src/HOL/Probability/Sigma_Algebra.thy
 author haftmann Sun Jun 23 21:16:07 2013 +0200 (2013-06-23) changeset 52435 6646bb548c6b parent 51683 baefa3b461c2 child 53374 a14d2a854c02 permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
1 (*  Title:      HOL/Probability/Sigma_Algebra.thy
2     Author:     Stefan Richter, Markus Wenzel, TU München
3     Author:     Johannes Hölzl, TU München
4     Plus material from the Hurd/Coble measure theory development,
5     translated by Lawrence Paulson.
6 *)
8 header {* Sigma Algebras *}
10 theory Sigma_Algebra
11 imports
12   Complex_Main
13   "~~/src/HOL/Library/Countable_Set"
14   "~~/src/HOL/Library/FuncSet"
15   "~~/src/HOL/Library/Indicator_Function"
16   "~~/src/HOL/Library/Extended_Real"
17 begin
19 text {* Sigma algebras are an elementary concept in measure
20   theory. To measure --- that is to integrate --- functions, we first have
21   to measure sets. Unfortunately, when dealing with a large universe,
22   it is often not possible to consistently assign a measure to every
23   subset. Therefore it is necessary to define the set of measurable
24   subsets of the universe. A sigma algebra is such a set that has
25   three very natural and desirable properties. *}
27 subsection {* Families of sets *}
29 locale subset_class =
30   fixes \<Omega> :: "'a set" and M :: "'a set set"
31   assumes space_closed: "M \<subseteq> Pow \<Omega>"
33 lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>"
34   by (metis PowD contra_subsetD space_closed)
36 subsection {* Semiring of sets *}
38 subsubsection {* Disjoint sets *}
40 definition "disjoint A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"
42 lemma disjointI:
43   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}) \<Longrightarrow> disjoint A"
44   unfolding disjoint_def by auto
46 lemma disjointD:
47   "disjoint A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}"
48   unfolding disjoint_def by auto
50 lemma disjoint_empty[iff]: "disjoint {}"
51   by (auto simp: disjoint_def)
53 lemma disjoint_union:
54   assumes C: "disjoint C" and B: "disjoint B" and disj: "\<Union>C \<inter> \<Union>B = {}"
55   shows "disjoint (C \<union> B)"
56 proof (rule disjointI)
57   fix c d assume sets: "c \<in> C \<union> B" "d \<in> C \<union> B" and "c \<noteq> d"
58   show "c \<inter> d = {}"
59   proof cases
60     assume "(c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B)"
61     then show ?thesis
62     proof
63       assume "c \<in> C \<and> d \<in> C" with `c \<noteq> d` C show "c \<inter> d = {}"
64         by (auto simp: disjoint_def)
65     next
66       assume "c \<in> B \<and> d \<in> B" with `c \<noteq> d` B show "c \<inter> d = {}"
67         by (auto simp: disjoint_def)
68     qed
69   next
70     assume "\<not> ((c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B))"
71     with sets have "(c \<subseteq> \<Union>C \<and> d \<subseteq> \<Union>B) \<or> (c \<subseteq> \<Union>B \<and> d \<subseteq> \<Union>C)"
72       by auto
73     with disj show "c \<inter> d = {}" by auto
74   qed
75 qed
77 locale semiring_of_sets = subset_class +
78   assumes empty_sets[iff]: "{} \<in> M"
79   assumes Int[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
80   assumes Diff_cover:
81     "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> \<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"
83 lemma (in semiring_of_sets) finite_INT[intro]:
84   assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
85   shows "(\<Inter>i\<in>I. A i) \<in> M"
86   using assms by (induct rule: finite_ne_induct) auto
88 lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x"
89   by (metis Int_absorb1 sets_into_space)
91 lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x"
92   by (metis Int_absorb2 sets_into_space)
94 lemma (in semiring_of_sets) sets_Collect_conj:
95   assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
96   shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M"
97 proof -
98   have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}"
99     by auto
100   with assms show ?thesis by auto
101 qed
103 lemma (in semiring_of_sets) sets_Collect_finite_All':
104   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}"
105   shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
106 proof -
107   have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
108     using `S \<noteq> {}` by auto
109   with assms show ?thesis by auto
110 qed
112 locale ring_of_sets = semiring_of_sets +
113   assumes Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
115 lemma (in ring_of_sets) finite_Union [intro]:
116   "finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> Union X \<in> M"
117   by (induct set: finite) (auto simp add: Un)
119 lemma (in ring_of_sets) finite_UN[intro]:
120   assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
121   shows "(\<Union>i\<in>I. A i) \<in> M"
122   using assms by induct auto
124 lemma (in ring_of_sets) Diff [intro]:
125   assumes "a \<in> M" "b \<in> M" shows "a - b \<in> M"
126   using Diff_cover[OF assms] by auto
128 lemma ring_of_setsI:
129   assumes space_closed: "M \<subseteq> Pow \<Omega>"
130   assumes empty_sets[iff]: "{} \<in> M"
131   assumes Un[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
132   assumes Diff[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M"
133   shows "ring_of_sets \<Omega> M"
134 proof
135   fix a b assume ab: "a \<in> M" "b \<in> M"
136   from ab show "\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"
137     by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def)
138   have "a \<inter> b = a - (a - b)" by auto
139   also have "\<dots> \<in> M" using ab by auto
140   finally show "a \<inter> b \<in> M" .
141 qed fact+
143 lemma ring_of_sets_iff: "ring_of_sets \<Omega> M \<longleftrightarrow> M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)"
144 proof
145   assume "ring_of_sets \<Omega> M"
146   then interpret ring_of_sets \<Omega> M .
147   show "M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)"
148     using space_closed by auto
149 qed (auto intro!: ring_of_setsI)
151 lemma (in ring_of_sets) insert_in_sets:
152   assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M"
153 proof -
154   have "{x} \<union> A \<in> M" using assms by (rule Un)
155   thus ?thesis by auto
156 qed
158 lemma (in ring_of_sets) sets_Collect_disj:
159   assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
160   shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M"
161 proof -
162   have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}"
163     by auto
164   with assms show ?thesis by auto
165 qed
167 lemma (in ring_of_sets) sets_Collect_finite_Ex:
168   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
169   shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M"
170 proof -
171   have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})"
172     by auto
173   with assms show ?thesis by auto
174 qed
176 locale algebra = ring_of_sets +
177   assumes top [iff]: "\<Omega> \<in> M"
179 lemma (in algebra) compl_sets [intro]:
180   "a \<in> M \<Longrightarrow> \<Omega> - a \<in> M"
181   by auto
183 lemma algebra_iff_Un:
184   "algebra \<Omega> M \<longleftrightarrow>
185     M \<subseteq> Pow \<Omega> \<and>
186     {} \<in> M \<and>
187     (\<forall>a \<in> M. \<Omega> - a \<in> M) \<and>
188     (\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un")
189 proof
190   assume "algebra \<Omega> M"
191   then interpret algebra \<Omega> M .
192   show ?Un using sets_into_space by auto
193 next
194   assume ?Un
195   then have "\<Omega> \<in> M" by auto
196   interpret ring_of_sets \<Omega> M
197   proof (rule ring_of_setsI)
198     show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
199       using `?Un` by auto
200     fix a b assume a: "a \<in> M" and b: "b \<in> M"
201     then show "a \<union> b \<in> M" using `?Un` by auto
202     have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)"
203       using \<Omega> a b by auto
204     then show "a - b \<in> M"
205       using a b  `?Un` by auto
206   qed
207   show "algebra \<Omega> M" proof qed fact
208 qed
210 lemma algebra_iff_Int:
211      "algebra \<Omega> M \<longleftrightarrow>
212        M \<subseteq> Pow \<Omega> & {} \<in> M &
213        (\<forall>a \<in> M. \<Omega> - a \<in> M) &
214        (\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int")
215 proof
216   assume "algebra \<Omega> M"
217   then interpret algebra \<Omega> M .
218   show ?Int using sets_into_space by auto
219 next
220   assume ?Int
221   show "algebra \<Omega> M"
222   proof (unfold algebra_iff_Un, intro conjI ballI)
223     show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
224       using `?Int` by auto
225     from `?Int` show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto
226     fix a b assume M: "a \<in> M" "b \<in> M"
227     hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))"
228       using \<Omega> by blast
229     also have "... \<in> M"
230       using M `?Int` by auto
231     finally show "a \<union> b \<in> M" .
232   qed
233 qed
235 lemma (in algebra) sets_Collect_neg:
236   assumes "{x\<in>\<Omega>. P x} \<in> M"
237   shows "{x\<in>\<Omega>. \<not> P x} \<in> M"
238 proof -
239   have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto
240   with assms show ?thesis by auto
241 qed
243 lemma (in algebra) sets_Collect_imp:
244   "{x\<in>\<Omega>. P x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x \<longrightarrow> P x} \<in> M"
245   unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)
247 lemma (in algebra) sets_Collect_const:
248   "{x\<in>\<Omega>. P} \<in> M"
249   by (cases P) auto
251 lemma algebra_single_set:
252   "X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }"
253   by (auto simp: algebra_iff_Int)
255 subsection {* Restricted algebras *}
257 abbreviation (in algebra)
258   "restricted_space A \<equiv> (op \<inter> A) ` M"
260 lemma (in algebra) restricted_algebra:
261   assumes "A \<in> M" shows "algebra A (restricted_space A)"
262   using assms by (auto simp: algebra_iff_Int)
264 subsection {* Sigma Algebras *}
266 locale sigma_algebra = algebra +
267   assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
269 lemma (in algebra) is_sigma_algebra:
270   assumes "finite M"
271   shows "sigma_algebra \<Omega> M"
272 proof
273   fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M"
274   then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)"
275     by auto
276   also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M"
277     using `finite M` by auto
278   finally show "(\<Union>i. A i) \<in> M" .
