pow A) ?M"
2219   proof safe
2220     fix F :: "nat \<Rightarrow> 'a set" assume "incseq F"
2221     show "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> ?M (\<Union>i. F i)"
2222     proof cases
2223       assume "\<exists>i. \<forall>j\<ge>i. F i = F j"
2224       then guess i .. note i = this
2225       { fix j from i \<open>incseq F\<close> have "F j \<subseteq> F i"
2226           by (cases "i \<le> j") (auto simp: incseq_def) }
2227       then have eq: "(\<Union>i. F i) = F i"
2228         by auto
2229       with i show ?thesis
2230         by (auto intro!: Lim_transform_eventually[OF _ tendsto_const] eventually_sequentiallyI[where c=i])
2231     next
2232       assume "\<not> (\<exists>i. \<forall>j\<ge>i. F i = F j)"
2233       then obtain f where f: "\<And>i. i \<le> f i" "\<And>i. F i \<noteq> F (f i)" by metis
2234       then have "\<And>i. F i \<subseteq> F (f i)" using \<open>incseq F\<close> by (auto simp: incseq_def)
2235       with f have *: "\<And>i. F i \<subset> F (f i)" by auto
2237       have "incseq (\<lambda>i. ?M (F i))"
2238         using \<open>incseq F\<close> unfolding incseq_def by (auto simp: card_mono dest: finite_subset)
2239       then have "(\<lambda>i. ?M (F i)) \<longlonglongrightarrow> (SUP n. ?M (F n))"
2240         by (rule LIMSEQ_SUP)
2242       moreover have "(SUP n. ?M (F n)) = top"
2243       proof (rule ennreal_SUP_eq_top)
2244         fix n :: nat show "\<exists>k::nat\<in>UNIV. of_nat n \<le> ?M (F k)"
2245         proof (induct n)
2246           case (Suc n)
2247           then guess k .. note k = this
2248           moreover have "finite (F k) \<Longrightarrow> finite (F (f k)) \<Longrightarrow> card (F k) < card (F (f k))"
2249             using \<open>F k \<subset> F (f k)\<close> by (simp add: psubset_card_mono)
2250           moreover have "finite (F (f k)) \<Longrightarrow> finite (F k)"
2251             using \<open>k \<le> f k\<close> \<open>incseq F\<close> by (auto simp: incseq_def dest: finite_subset)
2252           ultimately show ?case
2253             by (auto intro!: exI[of _ "f k"] simp del: of_nat_Suc)
2254         qed auto
2255       qed
2257       moreover
2258       have "inj (\<lambda>n. F ((f ^^ n) 0))"
2259         by (intro strict_mono_imp_inj_on strict_monoI_Suc) (simp add: *)
2260       then have 1: "infinite (range (\<lambda>i. F ((f ^^ i) 0)))"
2261         by (rule range_inj_infinite)
2262       have "infinite (Pow (\<Union>i. F i))"
2263         by (rule infinite_super[OF _ 1]) auto
2264       then have "infinite (\<Union>i. F i)"
2265         by auto
2267       ultimately show ?thesis by auto
2268     qed
2269   qed
2270 qed
2272 lemma distr_bij_count_space:
2273   assumes f: "bij_betw f A B"
2274   shows "distr (count_space A) (count_space B) f = count_space B"
2275 proof (rule measure_eqI)
2276   have f': "f \<in> measurable (count_space A) (count_space B)"
2277     using f unfolding Pi_def bij_betw_def by auto
2278   fix X assume "X \<in> sets (distr (count_space A) (count_space B) f)"
2279   then have X: "X \<in> sets (count_space B)" by auto
2280   moreover from X have "f - X \<inter> A = the_inv_into A f  X"
2281     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])
2282   moreover have "inj_on (the_inv_into A f) B"
2283     using X f by (auto simp: bij_betw_def inj_on_the_inv_into)
2284   with X have "inj_on (the_inv_into A f) X"
2285     by (auto intro: subset_inj_on)
2286   ultimately show "emeasure (distr (count_space A) (count_space B) f) X = emeasure (count_space B) X"
2287     using f unfolding emeasure_distr[OF f' X]
2288     by (subst (1 2) emeasure_count_space) (auto simp: card_image dest: finite_imageD)
2289 qed simp
2291 lemma emeasure_count_space_finite[simp]:
2292   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> emeasure (count_space A) X = of_nat (card X)"
2293   using emeasure_count_space[of X A] by simp
2295 lemma emeasure_count_space_infinite[simp]:
2296   "X \<subseteq> A \<Longrightarrow> infinite X \<Longrightarrow> emeasure (count_space A) X = \<infinity>"
2297   using emeasure_count_space[of X A] by simp
2299 lemma measure_count_space: "measure (count_space A) X = (if X \<subseteq> A then of_nat (card X) else 0)"
2300   by (cases "finite X") (auto simp: measure_notin_sets ennreal_of_nat_eq_real_of_nat
2301                                     measure_zero_top measure_eq_emeasure_eq_ennreal)
2303 lemma emeasure_count_space_eq_0:
2304   "emeasure (count_space A) X = 0 \<longleftrightarrow> (X \<subseteq> A \<longrightarrow> X = {})"
2305 proof cases
2306   assume X: "X \<subseteq> A"
2307   then show ?thesis
2308   proof (intro iffI impI)
2309     assume "emeasure (count_space A) X = 0"
2310     with X show "X = {}"
2311       by (subst (asm) emeasure_count_space) (auto split: if_split_asm)
2312   qed simp
2315 lemma null_sets_count_space: "null_sets (count_space A) = { {} }"
2316   unfolding null_sets_def by (auto simp add: emeasure_count_space_eq_0)
2318 lemma AE_count_space: "(AE x in count_space A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
2319   unfolding eventually_ae_filter by (auto simp add: null_sets_count_space)
2321 lemma sigma_finite_measure_count_space_countable:
2322   assumes A: "countable A"
2323   shows "sigma_finite_measure (count_space A)"
2324   proof qed (insert A, auto intro!: exI[of _ "(\<lambda>a. {a})  A"])
2326 lemma sigma_finite_measure_count_space:
2327   fixes A :: "'a::countable set" shows "sigma_finite_measure (count_space A)"
2328   by (rule sigma_finite_measure_count_space_countable) auto
2330 lemma finite_measure_count_space:
2331   assumes [simp]: "finite A"
2332   shows "finite_measure (count_space A)"
2333   by rule simp
2335 lemma sigma_finite_measure_count_space_finite:
2336   assumes A: "finite A" shows "sigma_finite_measure (count_space A)"
2337 proof -
2338   interpret finite_measure "count_space A" using A by (rule finite_measure_count_space)
2339   show "sigma_finite_measure (count_space A)" ..
2340 qed
2342 subsection \<open>Measure restricted to space\<close>
2344 lemma emeasure_restrict_space:
2345   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"
2346   shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"
2347 proof (cases "A \<in> sets M")
2348   case True
2349   show ?thesis
2350   proof (rule emeasure_measure_of[OF restrict_space_def])
2351     show "(\<inter>) \<Omega>  sets M \<subseteq> Pow (\<Omega> \<inter> space M)" "A \<in> sets (restrict_space M \<Omega>)"
2352       using \<open>A \<subseteq> \<Omega>\<close> \<open>A \<in> sets M\<close> sets.space_closed by (auto simp: sets_restrict_space)
2353     show "positive (sets (restrict_space M \<Omega>)) (emeasure M)"
2354       by (auto simp: positive_def)
2355     show "countably_additive (sets (restrict_space M \<Omega>)) (emeasure M)"
2357       fix A :: "nat \<Rightarrow> _" assume "range A \<subseteq> sets (restrict_space M \<Omega>)" "disjoint_family A"
2358       with assms have "\<And>i. A i \<in> sets M" "\<And>i. A i \<subseteq> space M" "disjoint_family A"
2359         by (fastforce simp: sets_restrict_space_iff[OF assms(1)] image_subset_iff
2360                       dest: sets.sets_into_space)+
2361       then show "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
2362         by (subst suminf_emeasure) (auto simp: disjoint_family_subset)
2363     qed
2364   qed
2365 next
2366   case False
2367   with assms have "A \<notin> sets (restrict_space M \<Omega>)"
2369   with False show ?thesis
2371 qed
2373 lemma measure_restrict_space:
2374   assumes "\<Omega> \<inter> space M \<in> sets M" "A \<subseteq> \<Omega>"
2375   shows "measure (restrict_space M \<Omega>) A = measure M A"
2376   using emeasure_restrict_space[OF assms] by (simp add: measure_def)
2378 lemma AE_restrict_space_iff:
2379   assumes "\<Omega> \<inter> space M \<in> sets M"
2380   shows "(AE x in restrict_space M \<Omega>. P x) \<longleftrightarrow> (AE x in M. x \<in> \<Omega> \<longrightarrow> P x)"
2381 proof -
2382   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)"
2383     by auto
2384   { fix X assume X: "X \<in> sets M" "emeasure M X = 0"
2385     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) \<le> emeasure M X"
2386       by (intro emeasure_mono) auto
2387     then have "emeasure M (\<Omega> \<inter> space M \<inter> X) = 0"
2388       using X by (auto intro!: antisym) }
2389   with assms show ?thesis
2390     unfolding eventually_ae_filter
2391     by (auto simp add: space_restrict_space null_sets_def sets_restrict_space_iff
2392                        emeasure_restrict_space cong: conj_cong
2393              intro!: ex_cong[where f="\<lambda>X. (\<Omega> \<inter> space M) \<inter> X"])
2394 qed
2396 lemma restrict_restrict_space:
2397   assumes "A \<inter> space M \<in> sets M" "B \<inter> space M \<in> sets M"
2398   shows "restrict_space (restrict_space M A) B = restrict_space M (A \<inter> B)" (is "?l = ?r")
2399 proof (rule measure_eqI[symmetric])
2400   show "sets ?r = sets ?l"
2401     unfolding sets_restrict_space image_comp by (intro image_cong) auto
2402 next
2403   fix X assume "X \<in> sets (restrict_space M (A \<inter> B))"
2404   then obtain Y where "Y \<in> sets M" "X = Y \<inter> A \<inter> B"
2405     by (auto simp: sets_restrict_space)
2406   with assms sets.Int[OF assms] show "emeasure ?r X = emeasure ?l X"
2407     by (subst (1 2) emeasure_restrict_space)
2408        (auto simp: space_restrict_space sets_restrict_space_iff emeasure_restrict_space ac_simps)
2409 qed
2411 lemma restrict_count_space: "restrict_space (count_space B) A = count_space (A \<inter> B)"
2412 proof (rule measure_eqI)
2413   show "sets (restrict_space (count_space B) A) = sets (count_space (A \<inter> B))"
2414     by (subst sets_restrict_space) auto
2415   moreover fix X assume "X \<in> sets (restrict_space (count_space B) A)"
2416   ultimately have "X \<subseteq> A \<inter> B" by auto
2417   then show "emeasure (restrict_space (count_space B) A) X = emeasure (count_space (A \<inter> B)) X"
2418     by (cases "finite X") (auto simp add: emeasure_restrict_space)
2419 qed
2421 lemma sigma_finite_measure_restrict_space:
2422   assumes "sigma_finite_measure M"
2423   and A: "A \<in> sets M"
2424   shows "sigma_finite_measure (restrict_space M A)"
2425 proof -
2426   interpret sigma_finite_measure M by fact
2427   from sigma_finite_countable obtain C
2428     where C: "countable C" "C \<subseteq> sets M" "(\<Union>C) = space M" "\<forall>a\<in>C. emeasure M a \<noteq> \<infinity>"
2429     by blast
2430   let ?C = "(\<inter>) A  C"
2431   from C have "countable ?C" "?C \<subseteq> sets (restrict_space M A)" "(\<Union>?C) = space (restrict_space M A)"
2432     by(auto simp add: sets_restrict_space space_restrict_space)
2433   moreover {
2434     fix a
2435     assume "a \<in> ?C"
2436     then obtain a' where "a = A \<inter> a'" "a' \<in> C" ..
