src/HOL/Probability/Sigma_Algebra.thy
 author hoelzl Fri Nov 02 14:23:40 2012 +0100 (2012-11-02) changeset 50002 ce0d316b5b44 parent 49834 b27bbb021df1 child 50003 8c213922ed49 permissions -rw-r--r--
add measurability prover; add support for Borel sets
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"
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 section {* 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_UN[intro]:
302   fixes A :: "'i::countable \<Rightarrow> 'a set"
303   assumes "A`X \<subseteq> M"
304   shows  "(\<Union>x\<in>X. A x) \<in> M"
305 proof -
306   let ?A = "\<lambda>i. if i \<in> X then A i else {}"
307   from assms have "range ?A \<subseteq> M" by auto
308   with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M]
309   have "(\<Union>x. ?A x) \<in> M" by auto
310   moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm)
311   ultimately show ?thesis by simp
312 qed
314 lemma (in sigma_algebra) countable_INT [intro]:
315   fixes A :: "'i::countable \<Rightarrow> 'a set"
316   assumes A: "A`X \<subseteq> M" "X \<noteq> {}"
317   shows "(\<Inter>i\<in>X. A i) \<in> M"
318 proof -
319   from A have "\<forall>i\<in>X. A i \<in> M" by fast
320   hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast
321   moreover
322   have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A
323     by blast
324   ultimately show ?thesis by metis
325 qed
327 lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)"
328   by (auto simp: ring_of_sets_iff)
330 lemma algebra_Pow: "algebra sp (Pow sp)"
331   by (auto simp: algebra_iff_Un)
333 lemma sigma_algebra_iff:
334   "sigma_algebra \<Omega> M \<longleftrightarrow>
335     algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
336   by (simp add: sigma_algebra_def sigma_algebra_axioms_def)
338 lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)"
339   by (auto simp: sigma_algebra_iff algebra_iff_Int)
341 lemma (in sigma_algebra) sets_Collect_countable_All:
342   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
343   shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M"
344 proof -
345   have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto
346   with assms show ?thesis by auto
347 qed
349 lemma (in sigma_algebra) sets_Collect_countable_Ex:
350   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
351   shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M"
352 proof -
353   have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto
354   with assms show ?thesis by auto
355 qed
357 lemmas (in sigma_algebra) sets_Collect =
358   sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const
359   sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All
361 lemma (in sigma_algebra) sets_Collect_countable_Ball:
362   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
363   shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M"
364   unfolding Ball_def by (intro sets_Collect assms)
366 lemma (in sigma_algebra) sets_Collect_countable_Bex:
367   assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
368   shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M"
369   unfolding Bex_def by (intro sets_Collect assms)
371 lemma sigma_algebra_single_set:
372   assumes "X \<subseteq> S"
373   shows "sigma_algebra S { {}, X, S - X, S }"
374   using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp
376 subsection {* Binary Unions *}
378 definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
379   where "binary a b =  (\<lambda>\<^isup>x. b)(0 := a)"
381 lemma range_binary_eq: "range(binary a b) = {a,b}"
382   by (auto simp add: binary_def)
384 lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"
385   by (simp add: SUP_def range_binary_eq)
387 lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"
388   by (simp add: INF_def range_binary_eq)
390 lemma sigma_algebra_iff2:
391      "sigma_algebra \<Omega> M \<longleftrightarrow>
392        M \<subseteq> Pow \<Omega> \<and>
393        {} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and>
394        (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
395   by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
396          algebra_iff_Un Un_range_binary)
398 subsection {* Initial Sigma Algebra *}
400 text {*Sigma algebras can naturally be created as the closure of any set of
401   M with regard to the properties just postulated.  *}
403 inductive_set
404   sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
405   for sp :: "'a set" and A :: "'a set set"
406   where
407     Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"
408   | Empty: "{} \<in> sigma_sets sp A"
409   | Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A"
410   | Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A"
412 lemma (in sigma_algebra) sigma_sets_subset:
413   assumes a: "a \<subseteq> M"
414   shows "sigma_sets \<Omega> a \<subseteq> M"
415 proof
416   fix x
417   assume "x \<in> sigma_sets \<Omega> a"
418   from this show "x \<in> M"
419     by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
420 qed
422 lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp"
423   by (erule sigma_sets.induct, auto)
425 lemma sigma_algebra_sigma_sets:
426      "a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
427   by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp
428            intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl)
430 lemma sigma_sets_least_sigma_algebra:
431   assumes "A \<subseteq> Pow S"
432   shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
433 proof safe
434   fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"
435     and X: "X \<in> sigma_sets S A"
436   from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X
437   show "X \<in> B" by auto
438 next
439   fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
440   then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B"
441      by simp
442   have "A \<subseteq> sigma_sets S A" using assms by auto
443   moreover have "sigma_algebra S (sigma_sets S A)"
444     using assms by (intro sigma_algebra_sigma_sets[of A]) auto
445   ultimately show "X \<in> sigma_sets S A" by auto
446 qed
448 lemma sigma_sets_top: "sp \<in> sigma_sets sp A"
449   by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)
451 lemma sigma_sets_Un:
452   "a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A"
453 apply (simp add: Un_range_binary range_binary_eq)
454 apply (rule Union, simp add: binary_def)
455 done
457 lemma sigma_sets_Inter:
458   assumes Asb: "A \<subseteq> Pow sp"
459   shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A"
460 proof -
461   assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A"
462   hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A"
463     by (rule sigma_sets.Compl)
464   hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
465     by (rule sigma_sets.Union)
466   hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
467     by (rule sigma_sets.Compl)
468   also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)"
469     by auto
470   also have "... = (\<Inter>i. a i)" using ai
471     by (blast dest: sigma_sets_into_sp [OF Asb])
472   finally show ?thesis .
473 qed
475 lemma sigma_sets_INTER:
476   assumes Asb: "A \<subseteq> Pow sp"
477       and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}"
478   shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A"
479 proof -
480   from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A"
481     by (simp add: sigma_sets.intros(2-) sigma_sets_top)
482   hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A"
483     by (rule sigma_sets_Inter [OF Asb])
484   also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)"
485     by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+
486   finally show ?thesis .
487 qed
489 lemma (in sigma_algebra) sigma_sets_eq:
490      "sigma_sets \<Omega> M = M"
491 proof
492   show "M \<subseteq> sigma_sets \<Omega> M"
493     by (metis Set.subsetI sigma_sets.Basic)
494   next
495   show "sigma_sets \<Omega> M \<subseteq> M"
496     by (metis sigma_sets_subset subset_refl)
497 qed
499 lemma sigma_sets_eqI:
500   assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"
501   assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A"
502   shows "sigma_sets M A = sigma_sets M B"
503 proof (intro set_eqI iffI)
504   fix a assume "a \<in> sigma_sets M A"
505   from this A show "a \<in> sigma_sets M B"
506     by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
507 next
508   fix b assume "b \<in> sigma_sets M B"
509   from this B show "b \<in> sigma_sets M A"
510     by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
511 qed
513 lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
514 proof
515   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
516     by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
517 qed
519 lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
520 proof
521   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
522     by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros(2-))
523 qed
525 lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
526 proof
527   fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
528     by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
529 qed
531 lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
532   by (auto intro: sigma_sets.Basic)
534 lemma (in sigma_algebra) restriction_in_sets:
535   fixes A :: "nat \<Rightarrow> 'a set"
536   assumes "S \<in> M"
537   and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r")
538   shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
539 proof -
540   { fix i have "A i \<in> ?r" using * by auto
541     hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto
542     hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto }
543   thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
544     by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
545 qed
547 lemma (in sigma_algebra) restricted_sigma_algebra:
548   assumes "S \<in> M"
549   shows "sigma_algebra S (restricted_space S)"
550   unfolding sigma_algebra_def sigma_algebra_axioms_def
551 proof safe
552   show "algebra S (restricted_space S)" using restricted_algebra[OF assms] .