279 qed
281 lemma countable_UN_eq:
282   fixes A :: "'i::countable \<Rightarrow> 'a set"
283   shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow>
284     (range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)"
285 proof -
286   let ?A' = "A \<circ> from_nat"
287   have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r")
288   proof safe
289     fix x i assume "x \<in> A i" thus "x \<in> ?l"
290       by (auto intro!: exI[of _ "to_nat i"])
291   next
292     fix x i assume "x \<in> ?A' i" thus "x \<in> ?r"
293       by (auto intro!: exI[of _ "from_nat i"])
294   qed
295   have **: "range ?A' = range A"
296     using surj_from_nat
297     by (auto simp: image_compose intro!: imageI)
298   show ?thesis unfolding * ** ..
299 qed
301 lemma (in sigma_algebra) countable_Union [intro]:
302   assumes "countable X" "X \<subseteq> M" shows "Union X \<in> M"
303 proof cases
304   assume "X \<noteq> {}"
305   hence "\<Union>X = (\<Union>n. from_nat_into X n)"
306     using assms by (auto intro: from_nat_into) (metis from_nat_into_surj)
307   also have "\<dots> \<in> M" using assms
308     by (auto intro!: countable_nat_UN) (metis `X \<noteq> {}` from_nat_into set_mp)
309   finally show ?thesis .
310 qed simp
312 lemma (in sigma_algebra) countable_UN[intro]:
313   fixes A :: "'i::countable \<Rightarrow> 'a set"
314   assumes "A`X \<subseteq> M"
315   shows  "(\<Union>x\<in>X. A x) \<in> M"
316 proof -
317   let ?A = "\<lambda>i. if i \<in> X then A i else {}"
318   from assms have "range ?A \<subseteq> M" by auto
319   with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M]
320   have "(\<Union>x. ?A x) \<in> M" by auto
321   moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm)
322   ultimately show ?thesis by simp
323 qed
325 lemma (in sigma_algebra) countable_UN':
326   fixes A :: "'i \<Rightarrow> 'a set"
327   assumes X: "countable X"
328   assumes A: "A`X \<subseteq> M"
329   shows  "(\<Union>x\<in>X. A x) \<in> M"
330 proof -
331   have "(\<Union>x\<in>X. A x) = (\<Union>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
332     using X by auto
333   also have "\<dots> \<in> M"
334     using A X
335     by (intro countable_UN) auto
336   finally show ?thesis .
337 qed
339 lemma (in sigma_algebra) countable_INT [intro]:
340   fixes A :: "'i::countable \<Rightarrow> 'a set"
341   assumes A: "A`X \<subseteq> M" "X \<noteq> {}"
342   shows "(\<Inter>i\<in>X. A i) \<in> M"
343 proof -
344   from A have "\<forall>i\<in>X. A i \<in> M" by fast
345   hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast
346   moreover
347   have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A
348     by blast
349   ultimately show ?thesis by metis
350 qed
352 lemma (in sigma_algebra) countable_INT':
353   fixes A :: "'i \<Rightarrow> 'a set"
354   assumes X: "countable X" "X \<noteq> {}"
355   assumes A: "A`X \<subseteq> M"
356   shows  "(\<Inter>x\<in>X. A x) \<in> M"
357 proof -
358   have "(\<Inter>x\<in>X. A x) = (\<Inter>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
359     using X by auto
360   also have "\<dots> \<in> M"
361     using A X
362     by (intro countable_INT) auto
363   finally show ?thesis .
364 qed
366 lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)"
367   by (auto simp: ring_of_sets_iff)
369 lemma algebra_Pow: "algebra sp (Pow sp)"
370   by (auto simp: algebra_iff_Un)
372 lemma sigma_algebra_iff:
373   "sigma_algebra \<Omega> M \<longleftrightarrow>
374     algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
375   by (simp add: sigma_algebra_def sigma_algebra_axioms_def)
377 lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)"
378   by (auto simp: sigma_algebra_iff algebra_iff_Int)
380 lemma (in sigma_algebra) sets_Collect_countable_All:
381   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
382   shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M"
383 proof -
384   have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto
385   with assms show ?thesis by auto
386 qed
388 lemma (in sigma_algebra) sets_Collect_countable_Ex:
389   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
390   shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M"
391 proof -
392   have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto
393   with assms show ?thesis by auto
394 qed
396 lemma (in sigma_algebra) sets_Collect_countable_Ex':
397   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
398   assumes "countable I"
399   shows "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} \<in> M"
400 proof -
401   have "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} = (\<Union>i\<in>I. {x\<in>\<Omega>. P i x})" by auto
402   with assms show ?thesis
403     by (auto intro!: countable_UN')
404 qed
406 lemmas (in sigma_algebra) sets_Collect =
407   sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const
408   sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All
410 lemma (in sigma_algebra) sets_Collect_countable_Ball:
411   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
412   shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M"
413   unfolding Ball_def by (intro sets_Collect assms)
415 lemma (in sigma_algebra) sets_Collect_countable_Bex:
416   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
417   shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M"
418   unfolding Bex_def by (intro sets_Collect assms)
420 lemma sigma_algebra_single_set:
421   assumes "X \<subseteq> S"
422   shows "sigma_algebra S { {}, X, S - X, S }"
423   using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp
425 subsection {* Binary Unions *}
427 definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
428   where "binary a b =  (\<lambda>x. b)(0 := a)"
430 lemma range_binary_eq: "range(binary a b) = {a,b}"
431   by (auto simp add: binary_def)
433 lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"
434   by (simp add: SUP_def range_binary_eq)
436 lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"
437   by (simp add: INF_def range_binary_eq)
439 lemma sigma_algebra_iff2:
440      "sigma_algebra \<Omega> M \<longleftrightarrow>
441        M \<subseteq> Pow \<Omega> \<and>
442        {} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and>
443        (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
444   by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
445          algebra_iff_Un Un_range_binary)
447 subsection {* Initial Sigma Algebra *}
449 text {*Sigma algebras can naturally be created as the closure of any set of
450   M with regard to the properties just postulated.  *}
452 inductive_set sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
453   for sp :: "'a set" and A :: "'a set set"
454   where
455     Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"
456   | Empty: "{} \<in> sigma_sets sp A"
457   | Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A"
458   | Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A"
460 lemma (in sigma_algebra) sigma_sets_subset:
461   assumes a: "a \<subseteq> M"
462   shows "sigma_sets \<Omega> a \<subseteq> M"
463 proof
464   fix x
465   assume "x \<in> sigma_sets \<Omega> a"
466   from this show "x \<in> M"
467     by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
468 qed
470 lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp"
471   by (erule sigma_sets.induct, auto)
473 lemma sigma_algebra_sigma_sets:
474      "a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
475   by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp
476            intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl)
478 lemma sigma_sets_least_sigma_algebra:
479   assumes "A \<subseteq> Pow S"
480   shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
481 proof safe
482   fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"
483     and X: "X \<in> sigma_sets S A"
484   from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X
485   show "X \<in> B" by auto
486 next
487   fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
488   then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B"
489      by simp
490   have "A \<subseteq> sigma_sets S A" using assms by auto
491   moreover have "sigma_algebra S (sigma_sets S A)"
492     using assms by (intro sigma_algebra_sigma_sets[of A]) auto
493   ultimately show "X \<in> sigma_sets S A" by auto
494 qed
496 lemma sigma_sets_top: "sp \<in> sigma_sets sp A"
497   by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)
499 lemma sigma_sets_Un:
500   "a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A"
501 apply (simp add: Un_range_binary range_binary_eq)
502 apply (rule Union, simp add: binary_def)
503 done
505 lemma sigma_sets_Inter:
506   assumes Asb: "A \<subseteq> Pow sp"
507   shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A"
508 proof -
509   assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A"
510   hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A"
511     by (rule sigma_sets.Compl)
512   hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
513     by (rule sigma_sets.Union)
514   hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
515     by (rule sigma_sets.Compl)
516   also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)"
517     by auto
518   also have "... = (\<Inter>i. a i)" using ai
519     by (blast dest: sigma_sets_into_sp [OF Asb])
520   finally show ?thesis .
521 qed
523 lemma sigma_sets_INTER:
524   assumes Asb: "A \<subseteq> Pow sp"
525       and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}"
526   shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A"
527 proof -
528   from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A"
529     by (simp add: sigma_sets.intros(2-) sigma_sets_top)
530   hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A"
531     by (rule sigma_sets_Inter [OF Asb])
532   also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)"
533     by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+
534   finally show ?thesis .