2437     then have "emeasure (restrict_space M A) a \<le> emeasure M a'"
2438       using A C by(auto simp add: emeasure_restrict_space intro: emeasure_mono)
2439     also have "\<dots> < \<infinity>" using C(4)[rule_format, of a'] \<open>a' \<in> C\<close> by (simp add: less_top)
2440     finally have "emeasure (restrict_space M A) a \<noteq> \<infinity>" by simp }
2441   ultimately show ?thesis
2442     by unfold_locales (rule exI conjI|assumption|blast)+
2443 qed
2445 lemma finite_measure_restrict_space:
2446   assumes "finite_measure M"
2447   and A: "A \<in> sets M"
2448   shows "finite_measure (restrict_space M A)"
2449 proof -
2450   interpret finite_measure M by fact
2451   show ?thesis
2452     by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space)
2453 qed
2455 lemma restrict_distr:
2456   assumes [measurable]: "f \<in> measurable M N"
2457   assumes [simp]: "\<Omega> \<inter> space N \<in> sets N" and restrict: "f \<in> space M \<rightarrow> \<Omega>"
2458   shows "restrict_space (distr M N f) \<Omega> = distr M (restrict_space N \<Omega>) f"
2459   (is "?l = ?r")
2460 proof (rule measure_eqI)
2461   fix A assume "A \<in> sets (restrict_space (distr M N f) \<Omega>)"
2462   with restrict show "emeasure ?l A = emeasure ?r A"
2463     by (subst emeasure_distr)
2464        (auto simp: sets_restrict_space_iff emeasure_restrict_space emeasure_distr
2465              intro!: measurable_restrict_space2)
2468 lemma measure_eqI_restrict_generator:
2469   assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
2470   assumes sets_eq: "sets M = sets N" and \<Omega>: "\<Omega> \<in> sets M"
2471   assumes "sets (restrict_space M \<Omega>) = sigma_sets \<Omega> E"
2472   assumes "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E"
2473   assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>"
2474   assumes A: "countable A" "A \<noteq> {}" "A \<subseteq> E" "\<Union>A = \<Omega>" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
2475   shows "M = N"
2476 proof (rule measure_eqI)
2477   fix X assume X: "X \<in> sets M"
2478   then have "emeasure M X = emeasure (restrict_space M \<Omega>) (X \<inter> \<Omega>)"
2479     using ae \<Omega> by (auto simp add: emeasure_restrict_space intro!: emeasure_eq_AE)
2480   also have "restrict_space M \<Omega> = restrict_space N \<Omega>"
2481   proof (rule measure_eqI_generator_eq)
2482     fix X assume "X \<in> E"
2483     then show "emeasure (restrict_space M \<Omega>) X = emeasure (restrict_space N \<Omega>) X"
2484       using E \<Omega> by (subst (1 2) emeasure_restrict_space) (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq])
2485   next
2486     show "range (from_nat_into A) \<subseteq> E" "(\<Union>i. from_nat_into A i) = \<Omega>"
2487       using A by (auto cong del: strong_SUP_cong)
2488   next
2489     fix i
2490     have "emeasure (restrict_space M \<Omega>) (from_nat_into A i) = emeasure M (from_nat_into A i)"
2491       using A \<Omega> by (subst emeasure_restrict_space)
2492                    (auto simp: sets_eq sets_eq[THEN sets_eq_imp_space_eq] intro: from_nat_into)
2493     with A show "emeasure (restrict_space M \<Omega>) (from_nat_into A i) \<noteq> \<infinity>"
2494       by (auto intro: from_nat_into)
2495   qed fact+
2496   also have "emeasure (restrict_space N \<Omega>) (X \<inter> \<Omega>) = emeasure N X"
2497     using X ae \<Omega> by (auto simp add: emeasure_restrict_space sets_eq intro!: emeasure_eq_AE)
2498   finally show "emeasure M X = emeasure N X" .
2499 qed fact
2501 subsection \<open>Null measure\<close>
2503 definition "null_measure M = sigma (space M) (sets M)"
2505 lemma space_null_measure[simp]: "space (null_measure M) = space M"
2508 lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M"
2511 lemma emeasure_null_measure[simp]: "emeasure (null_measure M) X = 0"
2512   by (cases "X \<in> sets M", rule emeasure_measure_of)
2513      (auto simp: positive_def countably_additive_def emeasure_notin_sets null_measure_def
2514            dest: sets.sets_into_space)
2516 lemma measure_null_measure[simp]: "measure (null_measure M) X = 0"
2517   by (intro measure_eq_emeasure_eq_ennreal) auto
2519 lemma null_measure_idem [simp]: "null_measure (null_measure M) = null_measure M"
2520   by(rule measure_eqI) simp_all
2522 subsection \<open>Scaling a measure\<close>
2524 definition scale_measure :: "ennreal \<Rightarrow> 'a measure \<Rightarrow> 'a measure"
2525 where
2526   "scale_measure r M = measure_of (space M) (sets M) (\<lambda>A. r * emeasure M A)"
2528 lemma space_scale_measure: "space (scale_measure r M) = space M"
2531 lemma sets_scale_measure [simp, measurable_cong]: "sets (scale_measure r M) = sets M"
2534 lemma emeasure_scale_measure [simp]:
2535   "emeasure (scale_measure r M) A = r * emeasure M A"
2536   (is "_ = ?\<mu> A")
2537 proof(cases "A \<in> sets M")
2538   case True
2539   show ?thesis unfolding scale_measure_def
2540   proof(rule emeasure_measure_of_sigma)
2541     show "sigma_algebra (space M) (sets M)" ..
2542     show "positive (sets M) ?\<mu>" by (simp add: positive_def)
2543     show "countably_additive (sets M) ?\<mu>"
2545       fix A :: "nat \<Rightarrow> _"  assume *: "range A \<subseteq> sets M" "disjoint_family A"
2546       have "(\<Sum>i. ?\<mu> (A i)) = r * (\<Sum>i. emeasure M (A i))"
2547         by simp
2548       also have "\<dots> = ?\<mu> (\<Union>i. A i)" using * by(simp add: suminf_emeasure)
2549       finally show "(\<Sum>i. ?\<mu> (A i)) = ?\<mu> (\<Union>i. A i)" .