553 next
554   fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S"
555   from restriction_in_sets[OF assms this[simplified]]
556   show "(\<Union>i. A i) \<in> restricted_space S" by simp
557 qed
559 lemma sigma_sets_Int:
560   assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp"
561   shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)"
562 proof (intro equalityI subsetI)
563   fix x assume "x \<in> op \<inter> A ` sigma_sets sp st"
564   then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto
565   then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)"
566   proof (induct arbitrary: x)
567     case (Compl a)
568     then show ?case
569       by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps)
570   next
571     case (Union a)
572     then show ?case
573       by (auto intro!: sigma_sets.Union
574                simp add: UN_extend_simps simp del: UN_simps)
575   qed (auto intro!: sigma_sets.intros(2-))
576   then show "x \<in> sigma_sets A (op \<inter> A ` st)"
577     using `A \<subseteq> sp` by (simp add: Int_absorb2)
578 next
579   fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)"
580   then show "x \<in> op \<inter> A ` sigma_sets sp st"
581   proof induct
582     case (Compl a)
583     then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto
584     then show ?case using `A \<subseteq> sp`
585       by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)
586   next
587     case (Union a)
588     then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x"
589       by (auto simp: image_iff Bex_def)
590     from choice[OF this] guess f ..
591     then show ?case
592       by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union
593                simp add: image_iff)
594   qed (auto intro!: sigma_sets.intros(2-))
595 qed
597 lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}"
598 proof (intro set_eqI iffI)
599   fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}"
600     by induct blast+
601 qed (auto intro: sigma_sets.Empty sigma_sets_top)
603 lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}"
604 proof (intro set_eqI iffI)
605   fix x assume "x \<in> sigma_sets A {A}"
606   then show "x \<in> {{}, A}"
607     by induct blast+
608 next
609   fix x assume "x \<in> {{}, A}"
610   then show "x \<in> sigma_sets A {A}"
611     by (auto intro: sigma_sets.Empty sigma_sets_top)
612 qed
614 lemma sigma_sets_sigma_sets_eq:
615   "M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M"
616   by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto
618 lemma sigma_sets_singleton:
619   assumes "X \<subseteq> S"
620   shows "sigma_sets S { X } = { {}, X, S - X, S }"
621 proof -
622   interpret sigma_algebra S "{ {}, X, S - X, S }"
623     by (rule sigma_algebra_single_set) fact
624   have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }"
625     by (rule sigma_sets_subseteq) simp
626   moreover have "\<dots> = { {}, X, S - X, S }"
627     using sigma_sets_eq by simp
628   moreover
629   { fix A assume "A \<in> { {}, X, S - X, S }"
630     then have "A \<in> sigma_sets S { X }"
631       by (auto intro: sigma_sets.intros(2-) sigma_sets_top) }
632   ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }"
633     by (intro antisym) auto
634   with sigma_sets_eq show ?thesis by simp
635 qed
637 lemma restricted_sigma:
638   assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>"
639   shows "algebra.restricted_space (sigma_sets \<Omega> M) S =
640     sigma_sets S (algebra.restricted_space M S)"
641 proof -
642   from S sigma_sets_into_sp[OF M]
643   have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto
644   from sigma_sets_Int[OF this]
645   show ?thesis by simp
646 qed
648 lemma sigma_sets_vimage_commute:
649   assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'"
650   shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'}
651        = sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R")
652 proof
653   show "?L \<subseteq> ?R"
654   proof clarify
655     fix A assume "A \<in> sigma_sets \<Omega>' M'"
656     then show "X -` A \<inter> \<Omega> \<in> ?R"
657     proof induct
658       case Empty then show ?case
659         by (auto intro!: sigma_sets.Empty)
660     next
661       case (Compl B)
662       have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)"
663         by (auto simp add: funcset_mem [OF X])
664       with Compl show ?case
665         by (auto intro!: sigma_sets.Compl)
666     next
667       case (Union F)
668       then show ?case
669         by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps
670                  intro!: sigma_sets.Union)
671     qed auto
672   qed
673   show "?R \<subseteq> ?L"
674   proof clarify
675     fix A assume "A \<in> ?R"
676     then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'"
677     proof induct
678       case (Basic B) then show ?case by auto
679     next
680       case Empty then show ?case
681         by (auto intro!: sigma_sets.Empty exI[of _ "{}"])
682     next
683       case (Compl B)
684       then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto
685       then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>"
686         by (auto simp add: funcset_mem [OF X])
687       with A(2) show ?case
688         by (auto intro: sigma_sets.Compl)
689     next
690       case (Union F)
691       then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto
692       from choice[OF this] guess A .. note A = this
693       with A show ?case
694         by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)
695     qed
696   qed
697 qed
699 section "Disjoint families"
701 definition
702   disjoint_family_on  where
703   "disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
705 abbreviation
706   "disjoint_family A \<equiv> disjoint_family_on A UNIV"
708 lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B"
709   by blast
711 lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
712   by blast
714 lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
715   by blast
717 lemma disjoint_family_subset:
718      "disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
719   by (force simp add: disjoint_family_on_def)
721 lemma disjoint_family_on_bisimulation:
722   assumes "disjoint_family_on f S"
723   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 = {}"
724   shows "disjoint_family_on g S"
725   using assms unfolding disjoint_family_on_def by auto
727 lemma disjoint_family_on_mono:
728   "A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"
729   unfolding disjoint_family_on_def by auto
731 lemma disjoint_family_Suc:
732   assumes Suc: "!!n. A n \<subseteq> A (Suc n)"
733   shows "disjoint_family (\<lambda>i. A (Suc i) - A i)"
734 proof -
735   {
736     fix m
737     have "!!n. A n \<subseteq> A (m+n)"
738     proof (induct m)
739       case 0 show ?case by simp
740     next
741       case (Suc m) thus ?case
742         by (metis Suc_eq_plus1 assms nat_add_commute nat_add_left_commute subset_trans)
743     qed
744   }
745   hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n"
747   thus ?thesis
748     by (auto simp add: disjoint_family_on_def)
749       (metis insert_absorb insert_subset le_SucE le_antisym not_leE)
750 qed
752 lemma setsum_indicator_disjoint_family:
753   fixes f :: "'d \<Rightarrow> 'e::semiring_1"
754   assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
755   shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
756 proof -
757   have "P \<inter> {i. x \<in> A i} = {j}"
758     using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
759     by auto
760   thus ?thesis
761     unfolding indicator_def
762     by (simp add: if_distrib setsum_cases[OF `finite P`])
763 qed
765 definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
766   where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
768 lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
769 proof (induct n)
770   case 0 show ?case by simp
771 next
772   case (Suc n)
773   thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
774 qed
776 lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
777   apply (rule UN_finite2_eq [where k=0])
778   apply (simp add: finite_UN_disjointed_eq)
779   done
781 lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
782   by (auto simp add: disjointed_def)
784 lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
785   by (simp add: disjoint_family_on_def)
786      (metis neq_iff Int_commute less_disjoint_disjointed)
788 lemma disjointed_subset: "disjointed A n \<subseteq> A n"
789   by (auto simp add: disjointed_def)
791 lemma (in ring_of_sets) UNION_in_sets:
792   fixes A:: "nat \<Rightarrow> 'a set"
793   assumes A: "range A \<subseteq> M"
794   shows  "(\<Union>i\<in>{0..<n}. A i) \<in> M"
795 proof (induct n)
796   case 0 show ?case by simp
797 next
798   case (Suc n)
799   thus ?case
800     by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
801 qed
803 lemma (in ring_of_sets) range_disjointed_sets:
804   assumes A: "range A \<subseteq> M"
805   shows  "range (disjointed A) \<subseteq> M"
806 proof (auto simp add: disjointed_def)
807   fix n
808   show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets
809     by (metis A Diff UNIV_I image_subset_iff)
810 qed
812 lemma (in algebra) range_disjointed_sets':
813   "range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M"
814   using range_disjointed_sets .