535 qed
537 lemma sigma_sets_UNION: "countable B \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A) \<Longrightarrow> (\<Union>B) \<in> sigma_sets X A"
538   using from_nat_into[of B] range_from_nat_into[of B] sigma_sets.Union[of "from_nat_into B" X A]
539   apply (cases "B = {}")
540   apply (simp add: sigma_sets.Empty)
541   apply (simp del: Union_image_eq add: Union_image_eq[symmetric])
542   done
544 lemma (in sigma_algebra) sigma_sets_eq:
545      "sigma_sets \<Omega> M = M"
546 proof
547   show "M \<subseteq> sigma_sets \<Omega> M"
548     by (metis Set.subsetI sigma_sets.Basic)
549   next
550   show "sigma_sets \<Omega> M \<subseteq> M"
551     by (metis sigma_sets_subset subset_refl)
552 qed
554 lemma sigma_sets_eqI:
555   assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"
556   assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A"
557   shows "sigma_sets M A = sigma_sets M B"
558 proof (intro set_eqI iffI)
559   fix a assume "a \<in> sigma_sets M A"
560   from this A show "a \<in> sigma_sets M B"
561     by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
562 next
563   fix b assume "b \<in> sigma_sets M B"
564   from this B show "b \<in> sigma_sets M A"
565     by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
566 qed
568 lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
569 proof
570   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
571     by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
572 qed
574 lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
575 proof
576   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
577     by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros(2-))
578 qed
580 lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
581 proof
582   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
583     by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
584 qed
586 lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
587   by (auto intro: sigma_sets.Basic)
589 lemma (in sigma_algebra) restriction_in_sets:
590   fixes A :: "nat \<Rightarrow> 'a set"
591   assumes "S \<in> M"
592   and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r")
593   shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
594 proof -
595   { fix i have "A i \<in> ?r" using * by auto
596     hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto
597     hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto }
598   thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
599     by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
600 qed
602 lemma (in sigma_algebra) restricted_sigma_algebra:
603   assumes "S \<in> M"
604   shows "sigma_algebra S (restricted_space S)"
605   unfolding sigma_algebra_def sigma_algebra_axioms_def
606 proof safe
607   show "algebra S (restricted_space S)" using restricted_algebra[OF assms] .
608 next
609   fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S"
610   from restriction_in_sets[OF assms this[simplified]]
611   show "(\<Union>i. A i) \<in> restricted_space S" by simp
612 qed
614 lemma sigma_sets_Int:
615   assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp"
616   shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)"
617 proof (intro equalityI subsetI)
618   fix x assume "x \<in> op \<inter> A ` sigma_sets sp st"
619   then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto
620   then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)"
621   proof (induct arbitrary: x)
622     case (Compl a)
623     then show ?case
624       by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps)
625   next
626     case (Union a)
627     then show ?case
628       by (auto intro!: sigma_sets.Union
629                simp add: UN_extend_simps simp del: UN_simps)
630   qed (auto intro!: sigma_sets.intros(2-))
631   then show "x \<in> sigma_sets A (op \<inter> A ` st)"
632     using `A \<subseteq> sp` by (simp add: Int_absorb2)
633 next
634   fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)"
635   then show "x \<in> op \<inter> A ` sigma_sets sp st"
636   proof induct
637     case (Compl a)
638     then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto
639     then show ?case using `A \<subseteq> sp`
640       by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)
641   next
642     case (Union a)
643     then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x"
644       by (auto simp: image_iff Bex_def)
645     from choice[OF this] guess f ..
646     then show ?case
647       by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union
648                simp add: image_iff)
649   qed (auto intro!: sigma_sets.intros(2-))
650 qed
652 lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}"
653 proof (intro set_eqI iffI)
654   fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}"
655     by induct blast+
656 qed (auto intro: sigma_sets.Empty sigma_sets_top)
658 lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}"
659 proof (intro set_eqI iffI)
660   fix x assume "x \<in> sigma_sets A {A}"
661   then show "x \<in> {{}, A}"
662     by induct blast+
663 next
664   fix x assume "x \<in> {{}, A}"
665   then show "x \<in> sigma_sets A {A}"
666     by (auto intro: sigma_sets.Empty sigma_sets_top)
667 qed
669 lemma sigma_sets_sigma_sets_eq:
670   "M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M"
671   by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto
673 lemma sigma_sets_singleton:
674   assumes "X \<subseteq> S"
675   shows "sigma_sets S { X } = { {}, X, S - X, S }"
676 proof -
677   interpret sigma_algebra S "{ {}, X, S - X, S }"
678     by (rule sigma_algebra_single_set) fact
679   have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }"
680     by (rule sigma_sets_subseteq) simp
681   moreover have "\<dots> = { {}, X, S - X, S }"
682     using sigma_sets_eq by simp
683   moreover
684   { fix A assume "A \<in> { {}, X, S - X, S }"
685     then have "A \<in> sigma_sets S { X }"
686       by (auto intro: sigma_sets.intros(2-) sigma_sets_top) }
687   ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }"
688     by (intro antisym) auto
689   with sigma_sets_eq show ?thesis by simp
690 qed
692 lemma restricted_sigma:
693   assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>"
694   shows "algebra.restricted_space (sigma_sets \<Omega> M) S =
695     sigma_sets S (algebra.restricted_space M S)"
696 proof -
697   from S sigma_sets_into_sp[OF M]
698   have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto
699   from sigma_sets_Int[OF this]
700   show ?thesis by simp
701 qed
703 lemma sigma_sets_vimage_commute:
704   assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'"
705   shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'}
706        = sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R")
707 proof
708   show "?L \<subseteq> ?R"
709   proof clarify
710     fix A assume "A \<in> sigma_sets \<Omega>' M'"
711     then show "X -` A \<inter> \<Omega> \<in> ?R"
712     proof induct
713       case Empty then show ?case
714         by (auto intro!: sigma_sets.Empty)
715     next
716       case (Compl B)
717       have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)"
718         by (auto simp add: funcset_mem [OF X])
719       with Compl show ?case
720         by (auto intro!: sigma_sets.Compl)
721     next
722       case (Union F)
723       then show ?case
724         by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps
725                  intro!: sigma_sets.Union)
726     qed auto
727   qed
728   show "?R \<subseteq> ?L"
729   proof clarify
730     fix A assume "A \<in> ?R"
731     then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'"
732     proof induct
733       case (Basic B) then show ?case by auto
734     next
735       case Empty then show ?case
736         by (auto intro!: sigma_sets.Empty exI[of _ "{}"])
737     next
738       case (Compl B)
739       then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto
740       then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>"
741         by (auto simp add: funcset_mem [OF X])
742       with A(2) show ?case
743         by (auto intro: sigma_sets.Compl)
744     next
745       case (Union F)
746       then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto
747       from choice[OF this] guess A .. note A = this
748       with A show ?case
749         by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)
750     qed
751   qed
752 qed
754 subsection "Disjoint families"
756 definition
757   disjoint_family_on  where
758   "disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
760 abbreviation
761   "disjoint_family A \<equiv> disjoint_family_on A UNIV"
763 lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B"
764   by blast
766 lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
767   by blast
769 lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
770   by blast
772 lemma disjoint_family_subset:
773      "disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
774   by (force simp add: disjoint_family_on_def)
776 lemma disjoint_family_on_bisimulation:
777   assumes "disjoint_family_on f S"
778   and "\<And>n m. n \<in> S \<Longrightarrow> m \<in> S \<Longrightarrow> n \<noteq> m \<Longrightarrow> f n \<inter> f m = {} \<Longrightarrow> g n \<inter> g m = {}"
779   shows "disjoint_family_on g S"
780   using assms unfolding disjoint_family_on_def by auto
782 lemma disjoint_family_on_mono:
783   "A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"
784   unfolding disjoint_family_on_def by auto
786 lemma disjoint_family_Suc:
787   assumes Suc: "!!n. A n \<subseteq> A (Suc n)"
788   shows "disjoint_family (\<lambda>i. A (Suc i) - A i)"
789 proof -
790   {
791     fix m
792     have "!!n. A n \<subseteq> A (m+n)"
793     proof (induct m)
794       case 0 show ?case by simp
795     next
796       case (Suc m) thus ?case
797         by (metis Suc_eq_plus1 assms nat_add_commute nat_add_left_commute subset_trans)
798     qed
799   }
800   hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n"
802   thus ?thesis
803     by (auto simp add: disjoint_family_on_def)
804       (metis insert_absorb insert_subset le_SucE le_antisym not_leE)
805 qed
807 lemma setsum_indicator_disjoint_family:
808   fixes f :: "'d \<Rightarrow> 'e::semiring_1"
809   assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
810   shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
811 proof -
812   have "P \<inter> {i. x \<in> A i} = {j}"
813     using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
814     by auto
815   thus ?thesis
816     unfolding indicator_def
817     by (simp add: if_distrib setsum_cases[OF `finite P`])
818 qed
820 definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
821   where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
823 lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
824 proof (induct n)
825   case 0 show ?case by simp
826 next
827   case (Suc n)
828   thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
829 qed
831 lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
832   apply (rule UN_finite2_eq [where k=0])
833   apply (simp add: finite_UN_disjointed_eq)
834   done
836 lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
837   by (auto simp add: disjointed_def)
839 lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
840   by (simp add: disjoint_family_on_def)
841      (metis neq_iff Int_commute less_disjoint_disjointed)
843 lemma disjointed_subset: "disjointed A n \<subseteq> A n"
844   by (auto simp add: disjointed_def)
846 lemma (in ring_of_sets) UNION_in_sets:
847   fixes A:: "nat \<Rightarrow> 'a set"
848   assumes A: "range A \<subseteq> M"
849   shows  "(\<Union>i\<in>{0..<n}. A i) \<in> M"
850 proof (induct n)
851   case 0 show ?case by simp
852 next
853   case (Suc n)
854   thus ?case
855     by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
856 qed
858 lemma (in ring_of_sets) range_disjointed_sets:
859   assumes A: "range A \<subseteq> M"
860   shows  "range (disjointed A) \<subseteq> M"
861 proof (auto simp add: disjointed_def)
862   fix n
863   show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets
864     by (metis A Diff UNIV_I image_subset_iff)
865 qed
867 lemma (in algebra) range_disjointed_sets':
868   "range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M"
869   using range_disjointed_sets .