2550     qed
2551   qed(fact True)
2554 lemma scale_measure_1 [simp]: "scale_measure 1 M = M"
2555   by(rule measure_eqI) simp_all
2557 lemma scale_measure_0[simp]: "scale_measure 0 M = null_measure M"
2558   by(rule measure_eqI) simp_all
2560 lemma measure_scale_measure [simp]: "0 \<le> r \<Longrightarrow> measure (scale_measure r M) A = r * measure M A"
2561   using emeasure_scale_measure[of r M A]
2562     emeasure_eq_ennreal_measure[of M A]
2563     measure_eq_emeasure_eq_ennreal[of _ "scale_measure r M" A]
2564   by (cases "emeasure (scale_measure r M) A = top")
2565      (auto simp del: emeasure_scale_measure
2566            simp: ennreal_top_eq_mult_iff ennreal_mult_eq_top_iff measure_zero_top ennreal_mult[symmetric])
2568 lemma scale_scale_measure [simp]:
2569   "scale_measure r (scale_measure r' M) = scale_measure (r * r') M"
2570   by (rule measure_eqI) (simp_all add: max_def mult.assoc)
2572 lemma scale_null_measure [simp]: "scale_measure r (null_measure M) = null_measure M"
2573   by (rule measure_eqI) simp_all
2576 subsection \<open>Complete lattice structure on measures\<close>
2578 lemma (in finite_measure) finite_measure_Diff':
2579   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A - B) = measure M A - measure M (A \<inter> B)"
2580   using finite_measure_Diff[of A "A \<inter> B"] by (auto simp: Diff_Int)
2582 lemma (in finite_measure) finite_measure_Union':
2583   "A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M (B - A)"
2584   using finite_measure_Union[of A "B - A"] by auto
2586 lemma finite_unsigned_Hahn_decomposition:
2587   assumes "finite_measure M" "finite_measure N" and [simp]: "sets N = sets M"
2588   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)"
2589 proof -
2590   interpret M: finite_measure M by fact
2591   interpret N: finite_measure N by fact
2593   define d where "d X = measure M X - measure N X" for X
2595   have [intro]: "bdd_above (dsets M)"
2596     using sets.sets_into_space[of _ M]
2597     by (intro bdd_aboveI[where M="measure M (space M)"])
2598        (auto simp: d_def field_simps subset_eq intro!: add_increasing M.finite_measure_mono)
2600   define \<gamma> where "\<gamma> = (SUP X:sets M. d X)"
2601   have le_\<gamma>[intro]: "X \<in> sets M \<Longrightarrow> d X \<le> \<gamma>" for X
2602     by (auto simp: \<gamma>_def intro!: cSUP_upper)
2604   have "\<exists>f. \<forall>n. f n \<in> sets M \<and> d (f n) > \<gamma> - 1 / 2^n"
2605   proof (intro choice_iff[THEN iffD1] allI)
2606     fix n
2607     have "\<exists>X\<in>sets M. \<gamma> - 1 / 2^n < d X"
2608       unfolding \<gamma>_def by (intro less_cSUP_iff[THEN iffD1]) auto
2609     then show "\<exists>y. y \<in> sets M \<and> \<gamma> - 1 / 2 ^ n < d y"
2610       by auto
2611   qed
2612   then obtain E where [measurable]: "E n \<in> sets M" and E: "d (E n) > \<gamma> - 1 / 2^n" for n
2613     by auto
2615   define F where "F m n = (if m \<le> n then \<Inter>i\<in>{m..n}. E i else space M)" for m n
2617   have [measurable]: "m \<le> n \<Longrightarrow> F m n \<in> sets M" for m n
2618     by (auto simp: F_def)
2620   have 1: "\<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)" if "m \<le> n" for m n
2621     using that
2622   proof (induct rule: dec_induct)
2623     case base with E[of m] show ?case
2624       by (simp add: F_def field_simps)
2625   next
2626     case (step i)
2627     have F_Suc: "F m (Suc i) = F m i \<inter> E (Suc i)"
2628       using \<open>m \<le> i\<close> by (auto simp: F_def le_Suc_eq)
2630     have "\<gamma> + (\<gamma> - 2 / 2^m + 1 / 2 ^ Suc i) \<le> (\<gamma> - 1 / 2^Suc i) + (\<gamma> - 2 / 2^m + 1 / 2^i)"
2632     also have "\<dots> \<le> d (E (Suc i)) + d (F m i)"
2633       using E[of "Suc i"] by (intro add_mono step) auto
2634     also have "\<dots> = d (E (Suc i)) + d (F m i - E (Suc i)) + d (F m (Suc i))"
2635       using \<open>m \<le> i\<close> by (simp add: d_def field_simps F_Suc M.finite_measure_Diff' N.finite_measure_Diff')
2636     also have "\<dots> = d (E (Suc i) \<union> F m i) + d (F m (Suc i))"
2637       using \<open>m \<le> i\<close> by (simp add: d_def field_simps M.finite_measure_Union' N.finite_measure_Union')
2638     also have "\<dots> \<le> \<gamma> + d (F m (Suc i))"
2639       using \<open>m \<le> i\<close> by auto
2640     finally show ?case
2641       by auto
2642   qed
2644   define F' where "F' m = (\<Inter>i\<in>{m..}. E i)" for m
2645   have F'_eq: "F' m = (\<Inter>i. F m (i + m))" for m
2646     by (fastforce simp: le_iff_add[of m] F'_def F_def)
2648   have [measurable]: "F' m \<in> sets M" for m
2649     by (auto simp: F'_def)
2651   have \<gamma>_le: "\<gamma> - 0 \<le> d (\<Union>m. F' m)"
2652   proof (rule LIMSEQ_le)
2653     show "(\<lambda>n. \<gamma> - 2 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 0"
2654       by (intro tendsto_intros LIMSEQ_divide_realpow_zero) auto
2655     have "incseq F'"
2656       by (auto simp: incseq_def F'_def)
2657     then show "(\<lambda>m. d (F' m)) \<longlonglongrightarrow> d (\<Union>m. F' m)"
2658       unfolding d_def
2659       by (intro tendsto_diff M.finite_Lim_measure_incseq N.finite_Lim_measure_incseq) auto
2661     have "\<gamma> - 2 / 2 ^ m + 0 \<le> d (F' m)" for m
2662     proof (rule LIMSEQ_le)
2663       have *: "decseq (\<lambda>n. F m (n + m))"
2664         by (auto simp: decseq_def F_def)
2665       show "(\<lambda>n. d (F m n)) \<longlonglongrightarrow> d (F' m)"
2666         unfolding d_def F'_eq
2667         by (rule LIMSEQ_offset[where k=m])
2668            (auto intro!: tendsto_diff M.finite_Lim_measure_decseq N.finite_Lim_measure_decseq *)
2669       show "(\<lambda>n. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n) \<longlonglongrightarrow> \<gamma> - 2 / 2 ^ m + 0"
2670         by (intro tendsto_add LIMSEQ_divide_realpow_zero tendsto_const) auto
2671       show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)"
2672         using 1[of m] by (intro exI[of _ m]) auto
2673     qed
2674     then show "\<exists>N. \<forall>n\<ge>N. \<gamma> - 2 / 2 ^ n \<le> d (F' n)"
2675       by auto
2676   qed
2678   show ?thesis
2679   proof (safe intro!: bexI[of _ "\<Union>m. F' m"])
2680     fix X assume [measurable]: "X \<in> sets M" and X: "X \<subseteq> (\<Union>m. F' m)"
2681     have "d (\<Union>m. F' m) - d X = d ((\<Union>m. F' m) - X)"
2682       using X by (auto simp: d_def M.finite_measure_Diff N.finite_measure_Diff)
2683     also have "\<dots> \<le> \<gamma>"
2684       by auto
2685     finally have "0 \<le> d X"
2686       using \<gamma>_le by auto
2687     then show "emeasure N X \<le> emeasure M X"
2688       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)
2689   next
2690     fix X assume [measurable]: "X \<in> sets M" and X: "X \<inter> (\<Union>m. F' m) = {}"
2691     then have "d (\<Union>m. F' m) + d X = d (X \<union> (\<Union>m. F' m))"
2692       by (auto simp: d_def M.finite_measure_Union N.finite_measure_Union)
2693     also have "\<dots> \<le> \<gamma>"
2694       by auto
2695     finally have "d X \<le> 0"
2696       using \<gamma>_le by auto
2697     then show "emeasure M X \<le> emeasure N X"
2698       by (auto simp: d_def M.emeasure_eq_measure N.emeasure_eq_measure)
2699   qed auto
2700 qed
2702 lemma unsigned_Hahn_decomposition:
2703   assumes [simp]: "sets N = sets M" and [measurable]: "A \<in> sets M"
2704     and [simp]: "emeasure M A \<noteq> top" "emeasure N A \<noteq> top"
2705   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)"
2706 proof -
2707   have "\<exists>Y\<in>sets (restrict_space M A).
2708     (\<forall>X\<in>sets (restrict_space M A). X \<subseteq> Y \<longrightarrow> (restrict_space N A) X \<le> (restrict_space M A) X) \<and>
2709     (\<forall>X\<in>sets (restrict_space M A). X \<inter> Y = {} \<longrightarrow> (restrict_space M A) X \<le> (restrict_space N A) X)"
2710   proof (rule finite_unsigned_Hahn_decomposition)
2711     show "finite_measure (restrict_space M A)" "finite_measure (restrict_space N A)"
2712       by (auto simp: space_restrict_space emeasure_restrict_space less_top intro!: finite_measureI)
2714   then guess Y ..
2715   then show ?thesis
2716     apply (intro bexI[of _ Y] conjI ballI conjI)
2717     apply (simp_all add: sets_restrict_space emeasure_restrict_space)
2718     apply safe
2719     subgoal for X Z
2720       by (erule ballE[of _ _ X]) (auto simp add: Int_absorb1)
2721     subgoal for X Z
2722       by (erule ballE[of _ _ X])  (auto simp add: Int_absorb1 ac_simps)
2723     apply auto
2724     done
2725 qed
2727 text \<open>
2728   Define a lexicographical order on @{type measure}, in the order space, sets and measure. The parts
2729   of the lexicographical order are point-wise ordered.
2730 \<close>
2732 instantiation measure :: (type) order_bot
2733 begin
2735 inductive less_eq_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where
2736   "space M \<subset> space N \<Longrightarrow> less_eq_measure M N"
2737 | "space M = space N \<Longrightarrow> sets M \<subset> sets N \<Longrightarrow> less_eq_measure M N"
2738 | "space M = space N \<Longrightarrow> sets M = sets N \<Longrightarrow> emeasure M \<le> emeasure N \<Longrightarrow> less_eq_measure M N"
2740 lemma le_measure_iff:
2741   "M \<le> N \<longleftrightarrow> (if space M = space N then
2742     if sets M = sets N then emeasure M \<le> emeasure N else sets M \<subseteq> sets N else space M \<subseteq> space N)"
2743   by (auto elim: less_eq_measure.cases intro: less_eq_measure.intros)
2745 definition less_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where
2746   "less_measure M N \<longleftrightarrow> (M \<le> N \<and> \<not> N \<le> M)"
2748 definition bot_measure :: "'a measure" where
2749   "bot_measure = sigma {} {}"
2751 lemma
2752   shows space_bot[simp]: "space bot = {}"
2753     and sets_bot[simp]: "sets bot = {{}}"
2754     and emeasure_bot[simp]: "emeasure bot X = 0"
2755   by (auto simp: bot_measure_def sigma_sets_empty_eq emeasure_sigma)
2757 instance
2758 proof standard
2759   show "bot \<le> a" for a :: "'a measure"
2760     by (simp add: le_measure_iff bot_measure_def sigma_sets_empty_eq emeasure_sigma le_fun_def)
2761 qed (auto simp: le_measure_iff less_measure_def split: if_split_asm intro: measure_eqI)
2763 end
2765 lemma le_measure: "sets M = sets N \<Longrightarrow> M \<le> N \<longleftrightarrow> (\<forall>A\<in>sets M. emeasure M A \<le> emeasure N A)"
2766   apply (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq)
2767   subgoal for X
2768     by (cases "X \<in> sets M") (auto simp: emeasure_notin_sets)
2769   done
2771 definition sup_measure' :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"
2772 where
2773   "sup_measure' A B = measure_of (space A) (sets A) (\<lambda>X. SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"
2775 lemma assumes [simp]: "sets B = sets A"
2776   shows space_sup_measure'[simp]: "space (sup_measure' A B) = space A"
2777     and sets_sup_measure'[simp]: "sets (sup_measure' A B) = sets A"
2778   using sets_eq_imp_space_eq[OF assms] by (simp_all add: sup_measure'_def)
2780 lemma emeasure_sup_measure':
2781   assumes sets_eq[simp]: "sets B = sets A" and [simp, intro]: "X \<in> sets A"
2782   shows "emeasure (sup_measure' A B) X = (SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"
2783     (is "_ = ?S X")
2784 proof -
2785   note sets_eq_imp_space_eq[OF sets_eq, simp]
2786   show ?thesis
2787     using sup_measure'_def
2788   proof (rule emeasure_measure_of)
2789     let ?d = "\<lambda>X Y. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)"
2790     show "countably_additive (sets (sup_measure' A B)) (\<lambda>X. SUP Y : sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"
2792       case (1 X)
2793       then have [measurable]: "\<And>i. X i \<in> sets A" and "disjoint_family X"
2794         by auto
2795       have "(\<Sum>i. ?S (X i)) = (SUP Y:sets A. \<Sum>i. ?d (X i) Y)"
2796       proof (rule ennreal_suminf_SUP_eq_directed)
2797         fix J :: "nat set" and a b assume "finite J" and [measurable]: "a \<in> sets A" "b \<in> sets A"
2798         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
2799         proof cases
2800           assume "emeasure A (X i) = top \<or> emeasure B (X i) = top"
2801           then show ?thesis
2802           proof
2803             assume "emeasure A (X i) = top" then show ?thesis
2804               by (intro bexI[of _ "X i"]) auto
2805           next
2806             assume "emeasure B (X i) = top" then show ?thesis
2807               by (intro bexI[of _ "{}"]) auto
2808           qed
2809         next
2810           assume finite: "\<not> (emeasure A (X i) = top \<or> emeasure B (X i) = top)"
2811           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)"
2812             using unsigned_Hahn_decomposition[of B A "X i"] by simp
2813           then obtain Y where [measurable]: "Y \<in> sets A" and [simp]: "Y \<subseteq> X i"
2814             and B_le_A: "\<And>C. C \<in> sets A \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> B C \<le> A C"
2815             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"
2816             by auto
2818           show ?thesis
2819           proof (intro bexI[of _ Y] ballI conjI)
2820             fix a assume [measurable]: "a \<in> sets A"
2821             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"
2822               for a Y by auto
2823             then have "?d (X i) a =
2824               (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))"
2825               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric])
2826             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))"
2827               by (intro add_mono order_refl B_le_A A_le_B) (auto simp: Diff_eq[symmetric])
2828             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))"
2830             also have "\<dots> \<le> A (X i \<inter> Y) + B (X i \<inter> - Y)"
2831               by (subst (1 2) plus_emeasure) (auto simp: Diff_eq[symmetric] *)
2832             finally show "?d (X i) a \<le> ?d (X i) Y" .