816 lemma disjointed_0[simp]: "disjointed A 0 = A 0"
817   by (simp add: disjointed_def)
819 lemma incseq_Un:
820   "incseq A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n"
821   unfolding incseq_def by auto
823 lemma disjointed_incseq:
824   "incseq A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n"
825   using incseq_Un[of A]
826   by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
828 lemma sigma_algebra_disjoint_iff:
829   "sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and>
830     (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
831 proof (auto simp add: sigma_algebra_iff)
832   fix A :: "nat \<Rightarrow> 'a set"
833   assume M: "algebra \<Omega> M"
834      and A: "range A \<subseteq> M"
835      and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M"
836   hence "range (disjointed A) \<subseteq> M \<longrightarrow>
837          disjoint_family (disjointed A) \<longrightarrow>
838          (\<Union>i. disjointed A i) \<in> M" by blast
839   hence "(\<Union>i. disjointed A i) \<in> M"
840     by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed)
841   thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)
842 qed
844 lemma disjoint_family_on_disjoint_image:
845   "disjoint_family_on A I \<Longrightarrow> disjoint (A ` I)"
846   unfolding disjoint_family_on_def disjoint_def by force
848 lemma disjoint_image_disjoint_family_on:
849   assumes d: "disjoint (A ` I)" and i: "inj_on A I"
850   shows "disjoint_family_on A I"
851   unfolding disjoint_family_on_def
852 proof (intro ballI impI)
853   fix n m assume nm: "m \<in> I" "n \<in> I" and "n \<noteq> m"
854   with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}"
855     by (intro disjointD[OF d]) auto
856 qed
858 section {* Ring generated by a semiring *}
860 definition (in semiring_of_sets)
861   "generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"
863 lemma (in semiring_of_sets) generated_ringE[elim?]:
864   assumes "a \<in> generated_ring"
865   obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
866   using assms unfolding generated_ring_def by auto
868 lemma (in semiring_of_sets) generated_ringI[intro?]:
869   assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
870   shows "a \<in> generated_ring"
871   using assms unfolding generated_ring_def by auto
873 lemma (in semiring_of_sets) generated_ringI_Basic:
874   "A \<in> M \<Longrightarrow> A \<in> generated_ring"
875   by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def)
877 lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:
878   assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
879   and "a \<inter> b = {}"
880   shows "a \<union> b \<in> generated_ring"
881 proof -
882   from a guess Ca .. note Ca = this
883   from b guess Cb .. note Cb = this
884   show ?thesis
885   proof
886     show "disjoint (Ca \<union> Cb)"
887       using `a \<inter> b = {}` Ca Cb by (auto intro!: disjoint_union)
888   qed (insert Ca Cb, auto)
889 qed
891 lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring"
892   by (auto simp: generated_ring_def disjoint_def)
894 lemma (in semiring_of_sets) generated_ring_disjoint_Union:
895   assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring"
896   using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty)
898 lemma (in semiring_of_sets) generated_ring_disjoint_UNION:
899   "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"
900   unfolding SUP_def by (intro generated_ring_disjoint_Union) auto
902 lemma (in semiring_of_sets) generated_ring_Int:
903   assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
904   shows "a \<inter> b \<in> generated_ring"
905 proof -
906   from a guess Ca .. note Ca = this
907   from b guess Cb .. note Cb = this
908   def C \<equiv> "(\<lambda>(a,b). a \<inter> b)` (Ca\<times>Cb)"
909   show ?thesis
910   proof
911     show "disjoint C"
912     proof (simp add: disjoint_def C_def, intro ballI impI)
913       fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"
914       assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"
915       then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto
916       then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"
917       proof
918         assume "a1 \<noteq> a2"
919         with sets Ca have "a1 \<inter> a2 = {}"
920           by (auto simp: disjoint_def)
921         then show ?thesis by auto
922       next
923         assume "b1 \<noteq> b2"
924         with sets Cb have "b1 \<inter> b2 = {}"
925           by (auto simp: disjoint_def)
926         then show ?thesis by auto
927       qed
928     qed
929   qed (insert Ca Cb, auto simp: C_def)
930 qed
932 lemma (in semiring_of_sets) generated_ring_Inter:
933   assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"
934   using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)
936 lemma (in semiring_of_sets) generated_ring_INTER:
937   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring"
938   unfolding INF_def by (intro generated_ring_Inter) auto
940 lemma (in semiring_of_sets) generating_ring:
941   "ring_of_sets \<Omega> generated_ring"
942 proof (rule ring_of_setsI)
943   let ?R = generated_ring
944   show "?R \<subseteq> Pow \<Omega>"
945     using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)
946   show "{} \<in> ?R" by (rule generated_ring_empty)
948   { fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this
949     fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this
951     show "a - b \<in> ?R"
952     proof cases
953       assume "Cb = {}" with Cb `a \<in> ?R` show ?thesis
954         by simp
955     next
956       assume "Cb \<noteq> {}"
957       with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto
958       also have "\<dots> \<in> ?R"
959       proof (intro generated_ring_INTER generated_ring_disjoint_UNION)
960         fix a b assume "a \<in> Ca" "b \<in> Cb"
961         with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"
962           by (auto simp add: generated_ring_def)
963       next
964         show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)"
965           using Ca by (auto simp add: disjoint_def `Cb \<noteq> {}`)
966       next
967         show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+
968       qed
969       finally show "a - b \<in> ?R" .
970     qed }
971   note Diff = this
973   fix a b assume sets: "a \<in> ?R" "b \<in> ?R"
974   have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto
975   also have "\<dots> \<in> ?R"
976     by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto
977   finally show "a \<union> b \<in> ?R" .
978 qed
980 lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"
981 proof
982   interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"
983     using space_closed by (rule sigma_algebra_sigma_sets)
984   show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"
985     by (blast intro!: sigma_sets_mono elim: generated_ringE)
986 qed (auto intro!: generated_ringI_Basic sigma_sets_mono)
988 section {* Measure type *}
990 definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
991   "positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"
993 definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
994   "countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
995     (\<Sum>i. f (A i)) = f (\<Union>i. A i))"
997 definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
998   "measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"
1000 typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
1001 proof
1002   have "sigma_algebra UNIV {{}, UNIV}"
1003     by (auto simp: sigma_algebra_iff2)
1004   then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "
1005     by (auto simp: measure_space_def positive_def countably_additive_def)
1006 qed
1008 definition space :: "'a measure \<Rightarrow> 'a set" where
1009   "space M = fst (Rep_measure M)"
1011 definition sets :: "'a measure \<Rightarrow> 'a set set" where
1012   "sets M = fst (snd (Rep_measure M))"
1014 definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where
1015   "emeasure M = snd (snd (Rep_measure M))"
1017 definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
1018   "measure M A = real (emeasure M A)"
1020 declare [[coercion sets]]
1022 declare [[coercion measure]]
1024 declare [[coercion emeasure]]
1026 lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
1027   by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)
1029 interpretation sigma_algebra "space M" "sets M" for M :: "'a measure"
1030   using measure_space[of M] by (auto simp: measure_space_def)
1032 definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
1033   "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, sigma_sets \<Omega> A,
1034     \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
1036 abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"
1038 lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
1039   unfolding measure_space_def
1040   by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)
1042 lemma (in ring_of_sets) positive_cong_eq:
1043   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"
1044   by (auto simp add: positive_def)
1046 lemma (in sigma_algebra) countably_additive_eq:
1047   "(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"
1049   by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq)
1051 lemma measure_space_eq:
1052   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
1053   shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
1054 proof -