871 lemma disjointed_0[simp]: "disjointed A 0 = A 0"
872   by (simp add: disjointed_def)
874 lemma incseq_Un:
875   "incseq A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n"
876   unfolding incseq_def by auto
878 lemma disjointed_incseq:
879   "incseq A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n"
880   using incseq_Un[of A]
881   by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
883 lemma sigma_algebra_disjoint_iff:
884   "sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and>
885     (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
886 proof (auto simp add: sigma_algebra_iff)
887   fix A :: "nat \<Rightarrow> 'a set"
888   assume M: "algebra \<Omega> M"
889      and A: "range A \<subseteq> M"
890      and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M"
891   hence "range (disjointed A) \<subseteq> M \<longrightarrow>
892          disjoint_family (disjointed A) \<longrightarrow>
893          (\<Union>i. disjointed A i) \<in> M" by blast
894   hence "(\<Union>i. disjointed A i) \<in> M"
895     by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed)
896   thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)
897 qed
899 lemma disjoint_family_on_disjoint_image:
900   "disjoint_family_on A I \<Longrightarrow> disjoint (A ` I)"
901   unfolding disjoint_family_on_def disjoint_def by force
903 lemma disjoint_image_disjoint_family_on:
904   assumes d: "disjoint (A ` I)" and i: "inj_on A I"
905   shows "disjoint_family_on A I"
906   unfolding disjoint_family_on_def
907 proof (intro ballI impI)
908   fix n m assume nm: "m \<in> I" "n \<in> I" and "n \<noteq> m"
909   with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}"
910     by (intro disjointD[OF d]) auto
911 qed
913 subsection {* Ring generated by a semiring *}
915 definition (in semiring_of_sets)
916   "generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"
918 lemma (in semiring_of_sets) generated_ringE[elim?]:
919   assumes "a \<in> generated_ring"
920   obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
921   using assms unfolding generated_ring_def by auto
923 lemma (in semiring_of_sets) generated_ringI[intro?]:
924   assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
925   shows "a \<in> generated_ring"
926   using assms unfolding generated_ring_def by auto
928 lemma (in semiring_of_sets) generated_ringI_Basic:
929   "A \<in> M \<Longrightarrow> A \<in> generated_ring"
930   by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def)
932 lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:
933   assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
934   and "a \<inter> b = {}"
935   shows "a \<union> b \<in> generated_ring"
936 proof -
937   from a guess Ca .. note Ca = this
938   from b guess Cb .. note Cb = this
939   show ?thesis
940   proof
941     show "disjoint (Ca \<union> Cb)"
942       using `a \<inter> b = {}` Ca Cb by (auto intro!: disjoint_union)
943   qed (insert Ca Cb, auto)
944 qed
946 lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring"
947   by (auto simp: generated_ring_def disjoint_def)
949 lemma (in semiring_of_sets) generated_ring_disjoint_Union:
950   assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring"
951   using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty)
953 lemma (in semiring_of_sets) generated_ring_disjoint_UNION:
954   "finite I \<Longrightarrow> disjoint (A ` I) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> UNION I A \<in> generated_ring"
955   unfolding SUP_def by (intro generated_ring_disjoint_Union) auto
957 lemma (in semiring_of_sets) generated_ring_Int:
958   assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
959   shows "a \<inter> b \<in> generated_ring"
960 proof -
961   from a guess Ca .. note Ca = this
962   from b guess Cb .. note Cb = this
963   def C \<equiv> "(\<lambda>(a,b). a \<inter> b)` (Ca\<times>Cb)"
964   show ?thesis
965   proof
966     show "disjoint C"
967     proof (simp add: disjoint_def C_def, intro ballI impI)
968       fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"
969       assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"
970       then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto
971       then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"
972       proof
973         assume "a1 \<noteq> a2"
974         with sets Ca have "a1 \<inter> a2 = {}"
975           by (auto simp: disjoint_def)
976         then show ?thesis by auto
977       next
978         assume "b1 \<noteq> b2"
979         with sets Cb have "b1 \<inter> b2 = {}"
980           by (auto simp: disjoint_def)
981         then show ?thesis by auto
982       qed
983     qed
984   qed (insert Ca Cb, auto simp: C_def)
985 qed
987 lemma (in semiring_of_sets) generated_ring_Inter:
988   assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"
989   using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)
991 lemma (in semiring_of_sets) generated_ring_INTER:
992   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring"
993   unfolding INF_def by (intro generated_ring_Inter) auto
995 lemma (in semiring_of_sets) generating_ring:
996   "ring_of_sets \<Omega> generated_ring"
997 proof (rule ring_of_setsI)
998   let ?R = generated_ring
999   show "?R \<subseteq> Pow \<Omega>"
1000     using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)
1001   show "{} \<in> ?R" by (rule generated_ring_empty)
1003   { fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this
1004     fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this
1006     show "a - b \<in> ?R"
1007     proof cases
1008       assume "Cb = {}" with Cb `a \<in> ?R` show ?thesis
1009         by simp
1010     next
1011       assume "Cb \<noteq> {}"
1012       with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto
1013       also have "\<dots> \<in> ?R"
1014       proof (intro generated_ring_INTER generated_ring_disjoint_UNION)
1015         fix a b assume "a \<in> Ca" "b \<in> Cb"
1016         with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"
1017           by (auto simp add: generated_ring_def)
1018       next
1019         show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)"
1020           using Ca by (auto simp add: disjoint_def `Cb \<noteq> {}`)
1021       next
1022         show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+
1023       qed
1024       finally show "a - b \<in> ?R" .
1025     qed }
1026   note Diff = this
1028   fix a b assume sets: "a \<in> ?R" "b \<in> ?R"
1029   have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto
1030   also have "\<dots> \<in> ?R"
1031     by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto
1032   finally show "a \<union> b \<in> ?R" .
1033 qed
1035 lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"
1036 proof
1037   interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"
1038     using space_closed by (rule sigma_algebra_sigma_sets)
1039   show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"
1040     by (blast intro!: sigma_sets_mono elim: generated_ringE)
1041 qed (auto intro!: generated_ringI_Basic sigma_sets_mono)
1043 subsection {* Measure type *}
1045 definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
1046   "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"
1048 definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
1049   "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
1050     (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
1052 definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
1053   "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"
1055 typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
1056 proof
1057   have "sigma_algebra UNIV {{}, UNIV}"
1058     by (auto simp: sigma_algebra_iff2)
1059   then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "
1060     by (auto simp: measure_space_def positive_def countably_additive_def)
1061 qed
1063 definition space :: "'a measure \<Rightarrow> 'a set" where
1064   "space M = fst (Rep_measure M)"
1066 definition sets :: "'a measure \<Rightarrow> 'a set set" where
1067   "sets M = fst (snd (Rep_measure M))"
1069 definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where
1070   "emeasure M = snd (snd (Rep_measure M))"
1072 definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
1073   "measure M A = real (emeasure M A)"
1075 declare [[coercion sets]]
1077 declare [[coercion measure]]
1079 declare [[coercion emeasure]]
1081 lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
1082   by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)
1084 interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure"
1085   using measure_space[of M] by (auto simp: measure_space_def)
1087 definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
1088   "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, sigma_sets \<Omega> A,
1089     \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
1091 abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"
1093 lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
1094   unfolding measure_space_def
1095   by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)
1097 lemma (in ring_of_sets) positive_cong_eq:
1098   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"
1099   by (auto simp add: positive_def)
1101 lemma (in sigma_algebra) countably_additive_eq:
1102   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"
1104   by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)
1106 lemma measure_space_eq:
1107   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
1108   shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
1109 proof -
1110   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
1111   from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
1112     by (auto simp: measure_space_def)
1113 qed
1115 lemma measure_of_eq:
1116   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
1117   shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
1118 proof -
1119   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
1120     using assms by (rule measure_space_eq)
1121   with eq show ?thesis
1122     by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
1123 qed
1125 lemma
1126   assumes A: "A \<subseteq> Pow \<Omega>"
1127   shows sets_measure_of[simp]: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A" (is ?sets)
1128     and space_measure_of[simp]: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
1129 proof -
1130   have "?sets \<and> ?space"
1131   proof cases
1132     assume "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
1133     moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
1134        (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
1135       using A by (rule measure_space_eq) auto
1136     ultimately show "?sets \<and> ?space"
1137       by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def)
1138   next
1139     assume "\<not> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
1140     with A show "?sets \<and> ?space"
1141       by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def measure_space_0)
1142   qed
1143   then show ?sets ?space by auto
1144 qed
1146 lemma (in sigma_algebra) sets_measure_of_eq[simp]:
1147   "sets (measure_of \<Omega> M \<mu>) = M"
1148   using space_closed by (auto intro!: sigma_sets_eq)
1150 lemma (in sigma_algebra) space_measure_of_eq[simp]:
1151   "space (measure_of \<Omega> M \<mu>) = \<Omega>"
1152   using space_closed by (auto intro!: sigma_sets_eq)
1154 lemma measure_of_subset:
1155   "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
1156   by (auto intro!: sigma_sets_subseteq)
1158 lemma sigma_sets_mono'':
1159   assumes "A \<in> sigma_sets C D"
1160   assumes "B \<subseteq> D"
1161   assumes "D \<subseteq> Pow C"
1162   shows "sigma_sets A B \<subseteq> sigma_sets C D"
1163 proof
1164   fix x assume "x \<in> sigma_sets A B"
1165   thus "x \<in> sigma_sets C D"
1166   proof induct
1167     case (Basic a) with assms have "a \<in> D" by auto
1168     thus ?case ..