2833           qed auto
2834         qed
2835         then obtain C where [measurable]: "C i \<in> sets A" and "C i \<subseteq> X i"
2836           and C: "\<And>a. a \<in> sets A \<Longrightarrow> ?d (X i) a \<le> ?d (X i) (C i)" for i
2837           by metis
2838         have *: "X i \<inter> (\<Union>i. C i) = X i \<inter> C i" for i
2839         proof safe
2840           fix x j assume "x \<in> X i" "x \<in> C j"
2841           moreover have "i = j \<or> X i \<inter> X j = {}"
2842             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)
2843           ultimately show "x \<in> C i"
2844             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto
2845         qed auto
2846         have **: "X i \<inter> - (\<Union>i. C i) = X i \<inter> - C i" for i
2847         proof safe
2848           fix x j assume "x \<in> X i" "x \<notin> C i" "x \<in> C j"
2849           moreover have "i = j \<or> X i \<inter> X j = {}"
2850             using \<open>disjoint_family X\<close> by (auto simp: disjoint_family_on_def)
2851           ultimately show False
2852             using \<open>C i \<subseteq> X i\<close> \<open>C j \<subseteq> X j\<close> by auto
2853         qed auto
2854         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"
2855           apply (intro bexI[of _ "\<Union>i. C i"])
2856           unfolding * **
2857           apply (intro C ballI conjI)
2858           apply auto
2859           done
2860       qed
2861       also have "\<dots> = ?S (\<Union>i. X i)"
2862         unfolding UN_extend_simps(4)
2864                  intro!: SUP_cong arg_cong2[where f="(+)"] suminf_emeasure
2865                          disjoint_family_on_bisimulation[OF \<open>disjoint_family X\<close>])
2866       finally show "(\<Sum>i. ?S (X i)) = ?S (\<Union>i. X i)" .
2867     qed
2868   qed (auto dest: sets.sets_into_space simp: positive_def intro!: SUP_const)
2869 qed
2871 lemma le_emeasure_sup_measure'1:
2872   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure A X \<le> emeasure (sup_measure' A B) X"
2873   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "X"] assms)
2875 lemma le_emeasure_sup_measure'2:
2876   assumes "sets B = sets A" "X \<in> sets A" shows "emeasure B X \<le> emeasure (sup_measure' A B) X"
2877   by (subst emeasure_sup_measure'[OF assms]) (auto intro!: SUP_upper2[of "{}"] assms)
2879 lemma emeasure_sup_measure'_le2:
2880   assumes [simp]: "sets B = sets C" "sets A = sets C" and [measurable]: "X \<in> sets C"
2881   assumes A: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure A Y \<le> emeasure C Y"
2882   assumes B: "\<And>Y. Y \<subseteq> X \<Longrightarrow> Y \<in> sets A \<Longrightarrow> emeasure B Y \<le> emeasure C Y"
2883   shows "emeasure (sup_measure' A B) X \<le> emeasure C X"
2884 proof (subst emeasure_sup_measure')
2885   show "(SUP Y:sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y)) \<le> emeasure C X"
2886     unfolding \<open>sets A = sets C\<close>
2887   proof (intro SUP_least)
2888     fix Y assume [measurable]: "Y \<in> sets C"
2889     have [simp]: "X \<inter> Y \<union> (X - Y) = X"
2890       by auto
2891     have "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C (X \<inter> Y) + emeasure C (X \<inter> - Y)"
2892       by (intro add_mono A B) (auto simp: Diff_eq[symmetric])
2893     also have "\<dots> = emeasure C X"
2894       by (subst plus_emeasure) (auto simp: Diff_eq[symmetric])
2895     finally show "emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y) \<le> emeasure C X" .
2896   qed
2897 qed simp_all
2899 definition sup_lexord :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b::order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"
2900 where
2901   "sup_lexord A B k s c =
2902     (if k A = k B then c else if \<not> k A \<le> k B \<and> \<not> k B \<le> k A then s else if k B \<le> k A then A else B)"
2904 lemma sup_lexord:
2905   "(k A < k B \<Longrightarrow> P B) \<Longrightarrow> (k B < k A \<Longrightarrow> P A) \<Longrightarrow> (k A = k B \<Longrightarrow> P c) \<Longrightarrow>
2906     (\<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)"
2907   by (auto simp: sup_lexord_def)
2909 lemmas le_sup_lexord = sup_lexord[where P="\<lambda>a. c \<le> a" for c]
2911 lemma sup_lexord1: "k A = k B \<Longrightarrow> sup_lexord A B k s c = c"
2914 lemma sup_lexord_commute: "sup_lexord A B k s c = sup_lexord B A k s c"
2915   by (auto simp: sup_lexord_def)
2917 lemma sigma_sets_le_sets_iff: "(sigma_sets (space x) \<A> \<subseteq> sets x) = (\<A> \<subseteq> sets x)"
2918   using sets.sigma_sets_subset[of \<A> x] by auto
2920 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))"
2921   by (cases "\<Omega> = space x")
2922      (simp_all add: eq_commute[of _ "sets x"] le_measure_iff emeasure_sigma le_fun_def
2923                     sigma_sets_superset_generator sigma_sets_le_sets_iff)
2925 instantiation measure :: (type) semilattice_sup
2926 begin
2928 definition sup_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"
2929 where
2930   "sup_measure A B =
2931     sup_lexord A B space (sigma (space A \<union> space B) {})
2932       (sup_lexord A B sets (sigma (space A) (sets A \<union> sets B)) (sup_measure' A B))"
2934 instance
2935 proof
2936   fix x y z :: "'a measure"
2937   show "x \<le> sup x y"
2938     unfolding sup_measure_def
2939   proof (intro le_sup_lexord)
2940     assume "space x = space y"
2941     then have *: "sets x \<union> sets y \<subseteq> Pow (space x)"
2942       using sets.space_closed by auto
2943     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"
2944     then have "sets x \<subset> sets x \<union> sets y"
2945       by auto
2946     also have "\<dots> \<le> sigma (space x) (sets x \<union> sets y)"
2947       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)
2948     finally show "x \<le> sigma (space x) (sets x \<union> sets y)"
2949       by (simp add: space_measure_of[OF *] less_eq_measure.intros(2))
2950   next
2951     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"
2952     then show "x \<le> sigma (space x \<union> space y) {}"
2953       by (intro less_eq_measure.intros) auto
2954   next
2955     assume "sets x = sets y" then show "x \<le> sup_measure' x y"
2956       by (simp add: le_measure le_emeasure_sup_measure'1)
2957   qed (auto intro: less_eq_measure.intros)
2958   show "y \<le> sup x y"
2959     unfolding sup_measure_def
2960   proof (intro le_sup_lexord)
2961     assume **: "space x = space y"
2962     then have *: "sets x \<union> sets y \<subseteq> Pow (space y)"
2963       using sets.space_closed by auto
2964     assume "\<not> sets y \<subseteq> sets x" "\<not> sets x \<subseteq> sets y"
2965     then have "sets y \<subset> sets x \<union> sets y"
2966       by auto
2967     also have "\<dots> \<le> sigma (space y) (sets x \<union> sets y)"
2968       by (subst sets_measure_of[OF *]) (rule sigma_sets_superset_generator)
2969     finally show "y \<le> sigma (space x) (sets x \<union> sets y)"
2970       by (simp add: ** space_measure_of[OF *] less_eq_measure.intros(2))
2971   next
2972     assume "\<not> space y \<subseteq> space x" "\<not> space x \<subseteq> space y"
2973     then show "y \<le> sigma (space x \<union> space y) {}"
2974       by (intro less_eq_measure.intros) auto
2975   next
2976     assume "sets x = sets y" then show "y \<le> sup_measure' x y"
2977       by (simp add: le_measure le_emeasure_sup_measure'2)
2978   qed (auto intro: less_eq_measure.intros)
2979   show "x \<le> y \<Longrightarrow> z \<le> y \<Longrightarrow> sup x z \<le> y"
2980     unfolding sup_measure_def
2981   proof (intro sup_lexord[where P="\<lambda>x. x \<le> y"])
2982     assume "x \<le> y" "z \<le> y" and [simp]: "space x = space z" "sets x = sets z"
2983     from \<open>x \<le> y\<close> show "sup_measure' x z \<le> y"
2984     proof cases
2985       case 1 then show ?thesis
2986         by (intro less_eq_measure.intros(1)) simp
2987     next
2988       case 2 then show ?thesis
2989         by (intro less_eq_measure.intros(2)) simp_all
2990     next
2991       case 3 with \<open>z \<le> y\<close> \<open>x \<le> y\<close> show ?thesis
2992         by (auto simp add: le_measure intro!: emeasure_sup_measure'_le2)
2993     qed
2994   next
2995     assume **: "x \<le> y" "z \<le> y" "space x = space z" "\<not> sets z \<subseteq> sets x" "\<not> sets x \<subseteq> sets z"
2996     then have *: "sets x \<union> sets z \<subseteq> Pow (space x)"
2997       using sets.space_closed by auto
2998     show "sigma (space x) (sets x \<union> sets z) \<le> y"
2999       unfolding sigma_le_iff[OF *] using ** by (auto simp: le_measure_iff split: if_split_asm)
3000   next
3001     assume "x \<le> y" "z \<le> y" "\<not> space z \<subseteq> space x" "\<not> space x \<subseteq> space z"
3002     then have "space x \<subseteq> space y" "space z \<subseteq> space y"
3003       by (auto simp: le_measure_iff split: if_split_asm)
3004     then show "sigma (space x \<union> space z) {} \<le> y"
3006   qed
3007 qed
3009 end
3011 lemma space_empty_eq_bot: "space a = {} \<longleftrightarrow> a = bot"
3012   using space_empty[of a] by (auto intro!: measure_eqI)
3014 lemma sets_eq_iff_bounded: "A \<le> B \<Longrightarrow> B \<le> C \<Longrightarrow> sets A = sets C \<Longrightarrow> sets B = sets A"
3015   by (auto dest: sets_eq_imp_space_eq simp add: le_measure_iff split: if_split_asm)
3017 lemma sets_sup: "sets A = sets M \<Longrightarrow> sets B = sets M \<Longrightarrow> sets (sup A B) = sets M"
3018   by (auto simp add: sup_measure_def sup_lexord_def dest: sets_eq_imp_space_eq)