1055   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
1056   from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
1057     by (auto simp: measure_space_def)
1058 qed
1060 lemma measure_of_eq:
1061   assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
1062   shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
1063 proof -
1064   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
1065     using assms by (rule measure_space_eq)
1066   with eq show ?thesis
1067     by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
1068 qed
1070 lemma
1071   assumes A: "A \<subseteq> Pow \<Omega>"
1072   shows sets_measure_of[simp]: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A" (is ?sets)
1073     and space_measure_of[simp]: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
1074 proof -
1075   have "?sets \<and> ?space"
1076   proof cases
1077     assume "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
1078     moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
1079        (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
1080       using A by (rule measure_space_eq) auto
1081     ultimately show "?sets \<and> ?space"
1082       by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def)
1083   next
1084     assume "\<not> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
1085     with A show "?sets \<and> ?space"
1086       by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def measure_space_0)
1087   qed
1088   then show ?sets ?space by auto
1089 qed
1091 lemma (in sigma_algebra) sets_measure_of_eq[simp]:
1092   "sets (measure_of \<Omega> M \<mu>) = M"
1093   using space_closed by (auto intro!: sigma_sets_eq)
1095 lemma (in sigma_algebra) space_measure_of_eq[simp]:
1096   "space (measure_of \<Omega> M \<mu>) = \<Omega>"
1097   using space_closed by (auto intro!: sigma_sets_eq)
1099 lemma measure_of_subset:
1100   "M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
1101   by (auto intro!: sigma_sets_subseteq)
1103 lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"
1104   by auto
1106 section {* Constructing simple @{typ "'a measure"} *}
1108 lemma emeasure_measure_of:
1109   assumes M: "M = measure_of \<Omega> A \<mu>"
1110   assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
1111   assumes X: "X \<in> sets M"
1112   shows "emeasure M X = \<mu> X"
1113 proof -
1114   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
1115   have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
1116     using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
1117   moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)
1118     = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
1119     using ms(1) by (rule measure_space_eq) auto
1120   moreover have "X \<in> sigma_sets \<Omega> A"
1121     using X M ms by simp
1122   ultimately show ?thesis
1123     unfolding emeasure_def measure_of_def M
1124     by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq)
1125 qed
1127 lemma emeasure_measure_of_sigma:
1128   assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
1129   assumes A: "A \<in> M"
1130   shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
1131 proof -
1132   interpret sigma_algebra \<Omega> M by fact
1133   have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
1134     using ms sigma_sets_eq by (simp add: measure_space_def)
1135   moreover have "measure_space \<Omega> (sigma_sets \<Omega> M) (\<lambda>a. if a \<in> sigma_sets \<Omega> M then \<mu> a else 0)
1136     = measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
1137     using space_closed by (rule measure_space_eq) auto
1138   ultimately show ?thesis using A
1139     unfolding emeasure_def measure_of_def
1140     by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq)
1141 qed
1143 lemma measure_cases[cases type: measure]:
1144   obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
1145   by atomize_elim (cases x, auto)
1147 lemma sets_eq_imp_space_eq:
1148   "sets M = sets M' \<Longrightarrow> space M = space M'"
1149   using top[of M] top[of M'] space_closed[of M] space_closed[of M']
1150   by blast
1152 lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
1153   by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)
1155 lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"
1156   by (simp add: measure_def emeasure_notin_sets)
1158 lemma measure_eqI:
1159   fixes M N :: "'a measure"
1160   assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
1161   shows "M = N"
1162 proof (cases M N rule: measure_cases[case_product measure_cases])
1163   case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
1164   interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
1165   interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
1166   have "A = sets M" "A' = sets N"
1167     using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
1168   with `sets M = sets N` have "A = A'" by simp
1169   moreover with M.top N.top M.space_closed N.space_closed have "\<Omega> = \<Omega>'" by auto
1170   moreover { fix B have "\<mu> B = \<mu>' B"
1171     proof cases
1172       assume "B \<in> A"
1173       with eq `A = sets M` have "emeasure M B = emeasure N B" by simp
1174       with measure_measure show "\<mu> B = \<mu>' B"
1175         by (simp add: emeasure_def Abs_measure_inverse)
1176     next
1177       assume "B \<notin> A"
1178       with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N"
1179         by auto
1180       then have "emeasure M B = 0" "emeasure N B = 0"
1181         by (simp_all add: emeasure_notin_sets)
1182       with measure_measure show "\<mu> B = \<mu>' B"
1183         by (simp add: emeasure_def Abs_measure_inverse)
1184     qed }
1185   then have "\<mu> = \<mu>'" by auto
1186   ultimately show "M = N"
1187     by (simp add: measure_measure)
1188 qed
1190 lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)"
1191   using measure_space_0[of A \<Omega>]
1192   by (simp add: measure_of_def emeasure_def Abs_measure_inverse)
1194 lemma sigma_eqI:
1195   assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
1196   shows "sigma \<Omega> M = sigma \<Omega> N"
1197   by (rule measure_eqI) (simp_all add: emeasure_sigma)
1199 section {* Measurable functions *}
1201 definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where
1202   "measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"
1204 lemma measurable_space:
1205   "f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
1206    unfolding measurable_def by auto
1208 lemma measurable_sets:
1209   "f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
1210    unfolding measurable_def by auto
1212 lemma measurable_sets_Collect:
1213   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"
1214 proof -
1215   have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"
1216     using measurable_space[OF f] by auto
1217   with measurable_sets[OF f P] show ?thesis
1218     by simp
1219 qed
1221 lemma measurable_sigma_sets:
1222   assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
1223       and f: "f \<in> space M \<rightarrow> \<Omega>"
1224       and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
1225   shows "f \<in> measurable M N"
1226 proof -
1227   interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
1228   from B top[of N] A.top space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force
1230   { fix X assume "X \<in> sigma_sets \<Omega> A"
1231     then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
1232       proof induct
1233         case (Basic a) then show ?case
1234           by (auto simp add: ba) (metis B(2) subsetD PowD)
1235       next
1236         case (Compl a)
1237         have [simp]: "f -` \<Omega> \<inter> space M = space M"
1238           by (auto simp add: funcset_mem [OF f])
1239         then show ?case
1240           by (auto simp add: vimage_Diff Diff_Int_distrib2 compl_sets Compl)
1241       next
1242         case (Union a)
1243         then show ?case
1244           by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
1245       qed auto }
1246   with f show ?thesis
1247     by (auto simp add: measurable_def B \<Omega>)
1248 qed
1250 lemma measurable_measure_of:
1251   assumes B: "N \<subseteq> Pow \<Omega>"
1252       and f: "f \<in> space M \<rightarrow> \<Omega>"
1253       and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
1254   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
1255 proof -
1256   have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
1257     using B by (rule sets_measure_of)
1258   from this assms show ?thesis by (rule measurable_sigma_sets)
1259 qed
1261 lemma measurable_iff_measure_of:
1262   assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
1263   shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
1264   by (metis assms in_measure_of measurable_measure_of assms measurable_sets)
1266 lemma measurable_cong:
1267   assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"
1268   shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
1269   unfolding measurable_def using assms
1270   by (simp cong: vimage_inter_cong Pi_cong)
1272 lemma measurable_eqI:
1273      "\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
1274         sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
1275       \<Longrightarrow> measurable m1 m2 = measurable m1' m2'"
1276   by (simp add: measurable_def sigma_algebra_iff2)
1278 lemma measurable_const[intro, simp]:
1279   "c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"
1280   by (auto simp add: measurable_def)