1169   next
1170     case Empty show ?case by (rule sigma_sets.Empty)
1171   next
1172     from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
1173     moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
1174     ultimately have "A - a \<in> sets (sigma C D)" ..
1175     thus ?case by (subst (asm) sets_measure_of[OF `D \<subseteq> Pow C`])
1176   next
1177     case (Union a)
1178     thus ?case by (intro sigma_sets.Union)
1179   qed
1180 qed
1182 lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"
1183   by auto
1185 subsection {* Constructing simple @{typ "'a measure"} *}
1187 lemma emeasure_measure_of:
1188   assumes M: "M = measure_of \<Omega> A \<mu>"
1189   assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
1190   assumes X: "X \<in> sets M"
1191   shows "emeasure M X = \<mu> X"
1192 proof -
1193   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
1194   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
1195     using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
1196   moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)
1197     = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
1198     using ms(1) by (rule measure_space_eq) auto
1199   moreover have "X \<in> sigma_sets \<Omega> A"
1200     using X M ms by simp
1201   ultimately show ?thesis
1202     unfolding emeasure_def measure_of_def M
1203     by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq)
1204 qed
1206 lemma emeasure_measure_of_sigma:
1207   assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
1208   assumes A: "A \<in> M"
1209   shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
1210 proof -
1211   interpret sigma_algebra \<Omega> M by fact
1212   have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
1213     using ms sigma_sets_eq by (simp add: measure_space_def)
1214   moreover have "measure_space \<Omega> (sigma_sets \<Omega> M) (\<lambda>a. if a \<in> sigma_sets \<Omega> M then \<mu> a else 0)
1215     = measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
1216     using space_closed by (rule measure_space_eq) auto
1217   ultimately show ?thesis using A
1218     unfolding emeasure_def measure_of_def
1219     by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq)
1220 qed
1222 lemma measure_cases[cases type: measure]:
1223   obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
1224   by atomize_elim (cases x, auto)
1226 lemma sets_eq_imp_space_eq:
1227   "sets M = sets M' \<Longrightarrow> space M = space M'"
1228   using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M']
1229   by blast
1231 lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
1232   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
1234 lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"
1235   by (simp add: measure_def emeasure_notin_sets)
1237 lemma measure_eqI:
1238   fixes M N :: "'a measure"
1239   assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
1240   shows "M = N"
1241 proof (cases M N rule: measure_cases[case_product measure_cases])
1242   case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
1243   interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
1244   interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
1245   have "A = sets M" "A' = sets N"
1246     using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
1247   with `sets M = sets N` have "A = A'" by simp
1248   moreover with M.top N.top M.space_closed N.space_closed have "\<Omega> = \<Omega>'" by auto
1249   moreover { fix B have "\<mu> B = \<mu>' B"
1250     proof cases
1251       assume "B \<in> A"
1252       with eq `A = sets M` have "emeasure M B = emeasure N B" by simp
1253       with measure_measure show "\<mu> B = \<mu>' B"
1254         by (simp add: emeasure_def Abs_measure_inverse)
1255     next
1256       assume "B \<notin> A"
1257       with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N"
1258         by auto
1259       then have "emeasure M B = 0" "emeasure N B = 0"
1260         by (simp_all add: emeasure_notin_sets)
1261       with measure_measure show "\<mu> B = \<mu>' B"
1262         by (simp add: emeasure_def Abs_measure_inverse)
1263     qed }
1264   then have "\<mu> = \<mu>'" by auto
1265   ultimately show "M = N"
1266     by (simp add: measure_measure)
1267 qed
1269 lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)"
1270   using measure_space_0[of A \<Omega>]
1271   by (simp add: measure_of_def emeasure_def Abs_measure_inverse)
1273 lemma sigma_eqI:
1274   assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
1275   shows "sigma \<Omega> M = sigma \<Omega> N"
1276   by (rule measure_eqI) (simp_all add: emeasure_sigma)
1278 subsection {* Measurable functions *}
1280 definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where
1281   "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
1283 lemma measurable_space:
1284   "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
1285    unfolding measurable_def by auto
1287 lemma measurable_sets:
1288   "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
1289    unfolding measurable_def by auto
1291 lemma measurable_sets_Collect:
1292   assumes f: "f \<in> measurable M N" and P: "{x\<in>space N. P x} \<in> sets N" shows "{x\<in>space M. P (f x)} \<in> sets M"
1293 proof -
1294   have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"
1295     using measurable_space[OF f] by auto
1296   with measurable_sets[OF f P] show ?thesis
1297     by simp
1298 qed
1300 lemma measurable_sigma_sets:
1301   assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
1302       and f: "f \<in> space M \<rightarrow> \<Omega>"
1303       and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
1304   shows "f \<in> measurable M N"
1305 proof -
1306   interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
1307   from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force
1309   { fix X assume "X \<in> sigma_sets \<Omega> A"
1310     then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
1311       proof induct
1312         case (Basic a) then show ?case
1313           by (auto simp add: ba) (metis B(2) subsetD PowD)
1314       next
1315         case (Compl a)
1316         have [simp]: "f -` \<Omega> \<inter> space M = space M"
1317           by (auto simp add: funcset_mem [OF f])
1318         then show ?case
1319           by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)
1320       next
1321         case (Union a)
1322         then show ?case
1323           by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
1324       qed auto }
1325   with f show ?thesis
1326     by (auto simp add: measurable_def B \<Omega>)
1327 qed
1329 lemma measurable_measure_of:
1330   assumes B: "N \<subseteq> Pow \<Omega>"
1331       and f: "f \<in> space M \<rightarrow> \<Omega>"
1332       and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
1333   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
1334 proof -
1335   have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
1336     using B by (rule sets_measure_of)
1337   from this assms show ?thesis by (rule measurable_sigma_sets)
1338 qed
1340 lemma measurable_iff_measure_of:
1341   assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
1342   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
1343   by (metis assms in_measure_of measurable_measure_of assms measurable_sets)
1345 lemma measurable_cong_sets:
1346   assumes sets: "sets M = sets M'" "sets N = sets N'"
1347   shows "measurable M N = measurable M' N'"
1348   using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)
1350 lemma measurable_cong:
1351   assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"
1352   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
1353   unfolding measurable_def using assms
1354   by (simp cong: vimage_inter_cong Pi_cong)
1356 lemma measurable_eqI:
1357      "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
1358         sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
1359       \<Longrightarrow> measurable m1 m2 = measurable m1' m2'"
1360   by (simp add: measurable_def sigma_algebra_iff2)
1362 lemma measurable_compose:
1363   assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"
1364   shows "(\<lambda>x. g (f x)) \<in> measurable M L"
1365 proof -
1366   have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M"
1367     using measurable_space[OF f] by auto
1368   with measurable_space[OF f] measurable_space[OF g] show ?thesis
1369     by (auto intro: measurable_sets[OF f] measurable_sets[OF g]
1370              simp del: vimage_Int simp add: measurable_def)
1371 qed
1373 lemma measurable_comp:
1374   "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"
1375   using measurable_compose[of f M N g L] by (simp add: comp_def)
1377 lemma measurable_const:
1378   "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"
1379   by (auto simp add: measurable_def)
1381 lemma measurable_If:
1382   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
1383   assumes P: "{x\<in>space M. P x} \<in> sets M"
1384   shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
1385   unfolding measurable_def
1386 proof safe
1387   fix x assume "x \<in> space M"
1388   thus "(if P x then f x else g x) \<in> space M'"
1389     using measure unfolding measurable_def by auto
1390 next
1391   fix A assume "A \<in> sets M'"
1392   hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
1393     ((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
1394     ((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
1395     using measure unfolding measurable_def by (auto split: split_if_asm)
1396   show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
1397     using `A \<in> sets M'` measure P unfolding * measurable_def
1398     by (auto intro!: sets.Un)
1399 qed
1401 lemma measurable_If_set:
1402   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
1403   assumes P: "A \<inter> space M \<in> sets M"
1404   shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
1405 proof (rule measurable_If[OF measure])
1406   have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
1407   thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto
1408 qed
1410 lemma measurable_ident: "id \<in> measurable M M"
1411   by (auto simp add: measurable_def)
1413 lemma measurable_ident_sets:
1414   assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"
1415   using measurable_ident[of M]
1416   unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .
1418 lemma sets_Least:
1419   assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
1420   shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
1421 proof -
1422   { fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
1423     proof cases
1424       assume i: "(LEAST j. False) = i"
1425       have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
1426         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x})) \<union> (space M - (\<Union>i. {x\<in>space M. P i x}))"
1427         by (simp add: set_eq_iff, safe)
1428            (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
1429       with meas show ?thesis
1430         by (auto intro!: sets.Int)
1431     next
1432       assume i: "(LEAST j. False) \<noteq> i"
1433       then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
1434         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
1435       proof (simp add: set_eq_iff, safe)
1436         fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
1437         have "\<exists>j. P j x"
1438           by (rule ccontr) (insert neq, auto)
1439         then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
1440       qed (auto dest: Least_le intro!: Least_equality)
1441       with meas show ?thesis
1442         by auto
1443     qed }
1444   then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
1445     by (intro sets.countable_UN) auto
1446   moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
1447     (\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
1448   ultimately show ?thesis by auto
1449 qed
1451 lemma measurable_strong:
1452   fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
1453   assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c"
1454       and t: "f ` (space a) \<subseteq> t"
1455       and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
1456   shows "(g o f) \<in> measurable a c"
1457 proof -
1458   have fab: "f \<in> (space a -> space b)"
1459    and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
1460      by (auto simp add: measurable_def)
1461   have eq: "\<And>y. f -` g -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
1462     by force
1463   show ?thesis
1464     apply (auto simp add: measurable_def vimage_compose)
1465     apply (metis funcset_mem fab g)
1466     apply (subst eq, metis ba cb)
1467     done
1468 qed
1470 lemma measurable_mono1:
1471   "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>
1472     measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
1473   using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)
1475 subsection {* Counting space *}
1477 definition count_space :: "'a set \<Rightarrow> 'a measure" where
1478   "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
1480 lemma
1481   shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
1482     and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
1483   using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
1484   by (auto simp: count_space_def)
1486 lemma measurable_count_space_eq1[simp]:
1487   "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
1488  unfolding measurable_def by simp
1490 lemma measurable_count_space_eq2:
1491   assumes "finite A"
1492   shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
1493 proof -
1494   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
1495     with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X"
1496       by (auto dest: finite_subset)
1497     moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
1498     ultimately have "f -` X \<inter> space M \<in> sets M"
1499       using `X \<subseteq> A` by (auto intro!: sets.finite_UN simp del: UN_simps) }
1500   then show ?thesis
1501     unfolding measurable_def by auto
1502 qed
1504 lemma measurable_compose_countable:
1505   assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"
1506   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"
1507   unfolding measurable_def
1508 proof safe
1509   fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"
1510     using f[THEN measurable_space] g[THEN measurable_space] by auto
1511 next
1512   fix A assume A: "A \<in> sets N"
1513   have "(\<lambda>x. f (g x) x) -` A \<inter> space M = (\<Union>i. (g -` {i} \<inter> space M) \<inter> (f i -` A \<inter> space M))"
1514     by auto
1515   also have "\<dots> \<in> sets M" using f[THEN measurable_sets, OF A] g[THEN measurable_sets]
1516     by (auto intro!: sets.countable_UN measurable_sets)
1517   finally show "(\<lambda>x. f (g x) x) -` A \<inter> space M \<in> sets M" .
1518 qed
1520 lemma measurable_count_space_const:
1521   "(\<lambda>x. c) \<in> measurable M (count_space UNIV)"
1522   by (simp add: measurable_const)
1524 lemma measurable_count_space:
1525   "f \<in> measurable (count_space A) (count_space UNIV)"
1526   by simp
1528 lemma measurable_compose_rev:
1529   assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"
1530   shows "(\<lambda>x. f (g x)) \<in> measurable M N"
1531   using measurable_compose[OF g f] .
1534 subsection {* Extend measure *}
1536 definition "extend_measure \<Omega> I G \<mu> =
1537   (if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0)
1538       then measure_of \<Omega> (G`I) (SOME \<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>')
1539       else measure_of \<Omega> (G`I) (\<lambda>_. 0))"
1541 lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
1542   unfolding extend_measure_def by simp
1544 lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)"
1545   unfolding extend_measure_def by simp
1547 lemma emeasure_extend_measure:
1548   assumes M: "M = extend_measure \<Omega> I G \<mu>"
1549     and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"
1550     and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
1551     and "i \<in> I"
1552   shows "emeasure M (G i) = \<mu> i"
1553 proof cases
1554   assume *: "(\<forall>i\<in>I. \<mu> i = 0)"
1555   with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)"
1556    by (simp add: extend_measure_def)
1557   from measure_space_0[OF ms(1)] ms `i\<in>I`
1558   have "emeasure M (G i) = 0"
1559     by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)
1560   with `i\<in>I` * show ?thesis
1561     by simp
1562 next
1563   def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'"
1564   assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"
1565   moreover
1566   have "measure_space (space M) (sets M) \<mu>'"
1567     using ms unfolding measure_space_def by auto default
1568   with ms eq have "\<exists>\<mu>'. P \<mu>'"
1569     unfolding P_def
1570     by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)
1571   ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)"
1572     by (simp add: M extend_measure_def P_def[symmetric])
1574   from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex)
1575   show "emeasure M (G i) = \<mu> i"
1576   proof (subst emeasure_measure_of[OF M_eq])
1577     have sets_M: "sets M = sigma_sets \<Omega> (G`I)"
1578       using M_eq ms by (auto simp: sets_extend_measure)
1579     then show "G i \<in> sets M" using `i \<in> I` by auto
1580     show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"
1581       using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def)
1582   qed fact
1583 qed
1585 lemma emeasure_extend_measure_Pair:
1586   assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"
1587     and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"
1588     and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
1589     and "I i j"
1590   shows "emeasure M (G i j) = \<mu> i j"
1591   using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j`
1592   by (auto simp: subset_eq)
1594 subsection {* Sigma algebra generated by function preimages *}
1596 definition
1597   "vimage_algebra M S f = sigma S ((\<lambda>A. f -` A \<inter> S) ` sets M)"
1599 lemma sigma_algebra_preimages:
1600   fixes f :: "'x \<Rightarrow> 'a"
1601   assumes "f \<in> S \<rightarrow> space M"
1602   shows "sigma_algebra S ((\<lambda>A. f -` A \<inter> S) ` sets M)"
1603     (is "sigma_algebra _ (?F ` sets M)")
1604 proof (simp add: sigma_algebra_iff2, safe)
1605   show "{} \<in> ?F ` sets M" by blast
1606 next
1607   fix A assume "A \<in> sets M"
1608   moreover have "S - ?F A = ?F (space M - A)"
1609     using assms by auto
1610   ultimately show "S - ?F A \<in> ?F ` sets M"
1611     by blast
1612 next
1613   fix A :: "nat \<Rightarrow> 'x set" assume *: "range A \<subseteq> ?F ` M"
1614   have "\<forall>i. \<exists>b. b \<in> M \<and> A i = ?F b"
1615   proof safe
1616     fix i
1617     have "A i \<in> ?F ` M" using * by auto
1618     then show "\<exists>b. b \<in> M \<and> A i = ?F b" by auto
1619   qed
1620   from choice[OF this] obtain b where b: "range b \<subseteq> M" "\<And>i. A i = ?F (b i)"
1621     by auto
1622   then have "(\<Union>i. A i) = ?F (\<Union>i. b i)" by auto
1623   then show "(\<Union>i. A i) \<in> ?F ` M" using b(1) by blast
1624 qed
1626 lemma sets_vimage_algebra[simp]:
1627   "f \<in> S \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra M S f) = (\<lambda>A. f -` A \<inter> S) ` sets M"
1628   using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_preimages, of f S M]
1629   by (simp add: vimage_algebra_def)
1631 lemma space_vimage_algebra[simp]:
1632   "f \<in> S \<rightarrow> space M \<Longrightarrow> space (vimage_algebra M S f) = S"
1633   using sigma_algebra.space_measure_of_eq[OF sigma_algebra_preimages, of f S M]
1634   by (simp add: vimage_algebra_def)
1636 lemma in_vimage_algebra[simp]:
1637   "f \<in> S \<rightarrow> space M \<Longrightarrow> A \<in> sets (vimage_algebra M S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f -` B \<inter> S)"
1638   by (simp add: image_iff)
1640 lemma measurable_vimage_algebra:
1641   fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
1642   shows "f \<in> measurable (vimage_algebra M S f) M"
1643   unfolding measurable_def using assms by force
1645 lemma measurable_vimage:
1646   fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a"
1647   assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M"
1648   shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra M S f) M2"
1649 proof -
1650   note measurable_vimage_algebra[OF assms(2)]
1651   from measurable_comp[OF this assms(1)]
1652   show ?thesis by (simp add: comp_def)
1653 qed
1655 lemma sigma_sets_vimage:
1656   assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S"
1657   shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A"
1658 proof (intro set_eqI iffI)
1659   let ?F = "\<lambda>X. f -` X \<inter> S'"
1660   fix X assume "X \<in> sigma_sets S' (?F ` A)"
1661   then show "X \<in> ?F ` sigma_sets S A"
1662   proof induct
1663     case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A"
1664       by auto
1665     then show ?case by auto
1666   next
1667     case Empty then show ?case
1668       by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty)
1669   next
1670     case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A"
1671       by auto
1672     then have "S - X' \<in> sigma_sets S A"
1673       by (auto intro!: sigma_sets.Compl)
1674     then show ?case
1675       using X assms by (auto intro!: image_eqI[where x="S - X'"])
1676   next
1677     case (Union F)
1678     then have "\<forall>i. \<exists>F'.  F' \<in> sigma_sets S A \<and> F i = f -` F' \<inter> S'"
1679       by (auto simp: image_iff Bex_def)
1680     from choice[OF this] obtain F' where
1681       "\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f -` F' i \<inter> S'"
1682       by auto
1683     then show ?case
1684       by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"])
1685   qed
1686 next
1687   let ?F = "\<lambda>X. f -` X \<inter> S'"
1688   fix X assume "X \<in> ?F ` sigma_sets S A"
1689   then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto
1690   then show "X \<in> sigma_sets S' (?F ` A)"
1691   proof (induct arbitrary: X)
1692     case Empty then show ?case by (auto intro: sigma_sets.Empty)
1693   next
1694     case (Compl X')
1695     have "S' - (S' - X) \<in> sigma_sets S' (?F ` A)"
1696       apply (rule sigma_sets.Compl)
1697       using assms by (auto intro!: Compl.hyps simp: Compl.prems)
1698     also have "S' - (S' - X) = X"
1699       using assms Compl by auto
1700     finally show ?case .