3020 lemma le_measureD1: "A \<le> B \<Longrightarrow> space A \<le> space B"
3021   by (auto simp: le_measure_iff split: if_split_asm)
3023 lemma le_measureD2: "A \<le> B \<Longrightarrow> space A = space B \<Longrightarrow> sets A \<le> sets B"
3024   by (auto simp: le_measure_iff split: if_split_asm)
3026 lemma le_measureD3: "A \<le> B \<Longrightarrow> sets A = sets B \<Longrightarrow> emeasure A X \<le> emeasure B X"
3027   by (auto simp: le_measure_iff le_fun_def dest: sets_eq_imp_space_eq split: if_split_asm)
3029 lemma UN_space_closed: "UNION S sets \<subseteq> Pow (UNION S space)"
3030   using sets.space_closed by auto
3032 definition Sup_lexord :: "('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> ('a set \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> 'a"
3033 where
3034   "Sup_lexord k c s A = (let U = (SUP a:A. k a) in if \<exists>a\<in>A. k a = U then c {a\<in>A. k a = U} else s A)"
3036 lemma Sup_lexord:
3037   "(\<And>a S. a \<in> A \<Longrightarrow> k a = (SUP a: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:A. k a)) \<Longrightarrow> P (s A)) \<Longrightarrow>
3038     P (Sup_lexord k c s A)"
3039   by (auto simp: Sup_lexord_def Let_def)
3041 lemma Sup_lexord1:
3042   assumes A: "A \<noteq> {}" "(\<And>a. a \<in> A \<Longrightarrow> k a = (\<Union>a\<in>A. k a))" "P (c A)"
3043   shows "P (Sup_lexord k c s A)"
3044   unfolding Sup_lexord_def Let_def
3045 proof (clarsimp, safe)
3046   show "\<forall>a\<in>A. k a \<noteq> (\<Union>x\<in>A. k x) \<Longrightarrow> P (s A)"
3047     by (metis assms(1,2) ex_in_conv)
3048 next
3049   fix a assume "a \<in> A" "k a = (\<Union>x\<in>A. k x)"
3050   then have "{a \<in> A. k a = (\<Union>x\<in>A. k x)} = {a \<in> A. k a = k a}"
3051     by (metis A(2)[symmetric])
3052   then show "P (c {a \<in> A. k a = (\<Union>x\<in>A. k x)})"
3054 qed
3056 instantiation measure :: (type) complete_lattice
3057 begin
3059 interpretation sup_measure: comm_monoid_set sup "bot :: 'a measure"
3060   by standard (auto intro!: antisym)
3062 lemma sup_measure_F_mono':
3063   "finite J \<Longrightarrow> finite I \<Longrightarrow> sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"
3064 proof (induction J rule: finite_induct)
3065   case empty then show ?case
3066     by simp
3067 next
3068   case (insert i J)
3069   show ?case
3070   proof cases
3071     assume "i \<in> I" with insert show ?thesis
3072       by (auto simp: insert_absorb)
3073   next
3074     assume "i \<notin> I"
3075     have "sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"
3076       by (intro insert)
3077     also have "\<dots> \<le> sup_measure.F id (insert i (I \<union> J))"
3078       using insert \<open>i \<notin> I\<close> by (subst sup_measure.insert) auto
3079     finally show ?thesis
3080       by auto
3081   qed
3082 qed
3084 lemma sup_measure_F_mono: "finite I \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sup_measure.F id J \<le> sup_measure.F id I"
3085   using sup_measure_F_mono'[of I J] by (auto simp: finite_subset Un_absorb1)
3087 lemma sets_sup_measure_F:
3088   "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"
3089   by (induction I rule: finite_ne_induct) (simp_all add: sets_sup)
3091 definition Sup_measure' :: "'a measure set \<Rightarrow> 'a measure"
3092 where
3093   "Sup_measure' M = measure_of (\<Union>a\<in>M. space a) (\<Union>a\<in>M. sets a)
3094     (\<lambda>X. (SUP P:{P. finite P \<and> P \<subseteq> M }. sup_measure.F id P X))"
3096 lemma space_Sup_measure'2: "space (Sup_measure' M) = (\<Union>m\<in>M. space m)"
3097   unfolding Sup_measure'_def by (intro space_measure_of[OF UN_space_closed])
3099 lemma sets_Sup_measure'2: "sets (Sup_measure' M) = sigma_sets (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)"
3100   unfolding Sup_measure'_def by (intro sets_measure_of[OF UN_space_closed])
3102 lemma sets_Sup_measure':
3103   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"
3104   shows "sets (Sup_measure' M) = sets A"
3105   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close> by (simp add: Sup_measure'_def)
3107 lemma space_Sup_measure':
3108   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "M \<noteq> {}"
3109   shows "space (Sup_measure' M) = space A"
3110   using sets_eq[THEN sets_eq_imp_space_eq, simp] \<open>M \<noteq> {}\<close>
3111   by (simp add: Sup_measure'_def )
3113 lemma emeasure_Sup_measure':
3114   assumes sets_eq[simp]: "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A" and "X \<in> sets A" "M \<noteq> {}"
3115   shows "emeasure (Sup_measure' M) X = (SUP P:{P. finite P \<and> P \<subseteq> M}. sup_measure.F id P X)"
3116     (is "_ = ?S X")
3117   using Sup_measure'_def
3118 proof (rule emeasure_measure_of)
3119   note sets_eq[THEN sets_eq_imp_space_eq, simp]
3120   have *: "sets (Sup_measure' M) = sets A" "space (Sup_measure' M) = space A"
3121     using \<open>M \<noteq> {}\<close> by (simp_all add: Sup_measure'_def)
3122   let ?\<mu> = "sup_measure.F id"
3123   show "countably_additive (sets (Sup_measure' M)) ?S"
3125     case (1 F)
3126     then have **: "range F \<subseteq> sets A"
3127       by (auto simp: *)
3128     show "(\<Sum>i. ?S (F i)) = ?S (\<Union>i. F i)"
3129     proof (subst ennreal_suminf_SUP_eq_directed)
3130       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}"
3131       have "(i \<noteq> {} \<longrightarrow> sets (?\<mu> i) = sets A) \<and> (j \<noteq> {} \<longrightarrow> sets (?\<mu> j) = sets A) \<and>
3132         (i \<noteq> {} \<or> j \<noteq> {} \<longrightarrow> sets (?\<mu> (i \<union> j)) = sets A)"
3133         using ij by (intro impI sets_sup_measure_F conjI) auto
3134       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
3135         using ij
3136         by (cases "i = {}"; cases "j = {}")
3137            (auto intro!: le_measureD3 sup_measure_F_mono simp: sets_sup_measure_F
3138                  simp del: id_apply)
3139       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)"
3140         by (safe intro!: bexI[of _ "i \<union> j"]) auto
3141     next
3142       show "(SUP P : {P. finite P \<and> P \<subseteq> M}. \<Sum>n. ?\<mu> P (F n)) = (SUP P : {P. finite P \<and> P \<subseteq> M}. ?\<mu> P (UNION UNIV F))"
3143       proof (intro SUP_cong refl)
3144         fix i assume i: "i \<in> {P. finite P \<and> P \<subseteq> M}"
3145         show "(\<Sum>n. ?\<mu> i (F n)) = ?\<mu> i (UNION UNIV F)"
3146         proof cases
3147           assume "i \<noteq> {}" with i ** show ?thesis
3148             apply (intro suminf_emeasure \<open>disjoint_family F\<close>)
3149             apply (subst sets_sup_measure_F[OF _ _ sets_eq])
3150             apply auto
3151             done
3152         qed simp
3153       qed
3154     qed
3155   qed
3156   show "positive (sets (Sup_measure' M)) ?S"
3157     by (auto simp: positive_def bot_ennreal[symmetric])
3158   show "X \<in> sets (Sup_measure' M)"
3159     using assms * by auto
3160 qed (rule UN_space_closed)
3162 definition Sup_measure :: "'a measure set \<Rightarrow> 'a measure"
3163 where
3164   "Sup_measure = Sup_lexord space (Sup_lexord sets Sup_measure'
3165     (\<lambda>U. sigma (\<Union>u\<in>U. space u) (\<Union>u\<in>U. sets u))) (\<lambda>U. sigma (\<Union>u\<in>U. space u) {})"
3167 definition Inf_measure :: "'a measure set \<Rightarrow> 'a measure"
3168 where
3169   "Inf_measure A = Sup {x. \<forall>a\<in>A. x \<le> a}"
3171 definition inf_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"
3172 where
3173   "inf_measure a b = Inf {a, b}"
3175 definition top_measure :: "'a measure"
3176 where
3177   "top_measure = Inf {}"
3179 instance
3180 proof
3181   note UN_space_closed [simp]
3182   show upper: "x \<le> Sup A" if x: "x \<in> A" for x :: "'a measure" and A
3183     unfolding Sup_measure_def
3184   proof (intro Sup_lexord[where P="\<lambda>y. x \<le> y"])
3185     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"
3186     from this[OF \<open>x \<in> A\<close>] \<open>x \<in> A\<close> show "x \<le> sigma (\<Union>a\<in>A. space a) {}"
3187       by (intro less_eq_measure.intros) auto
3188   next
3189     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}"
3190       and neq: "\<And>aa. aa \<in> S \<Longrightarrow> sets aa \<noteq> (\<Union>a\<in>S. sets a)"
3191     have sp_a: "space a = (UNION S space)"
3192       using \<open>a\<in>A\<close> by (auto simp: S)
3193     show "x \<le> sigma (UNION S space) (UNION S sets)"
3194     proof cases
3195       assume [simp]: "space x = space a"
3196       have "sets x \<subset> (\<Union>a\<in>S. sets a)"
3197         using \<open>x\<in>A\<close> neq[of x] by (auto simp: S)
3198       also have "\<dots> \<subseteq> sigma_sets (\<Union>x\<in>S. space x) (\<Union>x\<in>S. sets x)"
3199         by (rule sigma_sets_superset_generator)
3200       finally show ?thesis
3201         by (intro less_eq_measure.intros(2)) (simp_all add: sp_a)
3202     next
3203       assume "space x \<noteq> space a"
3204       moreover have "space x \<le> space a"
3205         unfolding a using \<open>x\<in>A\<close> by auto
3206       ultimately show ?thesis
3207         by (intro less_eq_measure.intros) (simp add: less_le sp_a)
3208     qed
3209   next
3210     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}"
3211       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"
3212     then have "S' \<noteq> {}" "space b = space a"
3213       by auto
3214     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"
3215       by (auto simp: S')
3216     note sets_eq[THEN sets_eq_imp_space_eq, simp]
3217     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"