1282 lemma measurable_If:
1283   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
1284   assumes P: "{x\<in>space M. P x} \<in> sets M"
1285   shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
1286   unfolding measurable_def
1287 proof safe
1288   fix x assume "x \<in> space M"
1289   thus "(if P x then f x else g x) \<in> space M'"
1290     using measure unfolding measurable_def by auto
1291 next
1292   fix A assume "A \<in> sets M'"
1293   hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
1294     ((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
1295     ((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
1296     using measure unfolding measurable_def by (auto split: split_if_asm)
1297   show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
1298     using `A \<in> sets M'` measure P unfolding * measurable_def
1299     by (auto intro!: Un)
1300 qed
1302 lemma measurable_If_set:
1303   assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
1304   assumes P: "A \<inter> space M \<in> sets M"
1305   shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
1306 proof (rule measurable_If[OF measure])
1307   have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
1308   thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto
1309 qed
1311 lemma measurable_ident[intro, simp]: "id \<in> measurable M M"
1312   by (auto simp add: measurable_def)
1314 lemma measurable_ident'[intro, simp]: "(\<lambda>x. x) \<in> measurable M M"
1315   by (auto simp add: measurable_def)
1317 lemma measurable_comp[intro]:
1318   fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
1319   shows "f \<in> measurable a b \<Longrightarrow> g \<in> measurable b c \<Longrightarrow> (g o f) \<in> measurable a c"
1320   apply (auto simp add: measurable_def vimage_compose)
1321   apply (subgoal_tac "f -` g -` y \<inter> space a = f -` (g -` y \<inter> space b) \<inter> space a")
1322   apply force+
1323   done
1325 lemma measurable_compose:
1326   "f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> (\<lambda>x. g (f x)) \<in> measurable M L"
1327   using measurable_comp[of f M N g L] by (simp add: comp_def)
1329 lemma sets_Least:
1330   assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
1331   shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
1332 proof -
1333   { fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
1334     proof cases
1335       assume i: "(LEAST j. False) = i"
1336       have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
1337         {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}))"
1338         by (simp add: set_eq_iff, safe)
1339            (insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
1340       with meas show ?thesis
1341         by (auto intro!: Int)
1342     next
1343       assume i: "(LEAST j. False) \<noteq> i"
1344       then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
1345         {x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
1346       proof (simp add: set_eq_iff, safe)
1347         fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
1348         have "\<exists>j. P j x"
1349           by (rule ccontr) (insert neq, auto)
1350         then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
1351       qed (auto dest: Least_le intro!: Least_equality)
1352       with meas show ?thesis
1353         by auto
1354     qed }
1355   then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
1356     by (intro countable_UN) auto
1357   moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
1358     (\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
1359   ultimately show ?thesis by auto
1360 qed
1362 lemma measurable_strong:
1363   fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
1364   assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c"
1365       and t: "f ` (space a) \<subseteq> t"
1366       and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
1367   shows "(g o f) \<in> measurable a c"
1368 proof -
1369   have fab: "f \<in> (space a -> space b)"
1370    and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
1371      by (auto simp add: measurable_def)
1372   have eq: "\<And>y. f -` g -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
1373     by force
1374   show ?thesis
1375     apply (auto simp add: measurable_def vimage_compose)
1376     apply (metis funcset_mem fab g)
1377     apply (subst eq, metis ba cb)
1378     done
1379 qed
1381 lemma measurable_mono1:
1382   "M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>
1383     measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
1384   using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)
1386 section {* Counting space *}
1388 definition count_space :: "'a set \<Rightarrow> 'a measure" where
1389   "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"
1391 lemma
1392   shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
1393     and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
1394   using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
1395   by (auto simp: count_space_def)
1397 lemma measurable_count_space_eq1[simp]:
1398   "f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
1399  unfolding measurable_def by simp
1401 lemma measurable_count_space_eq2:
1402   assumes "finite A"
1403   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))"
1404 proof -
1405   { fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
1406     with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X"
1407       by (auto dest: finite_subset)
1408     moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
1409     ultimately have "f -` X \<inter> space M \<in> sets M"
1410       using `X \<subseteq> A` by (auto intro!: finite_UN simp del: UN_simps) }
1411   then show ?thesis
1412     unfolding measurable_def by auto
1413 qed
1415 lemma measurable_compose_countable:
1416   assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"
1417   shows "(\<lambda>x. f (g x) x) \<in> measurable M N"
1418   unfolding measurable_def
1419 proof safe
1420   fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"
1421     using f[THEN measurable_space] g[THEN measurable_space] by auto
1422 next
1423   fix A assume A: "A \<in> sets N"
1424   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))"
1425     by auto
1426   also have "\<dots> \<in> sets M" using f[THEN measurable_sets, OF A] g[THEN measurable_sets]
1427     by (auto intro!: countable_UN measurable_sets)
1428   finally show "(\<lambda>x. f (g x) x) -` A \<inter> space M \<in> sets M" .
1429 qed
1431 subsection {* Measurable method *}
1433 lemma (in algebra) sets_Collect_finite_All:
1434   assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
1435   shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
1436 proof -
1437   have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (if S = {} then \<Omega> else \<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
1438     by auto
1439   with assms show ?thesis by (auto intro!: sets_Collect_finite_All')
1440 qed
1442 abbreviation "pred M P \<equiv> P \<in> measurable M (count_space (UNIV::bool set))"
1444 lemma pred_def: "pred M P \<longleftrightarrow> {x\<in>space M. P x} \<in> sets M"
1445 proof
1446   assume "pred M P"
1447   then have "P -` {True} \<inter> space M \<in> sets M"
1448     by (auto simp: measurable_count_space_eq2)
1449   also have "P -` {True} \<inter> space M = {x\<in>space M. P x}" by auto
1450   finally show "{x\<in>space M. P x} \<in> sets M" .
1451 next
1452   assume P: "{x\<in>space M. P x} \<in> sets M"
1453   moreover
1454   { fix X
1455     have "X \<in> Pow (UNIV :: bool set)" by simp
1456     then have "P -` X \<inter> space M = {x\<in>space M. ((X = {True} \<longrightarrow> P x) \<and> (X = {False} \<longrightarrow> \<not> P x) \<and> X \<noteq> {})}"
1457       unfolding UNIV_bool Pow_insert Pow_empty by auto
1458     then have "P -` X \<inter> space M \<in> sets M"
1459       by (auto intro!: sets_Collect_neg sets_Collect_imp sets_Collect_conj sets_Collect_const P) }
1460   then show "pred M P"
1461     by (auto simp: measurable_def)
1462 qed
1464 lemma pred_sets1: "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> f \<in> measurable N M \<Longrightarrow> pred N (\<lambda>x. P (f x))"
1465   by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def)
1467 lemma pred_sets2: "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A)"
1468   by (rule measurable_compose[where f=f and N=N]) (auto simp: pred_def Int_def[symmetric])
1470 lemma measurable_count_space_const:
1471   "(\<lambda>x. c) \<in> measurable M (count_space UNIV)"
1472   by auto
1474 lemma measurable_count_space:
1475   "f \<in> measurable (count_space A) (count_space UNIV)"
1476   by simp
1478 lemma measurable_compose_rev:
1479   assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"
1480   shows "(\<lambda>x. f (g x)) \<in> measurable M N"
1481   using measurable_compose[OF g f] .
1483 ML {*
1485 structure Measurable =
1486 struct
1488 datatype level = Concrete | Generic;
1490 structure Data = Generic_Data
1491 (
1492   type T = thm list * thm list;
1493   val empty = ([], []);
1494   val extend = I;
1495   val merge = fn ((a, b), (c, d)) => (a @ c, b @ d);
1496 );
1498 val debug =
1499   Attrib.setup_config_bool @{binding measurable_debug} (K false)
1501 val backtrack =
1502   Attrib.setup_config_int @{binding measurable_backtrack} (K 40)
1504 fun get lv = (case lv of Concrete => fst | Generic => snd) o Data.get o Context.Proof;