1701   next
1702     case (Union F)
1703     have "(\<Union>i. f -` F i \<inter> S') \<in> sigma_sets S' (?F ` A)"
1704       by (intro sigma_sets.Union Union.hyps) simp
1705     also have "(\<Union>i. f -` F i \<inter> S') = X"
1706       using assms Union by auto
1707     finally show ?case .
1708   qed auto
1709 qed
1711 subsection {* A Two-Element Series *}
1713 definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
1714   where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"
1716 lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
1717   apply (simp add: binaryset_def)
1718   apply (rule set_eqI)
1719   apply (auto simp add: image_iff)
1720   done
1722 lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
1723   by (simp add: SUP_def range_binaryset_eq)
1725 section {* Closed CDI *}
1727 definition closed_cdi where
1728   "closed_cdi \<Omega> M \<longleftrightarrow>
1729    M \<subseteq> Pow \<Omega> &
1730    (\<forall>s \<in> M. \<Omega> - s \<in> M) &
1731    (\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
1732         (\<Union>i. A i) \<in> M) &
1733    (\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
1735 inductive_set
1736   smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
1737   for \<Omega> M
1738   where
1739     Basic [intro]:
1740       "a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"
1741   | Compl [intro]:
1742       "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"
1743   | Inc:
1744       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
1745        \<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"
1746   | Disj:
1747       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A
1748        \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"
1750 lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"
1751   by auto
1753 lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"
1754   apply (rule subsetI)
1755   apply (erule smallest_ccdi_sets.induct)
1756   apply (auto intro: range_subsetD dest: sets_into_space)
1757   done
1759 lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"
1760   apply (auto simp add: closed_cdi_def smallest_ccdi_sets)
1761   apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +
1762   done
1764 lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"
1765   by (simp add: closed_cdi_def)
1767 lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"
1768   by (simp add: closed_cdi_def)
1770 lemma closed_cdi_Inc:
1771   "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow> (\<Union>i. A i) \<in> M"
1772   by (simp add: closed_cdi_def)
1774 lemma closed_cdi_Disj:
1775   "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
1776   by (simp add: closed_cdi_def)
1778 lemma closed_cdi_Un:
1779   assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"
1780       and A: "A \<in> M" and B: "B \<in> M"
1781       and disj: "A \<inter> B = {}"
1782     shows "A \<union> B \<in> M"
1783 proof -
1784   have ra: "range (binaryset A B) \<subseteq> M"
1785    by (simp add: range_binaryset_eq empty A B)
1786  have di:  "disjoint_family (binaryset A B)" using disj
1787    by (simp add: disjoint_family_on_def binaryset_def Int_commute)
1788  from closed_cdi_Disj [OF cdi ra di]
1789  show ?thesis
1790    by (simp add: UN_binaryset_eq)
1791 qed
1793 lemma (in algebra) smallest_ccdi_sets_Un:
1794   assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"
1795       and disj: "A \<inter> B = {}"
1796     shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"
1797 proof -
1798   have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"
1799     by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)
1800   have di:  "disjoint_family (binaryset A B)" using disj
1801     by (simp add: disjoint_family_on_def binaryset_def Int_commute)
1802   from Disj [OF ra di]
1803   show ?thesis
1804     by (simp add: UN_binaryset_eq)
1805 qed
1807 lemma (in algebra) smallest_ccdi_sets_Int1:
1808   assumes a: "a \<in> M"
1809   shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
1810 proof (induct rule: smallest_ccdi_sets.induct)
1811   case (Basic x)
1812   thus ?case
1813     by (metis a Int smallest_ccdi_sets.Basic)
1814 next
1815   case (Compl x)
1816   have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"
1817     by blast
1818   also have "... \<in> smallest_ccdi_sets \<Omega> M"
1819     by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2
1820            Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un
1821            smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)
1822   finally show ?case .
1823 next
1824   case (Inc A)
1825   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
1826     by blast
1827   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
1828     by blast
1829   moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"
1830     by (simp add: Inc)
1831   moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc
1832     by blast
1833   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
1834     by (rule smallest_ccdi_sets.Inc)
1835   show ?case
1836     by (metis 1 2)
1837 next
1838   case (Disj A)
1839   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
1840     by blast
1841   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
1842     by blast
1843   moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj
1844     by (auto simp add: disjoint_family_on_def)
1845   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
1846     by (rule smallest_ccdi_sets.Disj)
1847   show ?case
1848     by (metis 1 2)
1849 qed
1852 lemma (in algebra) smallest_ccdi_sets_Int:
1853   assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"
1854   shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
1855 proof (induct rule: smallest_ccdi_sets.induct)
1856   case (Basic x)
1857   thus ?case
1858     by (metis b smallest_ccdi_sets_Int1)
1859 next
1860   case (Compl x)
1861   have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"
1862     by blast
1863   also have "... \<in> smallest_ccdi_sets \<Omega> M"
1864     by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b
1865            smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)
1866   finally show ?case .
1867 next
1868   case (Inc A)
1869   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
1870     by blast
1871   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
1872     by blast
1873   moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"
1874     by (simp add: Inc)
1875   moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc
1876     by blast
1877   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
1878     by (rule smallest_ccdi_sets.Inc)
1879   show ?case
1880     by (metis 1 2)
1881 next
1882   case (Disj A)
1883   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
1884     by blast
1885   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
1886     by blast
1887   moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj
1888     by (auto simp add: disjoint_family_on_def)
1889   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
1890     by (rule smallest_ccdi_sets.Disj)
1891   show ?case
1892     by (metis 1 2)
1893 qed
1895 lemma (in algebra) sigma_property_disjoint_lemma:
1896   assumes sbC: "M \<subseteq> C"
1897       and ccdi: "closed_cdi \<Omega> C"
1898   shows "sigma_sets \<Omega> M \<subseteq> C"
1899 proof -
1900   have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"
1901     apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int
1902             smallest_ccdi_sets_Int)
1903     apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)
1904     apply (blast intro: smallest_ccdi_sets.Disj)
1905     done
1906   hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"
1907     by clarsimp
1908        (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)
1909   also have "...  \<subseteq> C"
1910     proof
1911       fix x
1912       assume x: "x \<in> smallest_ccdi_sets \<Omega> M"
1913       thus "x \<in> C"
1914         proof (induct rule: smallest_ccdi_sets.induct)
1915           case (Basic x)
1916           thus ?case
1917             by (metis Basic subsetD sbC)
1918         next
1919           case (Compl x)
1920           thus ?case
1921             by (blast intro: closed_cdi_Compl [OF ccdi, simplified])
1922         next
1923           case (Inc A)
1924           thus ?case
1925                by (auto intro: closed_cdi_Inc [OF ccdi, simplified])
1926         next
1927           case (Disj A)
1928           thus ?case
1929                by (auto intro: closed_cdi_Disj [OF ccdi, simplified])
1930         qed
1931     qed
1932   finally show ?thesis .
1933 qed
1935 lemma (in algebra) sigma_property_disjoint:
1936   assumes sbC: "M \<subseteq> C"
1937       and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"
1938       and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
1939                      \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))
1940                      \<Longrightarrow> (\<Union>i. A i) \<in> C"
1941       and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
1942                       \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"
1943   shows "sigma_sets (\<Omega>) (M) \<subseteq> C"
1944 proof -
1945   have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
1946     proof (rule sigma_property_disjoint_lemma)
1947       show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
1948         by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
1949     next
1950       show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"
1951         by (simp add: closed_cdi_def compl inc disj)
1952            (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed
1953              IntE sigma_sets.Compl range_subsetD sigma_sets.Union)
1954     qed
1955   thus ?thesis
1956     by blast
1957 qed
1959 subsection {* Dynkin systems *}
1961 locale dynkin_system = subset_class +
1962   assumes space: "\<Omega> \<in> M"
1963     and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
1964     and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
1965                            \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
1967 lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"
1968   using space compl[of "\<Omega>"] by simp
1970 lemma (in dynkin_system) diff:
1971   assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"
1972   shows "E - D \<in> M"
1973 proof -
1974   let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"
1975   have "range ?f = {D, \<Omega> - E, {}}"
1976     by (auto simp: image_iff)
1977   moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"
1978     by (auto simp: image_iff split: split_if_asm)
1979   moreover
1980   then have "disjoint_family ?f" unfolding disjoint_family_on_def
1981     using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto
1982   ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"
1983     using sets by auto
1984   also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"
1985     using assms sets_into_space by auto
1986   finally show ?thesis .