3218       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)
3219     show "x \<le> Sup_measure' S'"
3220     proof cases
3221       assume "x \<in> S"
3222       with \<open>b \<in> S\<close> have "space x = space b"
3224       show ?thesis
3225       proof cases
3226         assume "x \<in> S'"
3227         show "x \<le> Sup_measure' S'"
3228         proof (intro le_measure[THEN iffD2] ballI)
3229           show "sets x = sets (Sup_measure' S')"
3230             using \<open>x\<in>S'\<close> * by (simp add: S')
3231           fix X assume "X \<in> sets x"
3232           show "emeasure x X \<le> emeasure (Sup_measure' S') X"
3233           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets x\<close>])
3234             show "emeasure x X \<le> (SUP P : {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X)"
3235               using \<open>x\<in>S'\<close> by (intro SUP_upper2[where i="{x}"]) auto
3236           qed (insert \<open>x\<in>S'\<close> S', auto)
3237         qed
3238       next
3239         assume "x \<notin> S'"
3240         then have "sets x \<noteq> sets b"
3241           using \<open>x\<in>S\<close> by (auto simp: S')
3242         moreover have "sets x \<le> sets b"
3243           using \<open>x\<in>S\<close> unfolding b by auto
3244         ultimately show ?thesis
3245           using * \<open>x \<in> S\<close>
3246           by (intro less_eq_measure.intros(2))
3247              (simp_all add: * \<open>space x = space b\<close> less_le)
3248       qed
3249     next
3250       assume "x \<notin> S"
3251       with \<open>x\<in>A\<close> \<open>x \<notin> S\<close> \<open>space b = space a\<close> show ?thesis
3252         by (intro less_eq_measure.intros)
3253            (simp_all add: * less_le a SUP_upper S)
3254     qed
3255   qed
3256   show least: "Sup A \<le> x" if x: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x" for x :: "'a measure" and A
3257     unfolding Sup_measure_def
3258   proof (intro Sup_lexord[where P="\<lambda>y. y \<le> x"])
3259     assume "\<And>a. a \<in> A \<Longrightarrow> space a \<noteq> (\<Union>a\<in>A. space a)"
3260     show "sigma (UNION A space) {} \<le> x"
3261       using x[THEN le_measureD1] by (subst sigma_le_iff) auto
3262   next
3263     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}"
3264       "\<And>a. a \<in> S \<Longrightarrow> sets a \<noteq> (\<Union>a\<in>S. sets a)"
3265     have "UNION S space \<subseteq> space x"
3266       using S le_measureD1[OF x] by auto
3267     moreover
3268     have "UNION S space = space a"
3269       using \<open>a\<in>A\<close> S by auto
3270     then have "space x = UNION S space \<Longrightarrow> UNION S sets \<subseteq> sets x"
3271       using \<open>a \<in> A\<close> le_measureD2[OF x] by (auto simp: S)
3272     ultimately show "sigma (UNION S space) (UNION S sets) \<le> x"
3273       by (subst sigma_le_iff) simp_all
3274   next
3275     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}"
3276       and "b \<in> S" and b: "sets b = (\<Union>a\<in>S. sets a)" and S': "S' = {a' \<in> S. sets a' = sets b}"
3277     then have "S' \<noteq> {}" "space b = space a"
3278       by auto
3279     have sets_eq: "\<And>x. x \<in> S' \<Longrightarrow> sets x = sets b"
3280       by (auto simp: S')
3281     note sets_eq[THEN sets_eq_imp_space_eq, simp]
3282     have *: "sets (Sup_measure' S') = sets b" "space (Sup_measure' S') = space b"
3283       using \<open>S' \<noteq> {}\<close> by (simp_all add: Sup_measure'_def sets_eq)
3284     show "Sup_measure' S' \<le> x"
3285     proof cases
3286       assume "space x = space a"
3287       show ?thesis
3288       proof cases
3289         assume **: "sets x = sets b"
3290         show ?thesis
3291         proof (intro le_measure[THEN iffD2] ballI)
3292           show ***: "sets (Sup_measure' S') = sets x"
3293             by (simp add: * **)
3294           fix X assume "X \<in> sets (Sup_measure' S')"
3295           show "emeasure (Sup_measure' S') X \<le> emeasure x X"
3296             unfolding ***
3297           proof (subst emeasure_Sup_measure'[OF _ \<open>X \<in> sets (Sup_measure' S')\<close>])
3298             show "(SUP P : {P. finite P \<and> P \<subseteq> S'}. emeasure (sup_measure.F id P) X) \<le> emeasure x X"
3299             proof (safe intro!: SUP_least)
3300               fix P assume P: "finite P" "P \<subseteq> S'"
3301               show "emeasure (sup_measure.F id P) X \<le> emeasure x X"
3302               proof cases
3303                 assume "P = {}" then show ?thesis
3304                   by auto
3305               next
3306                 assume "P \<noteq> {}"
3307                 from P have "finite P" "P \<subseteq> A"
3308                   unfolding S' S by (simp_all add: subset_eq)
3309                 then have "sup_measure.F id P \<le> x"
3310                   by (induction P) (auto simp: x)
3311                 moreover have "sets (sup_measure.F id P) = sets x"
3312                   using \<open>finite P\<close> \<open>P \<noteq> {}\<close> \<open>P \<subseteq> S'\<close> \<open>sets x = sets b\<close>
3313                   by (intro sets_sup_measure_F) (auto simp: S')
3314                 ultimately show "emeasure (sup_measure.F id P) X \<le> emeasure x X"
3315                   by (rule le_measureD3)
3316               qed
3317             qed
3318             show "m \<in> S' \<Longrightarrow> sets m = sets (Sup_measure' S')" for m
3319               unfolding * by (simp add: S')
3320           qed fact
3321         qed
3322       next
3323         assume "sets x \<noteq> sets b"
3324         moreover have "sets b \<le> sets x"
3325           unfolding b S using x[THEN le_measureD2] \<open>space x = space a\<close> by auto
3326         ultimately show "Sup_measure' S' \<le> x"
3327           using \<open>space x = space a\<close> \<open>b \<in> S\<close>
3328           by (intro less_eq_measure.intros(2)) (simp_all add: * S)
3329       qed
3330     next
3331       assume "space x \<noteq> space a"
3332       then have "space a < space x"
3333         using le_measureD1[OF x[OF \<open>a\<in>A\<close>]] by auto
3334       then show "Sup_measure' S' \<le> x"
3335         by (intro less_eq_measure.intros) (simp add: * \<open>space b = space a\<close>)
3336     qed
3337   qed
3338   show "Sup {} = (bot::'a measure)" "Inf {} = (top::'a measure)"
3339     by (auto intro!: antisym least simp: top_measure_def)
3340   show lower: "x \<in> A \<Longrightarrow> Inf A \<le> x" for x :: "'a measure" and A
3341     unfolding Inf_measure_def by (intro least) auto
3342   show greatest: "(\<And>z. z \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> x \<le> Inf A" for x :: "'a measure" and A
3343     unfolding Inf_measure_def by (intro upper) auto
3344   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"
3345     by (auto simp: inf_measure_def intro!: lower greatest)
3346 qed
3348 end
3350 lemma sets_SUP:
3351   assumes "\<And>x. x \<in> I \<Longrightarrow> sets (M x) = sets N"
3352   shows "I \<noteq> {} \<Longrightarrow> sets (SUP i:I. M i) = sets N"
3353   unfolding Sup_measure_def
3354   using assms assms[THEN sets_eq_imp_space_eq]
3355     sets_Sup_measure'[where A=N and M="MI"]
3356   by (intro Sup_lexord1[where P="\<lambda>x. sets x = sets N"]) auto
3358 lemma emeasure_SUP:
3359   assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N" "I \<noteq> {}"
3360   shows "emeasure (SUP i:I. M i) X = (SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emeasure (SUP i:J. M i) X)"
3361 proof -
3362   interpret sup_measure: comm_monoid_set sup "bot :: 'b measure"
3363     by standard (auto intro!: antisym)
3364   have eq: "finite J \<Longrightarrow> sup_measure.F id J = (SUP i:J. i)" for J :: "'b measure set"
3365     by (induction J rule: finite_induct) auto
3366   have 1: "J \<noteq> {} \<Longrightarrow> J \<subseteq> I \<Longrightarrow> sets (SUP x:J. M x) = sets N" for J
3367     by (intro sets_SUP sets) (auto )
3368   from \<open>I \<noteq> {}\<close> obtain i where "i\<in>I" by auto
3369   have "Sup_measure' (MI) X = (SUP P:{P. finite P \<and> P \<subseteq> MI}. sup_measure.F id P X)"
3370     using sets by (intro emeasure_Sup_measure') auto
3371   also have "Sup_measure' (MI) = (SUP i:I. M i)"
3372     unfolding Sup_measure_def using \<open>I \<noteq> {}\<close> sets sets(1)[THEN sets_eq_imp_space_eq]
3373     by (intro Sup_lexord1[where P="\<lambda>x. _ = x"]) auto
3374   also have "(SUP P:{P. finite P \<and> P \<subseteq> MI}. sup_measure.F id P X) =
3375     (SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. (SUP i:J. M i) X)"
3376   proof (intro SUP_eq)
3377     fix J assume "J \<in> {P. finite P \<and> P \<subseteq> MI}"
3378     then obtain J' where J': "J' \<subseteq> I" "finite J'" and J: "J = MJ'" and "finite J"
3379       using finite_subset_image[of J M I] by auto
3380     show "\<exists>j\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. sup_measure.F id J X \<le> (SUP i:j. M i) X"
3381     proof cases
3382       assume "J' = {}" with \<open>i \<in> I\<close> show ?thesis
3383         by (auto simp add: J)
3384     next
3385       assume "J' \<noteq> {}" with J J' show ?thesis
3386         by (intro bexI[of _ "J'"]) (auto simp add: eq simp del: id_apply)
3387     qed
3388   next
3389     fix J assume J: "J \<in> {P. P \<noteq> {} \<and> finite P \<and> P \<subseteq> I}"
3390     show "\<exists>J'\<in>{J. finite J \<and> J \<subseteq> MI}. (SUP i:J. M i) X \<le> sup_measure.F id J' X"
3391       using J by (intro bexI[of _ "MJ"]) (auto simp add: eq simp del: id_apply)
3392   qed
3393   finally show ?thesis .