1505 fun get_all ctxt = get Concrete ctxt @ get Generic ctxt;
1507 fun update f lv = Data.map (case lv of Concrete => apfst f | Generic => apsnd f);
1508 fun add thms' = update (fn thms => thms @ thms');
1510 fun TRYALL' tacs = fold_rev (curry op APPEND') tacs (K no_tac);
1512 fun is_too_generic thm =
1513   let
1514     val concl = concl_of thm
1515     val concl' = HOLogic.dest_Trueprop concl handle TERM _ => concl
1516   in is_Var (head_of concl') end
1518 fun import_theorem thm = if is_too_generic thm then [] else
1519   [thm] @ map_filter (try (fn th' => thm RS th'))
1520     [@{thm measurable_compose_rev}, @{thm pred_sets1}, @{thm pred_sets2}, @{thm sets_into_space}];
1522 fun add_thm (raw, lv) thm = add (if raw then [thm] else import_theorem thm) lv;
1524 fun debug_tac ctxt msg f = if Config.get ctxt debug then K (print_tac (msg ())) THEN' f else f
1526 fun TAKE n f thm = Seq.take n (f thm)
1528 fun nth_hol_goal thm i =
1529   HOLogic.dest_Trueprop (Logic.strip_imp_concl (strip_all_body (nth (prems_of thm) (i - 1))))
1531 fun dest_measurable_fun t =
1532   (case t of
1533     (Const (@{const_name "Set.member"}, _) \$ f \$ (Const (@{const_name "measurable"}, _) \$ _ \$ _)) => f
1534   | _ => raise (TERM ("not a measurability predicate", [t])))
1536 fun indep (Bound i) t b = i < b orelse t <= i
1537   | indep (f \$ t) top bot = indep f top bot andalso indep t top bot
1538   | indep (Abs (_,_,t)) top bot = indep t (top + 1) (bot + 1)
1539   | indep _ _ _ = true;
1541 fun cnt_prefixes ctxt (Abs (n, T, t)) = let
1542       fun is_countable t = Type.of_sort (Proof_Context.tsig_of ctxt) (t, @{sort countable})
1543       fun cnt_walk (Abs (ns, T, t)) Ts =
1544           map (fn (t', t'') => (Abs (ns, T, t'), t'')) (cnt_walk t (T::Ts))
1545         | cnt_walk (f \$ g) Ts = let
1546             val n = length Ts - 1
1547           in
1548             map (fn (f', t) => (f' \$ g, t)) (cnt_walk f Ts) @
1549             map (fn (g', t) => (f \$ g', t)) (cnt_walk g Ts) @
1550             (if is_countable (fastype_of1 (Ts, g)) andalso loose_bvar1 (g, n)
1551                 andalso indep g n 0 andalso g <> Bound n
1552               then [(f \$ Bound (n + 1), incr_boundvars (~ n) g)]
1553               else [])
1554           end
1555         | cnt_walk _ _ = []
1556     in map (fn (t1, t2) => let
1557         val T1 = fastype_of1 ([T], t2)
1558         val T2 = fastype_of1 ([T], t)
1559       in ([SOME (Abs (n, T1, Abs (n, T, t1))), NONE, NONE, SOME (Abs (n, T, t2))],
1560         [SOME T1, SOME T, SOME T2])
1561       end) (cnt_walk t [T])
1562     end
1563   | cnt_prefixes _ _ = []
1565 val split_fun_tac =
1566   Subgoal.FOCUS (fn {context = ctxt, ...} => SUBGOAL (fn (t, i) =>
1567     let
1568       val f = dest_measurable_fun (HOLogic.dest_Trueprop t)
1569       fun cert f = map (Option.map (f (Proof_Context.theory_of ctxt)))
1570       fun inst t (ts, Ts) = Drule.instantiate' (cert ctyp_of Ts) (cert cterm_of ts) t
1571       val cps = cnt_prefixes ctxt f |> map (inst @{thm measurable_compose_countable})
1572     in if null cps then no_tac else debug_tac ctxt (K "split fun") (resolve_tac cps) i end
1573     handle TERM _ => no_tac) 1)
1575 fun single_measurable_tac ctxt facts =
1576   debug_tac ctxt (fn () => "single + " ^ Pretty.str_of (Pretty.block (map (Syntax.pretty_term ctxt o prop_of) facts)))
1577   (resolve_tac ((maps (import_theorem o Simplifier.norm_hhf) facts) @ get_all ctxt)
1578     APPEND' (split_fun_tac ctxt));
1580 fun is_cond_formlua n thm = if length (prems_of thm) < n then false else
1581   (case nth_hol_goal thm n of
1582     (Const (@{const_name "Set.member"}, _) \$ _ \$ (Const (@{const_name "sets"}, _) \$ _)) => false
1583   | (Const (@{const_name "Set.member"}, _) \$ _ \$ (Const (@{const_name "measurable"}, _) \$ _ \$ _)) => false
1584   | _ => true)
1585   handle TERM _ => true;
1587 fun measurable_tac' ctxt ss facts n =
1588   TAKE (Config.get ctxt backtrack)
1589   ((single_measurable_tac ctxt facts THEN'
1590    REPEAT o (single_measurable_tac ctxt facts APPEND'
1591              SOLVED' (fn n => COND (is_cond_formlua n) (debug_tac ctxt (K "simp") (asm_full_simp_tac ss) n) no_tac))) n);
1593 fun measurable_tac ctxt = measurable_tac' ctxt (simpset_of ctxt);
1595 val attr_add = Thm.declaration_attribute o add_thm;
1597 val attr : attribute context_parser =
1598   Scan.lift (Scan.optional (Args.parens (Scan.optional (Args.\$\$\$ "raw" >> K true) false --
1599      Scan.optional (Args.\$\$\$ "generic" >> K Generic) Concrete)) (false, Concrete) >> attr_add);
1601 val method : (Proof.context -> Method.method) context_parser =
1602   Scan.lift (Scan.succeed (fn ctxt => METHOD (fn facts => measurable_tac ctxt facts 1)));
1604 fun simproc ss redex = let
1605     val ctxt = Simplifier.the_context ss;
1606     val t = HOLogic.mk_Trueprop (term_of redex);
1607     fun tac {context = ctxt, ...} =
1608       SOLVE (measurable_tac' ctxt ss (Simplifier.prems_of ss) 1);
1609   in try (fn () => Goal.prove ctxt [] [] t tac RS @{thm Eq_TrueI}) () end;
1611 end
1613 *}
1615 attribute_setup measurable = {* Measurable.attr *} "declaration of measurability theorems"
1616 method_setup measurable = {* Measurable.method *} "measurability prover"
1617 simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *}
1619 declare
1620   top[measurable]
1621   empty_sets[measurable (raw)]
1622   Un[measurable (raw)]
1623   Diff[measurable (raw)]
1625 declare
1626   measurable_count_space[measurable (raw)]
1627   measurable_ident[measurable (raw)]
1628   measurable_ident'[measurable (raw)]
1629   measurable_count_space_const[measurable (raw)]
1630   measurable_const[measurable (raw)]
1631   measurable_If[measurable (raw)]
1632   measurable_comp[measurable (raw)]
1633   measurable_sets[measurable (raw)]
1635 lemma predE[measurable (raw)]:
1636   "pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M"
1637   unfolding pred_def .
1639 lemma pred_intros_imp'[measurable (raw)]:
1640   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)"
1641   by (cases K) auto
1643 lemma pred_intros_conj1'[measurable (raw)]:
1644   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)"
1645   by (cases K) auto
1647 lemma pred_intros_conj2'[measurable (raw)]:
1648   "(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)"
1649   by (cases K) auto
1651 lemma pred_intros_disj1'[measurable (raw)]:
1652   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)"
1653   by (cases K) auto
1655 lemma pred_intros_disj2'[measurable (raw)]:
1656   "(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)"
1657   by (cases K) auto
1659 lemma pred_intros_logic[measurable (raw)]:
1660   "pred M (\<lambda>x. x \<in> space M)"
1661   "pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)"
1662   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)"
1663   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)"
1664   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)"
1665   "pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)"
1666   "pred M (\<lambda>x. f x \<in> UNIV)"
1667   "pred M (\<lambda>x. f x \<in> {})"
1668   "pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))"
1669   "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) - (B x))"
1670   "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<inter> (B x))"
1671   "pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<union> (B x))"
1672   "pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))"
1673   by (auto intro!: sets_Collect simp: iff_conv_conj_imp pred_def)
1675 lemma pred_intros_countable[measurable (raw)]:
1676   fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool"
1677   shows
1678     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)"
1679     "(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)"
1680   by (auto intro!: sets_Collect_countable_All sets_Collect_countable_Ex simp: pred_def)
1682 lemma pred_intros_countable_bounded[measurable (raw)]:
1683   fixes X :: "'i :: countable set"
1684   shows
1685     "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Inter>i\<in>X. N x i))"
1686     "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Union>i\<in>X. N x i))"
1687     "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i\<in>X. P x i)"
1688     "(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i\<in>X. P x i)"
1689   by (auto simp: Bex_def Ball_def)
1691 lemma pred_intros_finite[measurable (raw)]:
1692   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Inter>i\<in>I. N x i))"
1693   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Union>i\<in>I. N x i))"
1694   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i\<in>I. P x i)"
1695   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i\<in>I. P x i)"
1696   by (auto intro!: sets_Collect_finite_Ex sets_Collect_finite_All simp: iff_conv_conj_imp pred_def)
1698 lemma countable_Un_Int[measurable (raw)]:
1699   "(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M"
1700   "I \<noteq> {} \<Longrightarrow> (\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Inter>i\<in>I. N i) \<in> sets M"
1701   by auto
1703 declare
1704   finite_UN[measurable (raw)]
1705   finite_INT[measurable (raw)]
1707 lemma sets_Int_pred[measurable (raw)]:
1708   assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)"
1709   shows "A \<inter> B \<in> sets M"
1710 proof -
1711   have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto
1712   also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B"
1713     using space by auto
1714   finally show ?thesis .