1987 qed
1989 lemma dynkin_systemI:
1990   assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"
1991   assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
1992   assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
1993           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
1994   shows "dynkin_system \<Omega> M"
1995   using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)
1997 lemma dynkin_systemI':
1998   assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"
1999   assumes empty: "{} \<in> M"
2000   assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
2001   assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
2002           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
2003   shows "dynkin_system \<Omega> M"
2004 proof -
2005   from Diff[OF empty] have "\<Omega> \<in> M" by auto
2006   from 1 this Diff 2 show ?thesis
2007     by (intro dynkin_systemI) auto
2008 qed
2010 lemma dynkin_system_trivial:
2011   shows "dynkin_system A (Pow A)"
2012   by (rule dynkin_systemI) auto
2014 lemma sigma_algebra_imp_dynkin_system:
2015   assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"
2016 proof -
2017   interpret sigma_algebra \<Omega> M by fact
2018   show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
2019 qed
2021 subsection "Intersection stable algebras"
2023 definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"
2025 lemma (in algebra) Int_stable: "Int_stable M"
2026   unfolding Int_stable_def by auto
2028 lemma Int_stableI:
2029   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"
2030   unfolding Int_stable_def by auto
2032 lemma Int_stableD:
2033   "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
2034   unfolding Int_stable_def by auto
2036 lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
2037   "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"
2038 proof
2039   assume "sigma_algebra \<Omega> M" then show "Int_stable M"
2040     unfolding sigma_algebra_def using algebra.Int_stable by auto
2041 next
2042   assume "Int_stable M"
2043   show "sigma_algebra \<Omega> M"
2044     unfolding sigma_algebra_disjoint_iff algebra_iff_Un
2045   proof (intro conjI ballI allI impI)
2046     show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto
2047   next
2048     fix A B assume "A \<in> M" "B \<in> M"
2049     then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"
2050               "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"
2051       using sets_into_space by auto
2052     then show "A \<union> B \<in> M"
2053       using `Int_stable M` unfolding Int_stable_def by auto
2054   qed auto
2055 qed
2057 subsection "Smallest Dynkin systems"
2059 definition dynkin where
2060   "dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"
2062 lemma dynkin_system_dynkin:
2063   assumes "M \<subseteq> Pow (\<Omega>)"
2064   shows "dynkin_system \<Omega> (dynkin \<Omega> M)"
2065 proof (rule dynkin_systemI)
2066   fix A assume "A \<in> dynkin \<Omega> M"
2067   moreover
2068   { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"
2069     then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }
2070   moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
2071     using assms dynkin_system_trivial by fastforce
2072   ultimately show "A \<subseteq> \<Omega>"
2073     unfolding dynkin_def using assms
2074     by auto
2075 next
2076   show "\<Omega> \<in> dynkin \<Omega> M"
2077     unfolding dynkin_def using dynkin_system.space by fastforce
2078 next
2079   fix A assume "A \<in> dynkin \<Omega> M"
2080   then show "\<Omega> - A \<in> dynkin \<Omega> M"
2081     unfolding dynkin_def using dynkin_system.compl by force
2082 next
2083   fix A :: "nat \<Rightarrow> 'a set"
2084   assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"
2085   show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def
2086   proof (simp, safe)
2087     fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"
2088     with A have "(\<Union>i. A i) \<in> D"
2089       by (intro dynkin_system.UN) (auto simp: dynkin_def)
2090     then show "(\<Union>i. A i) \<in> D" by auto
2091   qed
2092 qed
2094 lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"
2095   unfolding dynkin_def by auto
2097 lemma (in dynkin_system) restricted_dynkin_system:
2098   assumes "D \<in> M"
2099   shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
2100 proof (rule dynkin_systemI, simp_all)
2101   have "\<Omega> \<inter> D = D"
2102     using `D \<in> M` sets_into_space by auto
2103   then show "\<Omega> \<inter> D \<in> M"
2104     using `D \<in> M` by auto
2105 next
2106   fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"
2107   moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"
2108     by auto
2109   ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"
2110     using  `D \<in> M` by (auto intro: diff)
2111 next
2112   fix A :: "nat \<Rightarrow> 'a set"
2113   assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
2114   then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"
2115     "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
2116     by ((fastforce simp: disjoint_family_on_def)+)
2117   then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"
2118     by (auto simp del: UN_simps)
2119 qed
2121 lemma (in dynkin_system) dynkin_subset:
2122   assumes "N \<subseteq> M"
2123   shows "dynkin \<Omega> N \<subseteq> M"
2124 proof -
2125   have "dynkin_system \<Omega> M" by default
2126   then have "dynkin_system \<Omega> M"
2127     using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
2128   with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def)
2129 qed
2131 lemma sigma_eq_dynkin:
2132   assumes sets: "M \<subseteq> Pow \<Omega>"
2133   assumes "Int_stable M"
2134   shows "sigma_sets \<Omega> M = dynkin \<Omega> M"
2135 proof -
2136   have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
2137     using sigma_algebra_imp_dynkin_system
2138     unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
2139   moreover
2140   interpret dynkin_system \<Omega> "dynkin \<Omega> M"
2141     using dynkin_system_dynkin[OF sets] .
2142   have "sigma_algebra \<Omega> (dynkin \<Omega> M)"
2143     unfolding sigma_algebra_eq_Int_stable Int_stable_def
2144   proof (intro ballI)
2145     fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"
2146     let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"
2147     have "M \<subseteq> ?D B"
2148     proof
2149       fix E assume "E \<in> M"
2150       then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
2151         using sets_into_space `Int_stable M` by (auto simp: Int_stable_def)
2152       then have "dynkin \<Omega> M \<subseteq> ?D E"
2153         using restricted_dynkin_system `E \<in> dynkin \<Omega> M`
2154         by (intro dynkin_system.dynkin_subset) simp_all
2155       then have "B \<in> ?D E"
2156         using `B \<in> dynkin \<Omega> M` by auto
2157       then have "E \<inter> B \<in> dynkin \<Omega> M"
2158         by (subst Int_commute) simp
2159       then show "E \<in> ?D B"
2160         using sets `E \<in> M` by auto
2161     qed
2162     then have "dynkin \<Omega> M \<subseteq> ?D B"
2163       using restricted_dynkin_system `B \<in> dynkin \<Omega> M`
2164       by (intro dynkin_system.dynkin_subset) simp_all
2165     then show "A \<inter> B \<in> dynkin \<Omega> M"
2166       using `A \<in> dynkin \<Omega> M` sets_into_space by auto
2167   qed
2168   from sigma_algebra.sigma_sets_subset[OF this, of "M"]
2169   have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
2170   ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto
2171   then show ?thesis
2172     by (auto simp: dynkin_def)
2173 qed
2175 lemma (in dynkin_system) dynkin_idem:
2176   "dynkin \<Omega> M = M"
2177 proof -
2178   have "dynkin \<Omega> M = M"
2179   proof
2180     show "M \<subseteq> dynkin \<Omega> M"
2181       using dynkin_Basic by auto
2182     show "dynkin \<Omega> M \<subseteq> M"
2183       by (intro dynkin_subset) auto
2184   qed
2185   then show ?thesis
2186     by (auto simp: dynkin_def)
2187 qed
2189 lemma (in dynkin_system) dynkin_lemma:
2190   assumes "Int_stable E"
2191   and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"
2192   shows "sigma_sets \<Omega> E = M"
2193 proof -
2194   have "E \<subseteq> Pow \<Omega>"
2195     using E sets_into_space by force
2196   then have "sigma_sets \<Omega> E = dynkin \<Omega> E"
2197     using `Int_stable E` by (rule sigma_eq_dynkin)
2198   moreover then have "dynkin \<Omega> E = M"
2199     using assms dynkin_subset[OF E(1)] by simp
2200   ultimately show ?thesis
2201     using assms by (auto simp: dynkin_def)
2202 qed
2204 lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
2205   assumes "Int_stable G"
2206     and closed: "G \<subseteq> Pow \<Omega>"
2207     and A: "A \<in> sigma_sets \<Omega> G"
2208   assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"
2209     and empty: "P {}"
2210     and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"
2211     and union: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> sigma_sets \<Omega> G \<Longrightarrow> (\<And>i. P (A i)) \<Longrightarrow> P (\<Union>i::nat. A i)"
2212   shows "P A"
2213 proof -
2214   let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"
2215   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"
2216     using closed by (rule sigma_algebra_sigma_sets)
2217   from compl[OF _ empty] closed have space: "P \<Omega>" by simp
2218   interpret dynkin_system \<Omega> ?D
2219     by default (auto dest: sets_into_space intro!: space compl union)
2220   have "sigma_sets \<Omega> G = ?D"
2221     by (rule dynkin_lemma) (auto simp: basic `Int_stable G`)
2222   with A show ?thesis by auto
2223 qed
2225 end