3394 qed
3396 lemma emeasure_SUP_chain:
3397   assumes sets: "\<And>i. i \<in> A \<Longrightarrow> sets (M i) = sets N" "X \<in> sets N"
3398   assumes ch: "Complete_Partial_Order.chain (\<le>) (M  A)" and "A \<noteq> {}"
3399   shows "emeasure (SUP i:A. M i) X = (SUP i:A. emeasure (M i) X)"
3400 proof (subst emeasure_SUP[OF sets \<open>A \<noteq> {}\<close>])
3401   show "(SUP J:{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}. emeasure (SUPREMUM J M) X) = (SUP i:A. emeasure (M i) X)"
3402   proof (rule SUP_eq)
3403     fix J assume "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> A}"
3404     then have J: "Complete_Partial_Order.chain (\<le>) (M  J)" "finite J" "J \<noteq> {}" and "J \<subseteq> A"
3405       using ch[THEN chain_subset, of "MJ"] by auto
3406     with in_chain_finite[OF J(1)] obtain j where "j \<in> J" "(SUP j:J. M j) = M j"
3407       by auto
3408     with \<open>J \<subseteq> A\<close> show "\<exists>j\<in>A. emeasure (SUPREMUM J M) X \<le> emeasure (M j) X"
3409       by auto
3410   next
3411     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 (SUPREMUM i M) X"
3412       by (intro bexI[of _ "{j}"]) auto
3413   qed
3414 qed
3416 subsubsection \<open>Supremum of a set of $\sigma$-algebras\<close>
3418 lemma space_Sup_eq_UN: "space (Sup M) = (\<Union>x\<in>M. space x)"
3419   unfolding Sup_measure_def
3420   apply (intro Sup_lexord[where P="\<lambda>x. space x = _"])
3421   apply (subst space_Sup_measure'2)
3422   apply auto []
3423   apply (subst space_measure_of[OF UN_space_closed])
3424   apply auto
3425   done
3427 lemma sets_Sup_eq:
3428   assumes *: "\<And>m. m \<in> M \<Longrightarrow> space m = X" and "M \<noteq> {}"
3429   shows "sets (Sup M) = sigma_sets X (\<Union>x\<in>M. sets x)"
3430   unfolding Sup_measure_def
3431   apply (rule Sup_lexord1)
3432   apply fact
3434   apply (rule Sup_lexord)
3435   subgoal premises that for a S
3436     unfolding that(3) that(2)[symmetric]
3437     using that(1)
3438     apply (subst sets_Sup_measure'2)
3439     apply (intro arg_cong2[where f=sigma_sets])
3440     apply (auto simp: *)
3441     done
3442   apply (subst sets_measure_of[OF UN_space_closed])
3444   done
3446 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)"
3447   by (subst sets_Sup_eq[where X=X]) auto
3449 lemma Sup_lexord_rel:
3450   assumes "\<And>i. i \<in> I \<Longrightarrow> k (A i) = k (B i)"
3451     "R (c (A  {a \<in> I. k (B a) = (SUP x:I. k (B x))})) (c (B  {a \<in> I. k (B a) = (SUP x:I. k (B x))}))"
3452     "R (s (AI)) (s (BI))"
3453   shows "R (Sup_lexord k c s (AI)) (Sup_lexord k c s (BI))"
3454 proof -
3455   have "A  {a \<in> I. k (B a) = (SUP x:I. k (B x))} =  {a \<in> A  I. k a = (SUP x:I. k (B x))}"
3456     using assms(1) by auto
3457   moreover have "B  {a \<in> I. k (B a) = (SUP x:I. k (B x))} =  {a \<in> B  I. k a = (SUP x:I. k (B x))}"
3458     by auto
3459   ultimately show ?thesis
3460     using assms by (auto simp: Sup_lexord_def Let_def)
3461 qed
3463 lemma sets_SUP_cong:
3464   assumes eq: "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (SUP i:I. M i) = sets (SUP i:I. N i)"
3465   unfolding Sup_measure_def
3466   using eq eq[THEN sets_eq_imp_space_eq]
3467   apply (intro Sup_lexord_rel[where R="\<lambda>x y. sets x = sets y"])
3468   apply simp
3469   apply simp
3471   apply (intro arg_cong2[where f="\<lambda>x y. sets (sigma x y)"])
3472   apply auto
3473   done
3475 lemma sets_Sup_in_sets:
3476   assumes "M \<noteq> {}"
3477   assumes "\<And>m. m \<in> M \<Longrightarrow> space m = space N"
3478   assumes "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N"
3479   shows "sets (Sup M) \<subseteq> sets N"
3480 proof -
3481   have *: "UNION M space = space N"
3482     using assms by auto
3483   show ?thesis
3484     unfolding * using assms by (subst sets_Sup_eq[of M "space N"]) (auto intro!: sets.sigma_sets_subset)
3485 qed
3487 lemma measurable_Sup1:
3488   assumes m: "m \<in> M" and f: "f \<in> measurable m N"
3489     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"
3490   shows "f \<in> measurable (Sup M) N"
3491 proof -
3492   have "space (Sup M) = space m"
3493     using m by (auto simp add: space_Sup_eq_UN dest: const_space)
3494   then show ?thesis
3495     using m f unfolding measurable_def by (auto intro: in_sets_Sup[OF const_space])
3496 qed
3498 lemma measurable_Sup2:
3499   assumes M: "M \<noteq> {}"
3500   assumes f: "\<And>m. m \<in> M \<Longrightarrow> f \<in> measurable N m"
3501     and const_space: "\<And>m n. m \<in> M \<Longrightarrow> n \<in> M \<Longrightarrow> space m = space n"
3502   shows "f \<in> measurable N (Sup M)"
3503 proof -
3504   from M obtain m where "m \<in> M" by auto
3505   have space_eq: "\<And>n. n \<in> M \<Longrightarrow> space n = space m"
3506     by (intro const_space \<open>m \<in> M\<close>)
3507   have "f \<in> measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m))"
3508   proof (rule measurable_measure_of)
3509     show "f \<in> space N \<rightarrow> UNION M space"
3510       using measurable_space[OF f] M by auto
3511   qed (auto intro: measurable_sets f dest: sets.sets_into_space)
3512   also have "measurable N (sigma (\<Union>m\<in>M. space m) (\<Union>m\<in>M. sets m)) = measurable N (Sup M)"
3513     apply (intro measurable_cong_sets refl)
3514     apply (subst sets_Sup_eq[OF space_eq M])
3515     apply simp
3516     apply (subst sets_measure_of[OF UN_space_closed])
3517     apply (simp add: space_eq M)
3518     done
3519   finally show ?thesis .