1715 qed
1717 lemma [measurable (raw generic)]:
1718   assumes f: "f \<in> measurable M N" and c: "{c} \<in> sets N"
1719   shows pred_eq_const1: "pred M (\<lambda>x. f x = c)"
1720     and pred_eq_const2: "pred M (\<lambda>x. c = f x)"
1721   using measurable_sets[OF f c] by (auto simp: Int_def conj_commute eq_commute pred_def)
1723 lemma pred_le_const[measurable (raw generic)]:
1724   assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)"
1725   using measurable_sets[OF f c]
1726   by (auto simp: Int_def conj_commute eq_commute pred_def)
1728 lemma pred_const_le[measurable (raw generic)]:
1729   assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)"
1730   using measurable_sets[OF f c]
1731   by (auto simp: Int_def conj_commute eq_commute pred_def)
1733 lemma pred_less_const[measurable (raw generic)]:
1734   assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)"
1735   using measurable_sets[OF f c]
1736   by (auto simp: Int_def conj_commute eq_commute pred_def)
1738 lemma pred_const_less[measurable (raw generic)]:
1739   assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)"
1740   using measurable_sets[OF f c]
1741   by (auto simp: Int_def conj_commute eq_commute pred_def)
1743 declare
1744   Int[measurable (raw)]
1746 hide_const (open) pred
1748 subsection {* Extend measure *}
1750 definition "extend_measure \<Omega> I G \<mu> =
1751   (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)
1752       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>')
1753       else measure_of \<Omega> (G`I) (\<lambda>_. 0))"
1755 lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
1756   unfolding extend_measure_def by simp
1758 lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)"
1759   unfolding extend_measure_def by simp
1761 lemma emeasure_extend_measure:
1762   assumes M: "M = extend_measure \<Omega> I G \<mu>"
1763     and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"
1764     and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
1765     and "i \<in> I"
1766   shows "emeasure M (G i) = \<mu> i"
1767 proof cases
1768   assume *: "(\<forall>i\<in>I. \<mu> i = 0)"
1769   with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)"
1770    by (simp add: extend_measure_def)
1771   from measure_space_0[OF ms(1)] ms `i\<in>I`
1772   have "emeasure M (G i) = 0"
1773     by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)
1774   with `i\<in>I` * show ?thesis
1775     by simp
1776 next
1777   def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'"
1778   assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"
1779   moreover
1780   have "measure_space (space M) (sets M) \<mu>'"
1781     using ms unfolding measure_space_def by auto default
1782   with ms eq have "\<exists>\<mu>'. P \<mu>'"
1783     unfolding P_def
1784     by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)
1785   ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)"
1786     by (simp add: M extend_measure_def P_def[symmetric])
1788   from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex)
1789   show "emeasure M (G i) = \<mu> i"
1790   proof (subst emeasure_measure_of[OF M_eq])
1791     have sets_M: "sets M = sigma_sets \<Omega> (G`I)"
1792       using M_eq ms by (auto simp: sets_extend_measure)
1793     then show "G i \<in> sets M" using `i \<in> I` by auto
1794     show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"
1795       using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def)
1796   qed fact
1797 qed
1799 lemma emeasure_extend_measure_Pair:
1800   assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"
1801     and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"
1802     and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
1803     and "I i j"
1804   shows "emeasure M (G i j) = \<mu> i j"
1805   using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j`
1806   by (auto simp: subset_eq)
1808 subsection {* Sigma algebra generated by function preimages *}
1810 definition
1811   "vimage_algebra M S f = sigma S ((\<lambda>A. f -` A \<inter> S) ` sets M)"
1813 lemma sigma_algebra_preimages:
1814   fixes f :: "'x \<Rightarrow> 'a"
1815   assumes "f \<in> S \<rightarrow> space M"
1816   shows "sigma_algebra S ((\<lambda>A. f -` A \<inter> S) ` sets M)"
1817     (is "sigma_algebra _ (?F ` sets M)")
1818 proof (simp add: sigma_algebra_iff2, safe)
1819   show "{} \<in> ?F ` sets M" by blast
1820 next
1821   fix A assume "A \<in> sets M"
1822   moreover have "S - ?F A = ?F (space M - A)"
1823     using assms by auto
1824   ultimately show "S - ?F A \<in> ?F ` sets M"
1825     by blast
1826 next
1827   fix A :: "nat \<Rightarrow> 'x set" assume *: "range A \<subseteq> ?F ` M"
1828   have "\<forall>i. \<exists>b. b \<in> M \<and> A i = ?F b"
1829   proof safe
1830     fix i
1831     have "A i \<in> ?F ` M" using * by auto
1832     then show "\<exists>b. b \<in> M \<and> A i = ?F b" by auto
1833   qed
1834   from choice[OF this] obtain b where b: "range b \<subseteq> M" "\<And>i. A i = ?F (b i)"
1835     by auto
1836   then have "(\<Union>i. A i) = ?F (\<Union>i. b i)" by auto
1837   then show "(\<Union>i. A i) \<in> ?F ` M" using b(1) by blast
1838 qed
1840 lemma sets_vimage_algebra[simp]:
1841   "f \<in> S \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra M S f) = (\<lambda>A. f -` A \<inter> S) ` sets M"
1842   using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_preimages, of f S M]
1843   by (simp add: vimage_algebra_def)
1845 lemma space_vimage_algebra[simp]:
1846   "f \<in> S \<rightarrow> space M \<Longrightarrow> space (vimage_algebra M S f) = S"
1847   using sigma_algebra.space_measure_of_eq[OF sigma_algebra_preimages, of f S M]
1848   by (simp add: vimage_algebra_def)
1850 lemma in_vimage_algebra[simp]:
1851   "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)"
1852   by (simp add: image_iff)
1854 lemma measurable_vimage_algebra:
1855   fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
1856   shows "f \<in> measurable (vimage_algebra M S f) M"
1857   unfolding measurable_def using assms by force
1859 lemma measurable_vimage:
1860   fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a"
1861   assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M"
1862   shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra M S f) M2"
1863 proof -
1864   note measurable_vimage_algebra[OF assms(2)]
1865   from measurable_comp[OF this assms(1)]
1866   show ?thesis by (simp add: comp_def)
1867 qed
1869 lemma sigma_sets_vimage:
1870   assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S"
1871   shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A"
1872 proof (intro set_eqI iffI)
1873   let ?F = "\<lambda>X. f -` X \<inter> S'"
1874   fix X assume "X \<in> sigma_sets S' (?F ` A)"
1875   then show "X \<in> ?F ` sigma_sets S A"
1876   proof induct
1877     case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A"
1878       by auto
1879     then show ?case by auto
1880   next
1881     case Empty then show ?case
1882       by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty)
1883   next
1884     case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A"
1885       by auto
1886     then have "S - X' \<in> sigma_sets S A"
1887       by (auto intro!: sigma_sets.Compl)
1888     then show ?case
1889       using X assms by (auto intro!: image_eqI[where x="S - X'"])
1890   next
1891     case (Union F)
1892     then have "\<forall>i. \<exists>F'.  F' \<in> sigma_sets S A \<and> F i = f -` F' \<inter> S'"
1893       by (auto simp: image_iff Bex_def)
1894     from choice[OF this] obtain F' where
1895       "\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f -` F' i \<inter> S'"
1896       by auto
1897     then show ?case
1898       by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"])
1899   qed
1900 next
1901   let ?F = "\<lambda>X. f -` X \<inter> S'"
1902   fix X assume "X \<in> ?F ` sigma_sets S A"
1903   then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto
1904   then show "X \<in> sigma_sets S' (?F ` A)"
1905   proof (induct arbitrary: X)
1906     case Empty then show ?case by (auto intro: sigma_sets.Empty)
1907   next
1908     case (Compl X')
1909     have "S' - (S' - X) \<in> sigma_sets S' (?F ` A)"
1910       apply (rule sigma_sets.Compl)
1911       using assms by (auto intro!: Compl.hyps simp: Compl.prems)
1912     also have "S' - (S' - X) = X"
1913       using assms Compl by auto
1914     finally show ?case .
1915   next
1916     case (Union F)
1917     have "(\<Union>i. f -` F i \<inter> S') \<in> sigma_sets S' (?F ` A)"
1918       by (intro sigma_sets.Union Union.hyps) simp
1919     also have "(\<Union>i. f -` F i \<inter> S') = X"
1920       using assms Union by auto
1921     finally show ?case .