3520 qed
3522 lemma measurable_SUP2:
3523   "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f \<in> measurable N (M i)) \<Longrightarrow>
3524     (\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> space (M i) = space (M j)) \<Longrightarrow> f \<in> measurable N (SUP i:I. M i)"
3525   by (auto intro!: measurable_Sup2)
3527 lemma sets_Sup_sigma:
3528   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"
3529   shows "sets (SUP m:M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"
3530 proof -
3531   { fix a m assume "a \<in> sigma_sets \<Omega> m" "m \<in> M"
3532     then have "a \<in> sigma_sets \<Omega> (\<Union>M)"
3533      by induction (auto intro: sigma_sets.intros(2-)) }
3534   then show "sets (SUP m:M. sigma \<Omega> m) = sets (sigma \<Omega> (\<Union>M))"
3535     apply (subst sets_Sup_eq[where X="\<Omega>"])
3536     apply (auto simp add: M) []
3537     apply auto []
3538     apply (simp add: space_measure_of_conv M Union_least)
3539     apply (rule sigma_sets_eqI)
3540     apply auto
3541     done
3542 qed
3544 lemma Sup_sigma:
3545   assumes [simp]: "M \<noteq> {}" and M: "\<And>m. m \<in> M \<Longrightarrow> m \<subseteq> Pow \<Omega>"
3546   shows "(SUP m:M. sigma \<Omega> m) = (sigma \<Omega> (\<Union>M))"
3547 proof (intro antisym SUP_least)
3548   have *: "\<Union>M \<subseteq> Pow \<Omega>"
3549     using M by auto
3550   show "sigma \<Omega> (\<Union>M) \<le> (SUP m:M. sigma \<Omega> m)"
3551   proof (intro less_eq_measure.intros(3))
3552     show "space (sigma \<Omega> (\<Union>M)) = space (SUP m:M. sigma \<Omega> m)"
3553       "sets (sigma \<Omega> (\<Union>M)) = sets (SUP m:M. sigma \<Omega> m)"
3554       using sets_Sup_sigma[OF assms] sets_Sup_sigma[OF assms, THEN sets_eq_imp_space_eq]
3555       by auto
3556   qed (simp add: emeasure_sigma le_fun_def)
3557   fix m assume "m \<in> M" then show "sigma \<Omega> m \<le> sigma \<Omega> (\<Union>M)"
3558     by (subst sigma_le_iff) (auto simp add: M *)
3559 qed
3561 lemma SUP_sigma_sigma:
3562   "M \<noteq> {} \<Longrightarrow> (\<And>m. m \<in> M \<Longrightarrow> f m \<subseteq> Pow \<Omega>) \<Longrightarrow> (SUP m:M. sigma \<Omega> (f m)) = sigma \<Omega> (\<Union>m\<in>M. f m)"
3563   using Sup_sigma[of "fM" \<Omega>] by auto
3565 lemma sets_vimage_Sup_eq:
3566   assumes *: "M \<noteq> {}" "f \<in> X \<rightarrow> Y" "\<And>m. m \<in> M \<Longrightarrow> space m = Y"
3567   shows "sets (vimage_algebra X f (Sup M)) = sets (SUP m : M. vimage_algebra X f m)"
3568   (is "?IS = ?SI")
3569 proof
3570   show "?IS \<subseteq> ?SI"
3571     apply (intro sets_image_in_sets measurable_Sup2)
3572     apply (simp add: space_Sup_eq_UN *)
3574     apply (intro measurable_Sup1)
3575     apply (rule imageI)
3576     apply assumption
3577     apply (rule measurable_vimage_algebra1)
3578     apply (auto simp: *)
3579     done
3580   show "?SI \<subseteq> ?IS"
3581     apply (intro sets_Sup_in_sets)
3582     apply (auto simp: *) []
3583     apply (auto simp: *) []
3584     apply (elim imageE)
3585     apply simp
3586     apply (rule sets_image_in_sets)
3587     apply simp
3589     apply (simp add: * space_Sup_eq_UN sets_vimage_algebra2)
3590     apply (auto intro: in_sets_Sup[OF *(3)])
3591     done
3592 qed
3594 lemma restrict_space_eq_vimage_algebra':
3595   "sets (restrict_space M \<Omega>) = sets (vimage_algebra (\<Omega> \<inter> space M) (\<lambda>x. x) M)"
3596 proof -
3597   have *: "{A \<inter> (\<Omega> \<inter> space M) |A. A \<in> sets M} = {A \<inter> \<Omega> |A. A \<in> sets M}"
3598     using sets.sets_into_space[of _ M] by blast
3600   show ?thesis
3601     unfolding restrict_space_def
3602     by (subst sets_measure_of)
3603        (auto simp add: image_subset_iff sets_vimage_algebra * dest: sets.sets_into_space intro!: arg_cong2[where f=sigma_sets])
3604 qed
3606 lemma sigma_le_sets:
3607   assumes [simp]: "A \<subseteq> Pow X" shows "sets (sigma X A) \<subseteq> sets N \<longleftrightarrow> X \<in> sets N \<and> A \<subseteq> sets N"
3608 proof
3609   have "X \<in> sigma_sets X A" "A \<subseteq> sigma_sets X A"
3610     by (auto intro: sigma_sets_top)
3611   moreover assume "sets (sigma X A) \<subseteq> sets N"
3612   ultimately show "X \<in> sets N \<and> A \<subseteq> sets N"
3613     by auto
3614 next
3615   assume *: "X \<in> sets N \<and> A \<subseteq> sets N"
3616   { fix Y assume "Y \<in> sigma_sets X A" from this * have "Y \<in> sets N"
3617       by induction auto }
3618   then show "sets (sigma X A) \<subseteq> sets N"
3619     by auto
3620 qed
3622 lemma measurable_iff_sets:
3623   "f \<in> measurable M N \<longleftrightarrow> (f \<in> space M \<rightarrow> space N \<and> sets (vimage_algebra (space M) f N) \<subseteq> sets M)"
3624 proof -
3625   have *: "{f - A \<inter> space M |A. A \<in> sets N} \<subseteq> Pow (space M)"
3626     by auto
3627   show ?thesis
3628     unfolding measurable_def
3629     by (auto simp add: vimage_algebra_def sigma_le_sets[OF *])
3630 qed
3632 lemma sets_vimage_algebra_space: "X \<in> sets (vimage_algebra X f M)"
3633   using sets.top[of "vimage_algebra X f M"] by simp
3635 lemma measurable_mono:
3636   assumes N: "sets N' \<le> sets N" "space N = space N'"
3637   assumes M: "sets M \<le> sets M'" "space M = space M'"
3638   shows "measurable M N \<subseteq> measurable M' N'"
3639   unfolding measurable_def
3640 proof safe
3641   fix f A assume "f \<in> space M \<rightarrow> space N" "A \<in> sets N'"
3642   moreover assume "\<forall>y\<in>sets N. f - y \<inter> space M \<in> sets M" note this[THEN bspec, of A]
3643   ultimately show "f - A \<inter> space M' \<in> sets M'"
3644     using assms by auto
3645 qed (insert N M, auto)
3647 lemma measurable_Sup_measurable:
3648   assumes f: "f \<in> space N \<rightarrow> A"
3649   shows "f \<in> measurable N (Sup {M. space M = A \<and> f \<in> measurable N M})"
3650 proof (rule measurable_Sup2)
3651   show "{M. space M = A \<and> f \<in> measurable N M} \<noteq> {}"
3652     using f unfolding ex_in_conv[symmetric]
3653     by (intro exI[of _ "sigma A {}"]) (auto intro!: measurable_measure_of)
3654 qed auto
3656 lemma (in sigma_algebra) sigma_sets_subset':
3657   assumes a: "a \<subseteq> M" "\<Omega>' \<in> M"
3658   shows "sigma_sets \<Omega>' a \<subseteq> M"
3659 proof
3660   show "x \<in> M" if x: "x \<in> sigma_sets \<Omega>' a" for x
3661     using x by (induct rule: sigma_sets.induct) (insert a, auto)
3662 qed
3664 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:I. M i)"
3665   by (intro in_sets_Sup[where X=Y]) auto
3667 lemma measurable_SUP1:
3668   "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>
3669     f \<in> measurable (SUP i:I. M i) N"
3670   by (auto intro: measurable_Sup1)
3672 lemma sets_image_in_sets':
3673   assumes X: "X \<in> sets N"
3674   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> f - A \<inter> X \<in> sets N"
3675   shows "sets (vimage_algebra X f M) \<subseteq> sets N"
3676   unfolding sets_vimage_algebra
3677   by (rule sets.sigma_sets_subset') (auto intro!: measurable_sets X f)
3679 lemma mono_vimage_algebra:
3680   "sets M \<le> sets N \<Longrightarrow> sets (vimage_algebra X f M) \<subseteq> sets (vimage_algebra X f N)"
3681   using sets.top[of "sigma X {f - A \<inter> X |A. A \<in> sets N}"]
3682   unfolding vimage_algebra_def
3683   apply (subst (asm) space_measure_of)
3684   apply auto []
3685   apply (subst sigma_le_sets)
3686   apply auto
3687   done
3689 lemma mono_restrict_space: "sets M \<le> sets N \<Longrightarrow> sets (restrict_space M X) \<subseteq> sets (restrict_space N X)"
3690   unfolding sets_restrict_space by (rule image_mono)
3692 lemma sets_eq_bot: "sets M = {{}} \<longleftrightarrow> M = bot"
3693   apply safe
3694   apply (intro measure_eqI)
3695   apply auto
3696   done
3698 lemma sets_eq_bot2: "{{}} = sets M \<longleftrightarrow> M = bot"
3699   using sets_eq_bot[of M] by blast
3702 lemma (in finite_measure) countable_support:
3703   "countable {x. measure M {x} \<noteq> 0}"
3704 proof cases
3705   assume "measure M (space M) = 0"
3706   with bounded_measure measure_le_0_iff have "{x. measure M {x} \<noteq> 0} = {}"
3707     by auto
3708   then show ?thesis
3709     by simp
3710 next
3711   let ?M = "measure M (space M)" and ?m = "\<lambda>x. measure M {x}"
3712   assume "?M \<noteq> 0"
3713   then have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. ?M / Suc n < ?m x})"
3714     using reals_Archimedean[of "?m x / ?M" for x]
3715     by (auto simp: field_simps not_le[symmetric] divide_le_0_iff measure_le_0_iff)
3716   have **: "\<And>n. finite {x. ?M / Suc n < ?m x}"
3717   proof (rule ccontr)
3718     fix n assume "infinite {x. ?M / Suc n < ?m x}" (is "infinite ?X")
3719     then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
3720       by (metis infinite_arbitrarily_large)
3721     from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> ?M / Suc n \<le> ?m x"
3722       by auto
3723     { fix x assume "x \<in> X"
3724       from \<open>?M \<noteq> 0\<close> *[OF this] have "?m x \<noteq> 0" by (auto simp: field_simps measure_le_0_iff)
3725       then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
3726     note singleton_sets = this
3727     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
3728       using \<open>?M \<noteq> 0\<close>
3729       by (simp add: \<open>card X = Suc (Suc n)\<close> field_simps less_le)
3730     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
3731       by (rule sum_mono) fact
3732     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
3733       using singleton_sets \<open>finite X\<close>
3734       by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
3735     finally have "?M < measure M (\<Union>x\<in>X. {x})" .
3736     moreover have "measure M (\<Union>x\<in>X. {x}) \<le> ?M"
3737       using singleton_sets[THEN sets.sets_into_space] by (intro finite_measure_mono) auto
3738     ultimately show False by simp
3739   qed
3740   show ?thesis
3741     unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
3742 qed
3744 end