1922   qed auto
1923 qed
1925 subsection {* A Two-Element Series *}
1927 definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
1928   where "binaryset A B = (\<lambda>\<^isup>x. {})(0 := A, Suc 0 := B)"
1930 lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
1931   apply (simp add: binaryset_def)
1932   apply (rule set_eqI)
1933   apply (auto simp add: image_iff)
1934   done
1936 lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
1937   by (simp add: SUP_def range_binaryset_eq)
1939 section {* Closed CDI *}
1941 definition closed_cdi where
1942   "closed_cdi \<Omega> M \<longleftrightarrow>
1943    M \<subseteq> Pow \<Omega> &
1944    (\<forall>s \<in> M. \<Omega> - s \<in> M) &
1945    (\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
1946         (\<Union>i. A i) \<in> M) &
1947    (\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
1949 inductive_set
1950   smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
1951   for \<Omega> M
1952   where
1953     Basic [intro]:
1954       "a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"
1955   | Compl [intro]:
1956       "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"
1957   | Inc:
1958       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
1959        \<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"
1960   | Disj:
1961       "range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A
1962        \<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"
1964 lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"
1965   by auto
1967 lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"
1968   apply (rule subsetI)
1969   apply (erule smallest_ccdi_sets.induct)
1970   apply (auto intro: range_subsetD dest: sets_into_space)
1971   done
1973 lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"
1974   apply (auto simp add: closed_cdi_def smallest_ccdi_sets)
1975   apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +
1976   done
1978 lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"
1979   by (simp add: closed_cdi_def)
1981 lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"
1982   by (simp add: closed_cdi_def)
1984 lemma closed_cdi_Inc:
1985   "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"
1986   by (simp add: closed_cdi_def)
1988 lemma closed_cdi_Disj:
1989   "closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
1990   by (simp add: closed_cdi_def)
1992 lemma closed_cdi_Un:
1993   assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"
1994       and A: "A \<in> M" and B: "B \<in> M"
1995       and disj: "A \<inter> B = {}"
1996     shows "A \<union> B \<in> M"
1997 proof -
1998   have ra: "range (binaryset A B) \<subseteq> M"
1999    by (simp add: range_binaryset_eq empty A B)
2000  have di:  "disjoint_family (binaryset A B)" using disj
2001    by (simp add: disjoint_family_on_def binaryset_def Int_commute)
2002  from closed_cdi_Disj [OF cdi ra di]
2003  show ?thesis
2004    by (simp add: UN_binaryset_eq)
2005 qed
2007 lemma (in algebra) smallest_ccdi_sets_Un:
2008   assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"
2009       and disj: "A \<inter> B = {}"
2010     shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"
2011 proof -
2012   have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"
2013     by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)
2014   have di:  "disjoint_family (binaryset A B)" using disj
2015     by (simp add: disjoint_family_on_def binaryset_def Int_commute)
2016   from Disj [OF ra di]
2017   show ?thesis
2018     by (simp add: UN_binaryset_eq)
2019 qed
2021 lemma (in algebra) smallest_ccdi_sets_Int1:
2022   assumes a: "a \<in> M"
2023   shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
2024 proof (induct rule: smallest_ccdi_sets.induct)
2025   case (Basic x)
2026   thus ?case
2027     by (metis a Int smallest_ccdi_sets.Basic)
2028 next
2029   case (Compl x)
2030   have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"
2031     by blast
2032   also have "... \<in> smallest_ccdi_sets \<Omega> M"
2033     by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2
2034            Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un
2035            smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)
2036   finally show ?case .
2037 next
2038   case (Inc A)
2039   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
2040     by blast
2041   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
2042     by blast
2043   moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"
2044     by (simp add: Inc)
2045   moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc
2046     by blast
2047   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
2048     by (rule smallest_ccdi_sets.Inc)
2049   show ?case
2050     by (metis 1 2)
2051 next
2052   case (Disj A)
2053   have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
2054     by blast
2055   have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
2056     by blast
2057   moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj
2058     by (auto simp add: disjoint_family_on_def)
2059   ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
2060     by (rule smallest_ccdi_sets.Disj)
2061   show ?case
2062     by (metis 1 2)
2063 qed
2066 lemma (in algebra) smallest_ccdi_sets_Int:
2067   assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"
2068   shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
2069 proof (induct rule: smallest_ccdi_sets.induct)
2070   case (Basic x)
2071   thus ?case
2072     by (metis b smallest_ccdi_sets_Int1)
2073 next
2074   case (Compl x)
2075   have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"
2076     by blast
2077   also have "... \<in> smallest_ccdi_sets \<Omega> M"
2078     by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b
2079            smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)
2080   finally show ?case .
2081 next
2082   case (Inc A)
2083   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
2084     by blast
2085   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
2086     by blast
2087   moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"
2088     by (simp add: Inc)
2089   moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc
2090     by blast
2091   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
2092     by (rule smallest_ccdi_sets.Inc)
2093   show ?case
2094     by (metis 1 2)
2095 next
2096   case (Disj A)
2097   have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
2098     by blast
2099   have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
2100     by blast
2101   moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj
2102     by (auto simp add: disjoint_family_on_def)
2103   ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
2104     by (rule smallest_ccdi_sets.Disj)
2105   show ?case
2106     by (metis 1 2)
2107 qed
2109 lemma (in algebra) sigma_property_disjoint_lemma:
2110   assumes sbC: "M \<subseteq> C"
2111       and ccdi: "closed_cdi \<Omega> C"
2112   shows "sigma_sets \<Omega> M \<subseteq> C"
2113 proof -
2114   have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"
2115     apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int
2116             smallest_ccdi_sets_Int)
2117     apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)
2118     apply (blast intro: smallest_ccdi_sets.Disj)
2119     done
2120   hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"
2121     by clarsimp
2122        (drule sigma_algebra.sigma_sets_subset [where a="M"], auto)
2123   also have "...  \<subseteq> C"
2124     proof
2125       fix x
2126       assume x: "x \<in> smallest_ccdi_sets \<Omega> M"
2127       thus "x \<in> C"
2128         proof (induct rule: smallest_ccdi_sets.induct)
2129           case (Basic x)
2130           thus ?case
2131             by (metis Basic subsetD sbC)
2132         next
2133           case (Compl x)
2134           thus ?case
2135             by (blast intro: closed_cdi_Compl [OF ccdi, simplified])
2136         next
2137           case (Inc A)
2138           thus ?case
2139                by (auto intro: closed_cdi_Inc [OF ccdi, simplified])
2140         next
2141           case (Disj A)
2142           thus ?case
2143                by (auto intro: closed_cdi_Disj [OF ccdi, simplified])
2144         qed
2145     qed
2146   finally show ?thesis .
2147 qed
2149 lemma (in algebra) sigma_property_disjoint:
2150   assumes sbC: "M \<subseteq> C"
2151       and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"
2152       and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
2153                      \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))
2154                      \<Longrightarrow> (\<Union>i. A i) \<in> C"
2155       and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
2156                       \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"
2157   shows "sigma_sets (\<Omega>) (M) \<subseteq> C"
2158 proof -
2159   have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
2160     proof (rule sigma_property_disjoint_lemma)
2161       show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
2162         by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
2163     next
2164       show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"
2165         by (simp add: closed_cdi_def compl inc disj)
2166            (metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed
2167              IntE sigma_sets.Compl range_subsetD sigma_sets.Union)
2168     qed
2169   thus ?thesis
2170     by blast
2171 qed
2173 section {* Dynkin systems *}
2175 locale dynkin_system = subset_class +
2176   assumes space: "\<Omega> \<in> M"
2177     and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
2178     and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
2179                            \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
2181 lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"
2182   using space compl[of "\<Omega>"] by simp
2184 lemma (in dynkin_system) diff:
2185   assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"
2186   shows "E - D \<in> M"
2187 proof -
2188   let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"
2189   have "range ?f = {D, \<Omega> - E, {}}"
2190     by (auto simp: image_iff)
2191   moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"
2192     by (auto simp: image_iff split: split_if_asm)
2193   moreover
2194   then have "disjoint_family ?f" unfolding disjoint_family_on_def
2195     using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto
2196   ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"
2197     using sets by auto
2198   also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"
2199     using assms sets_into_space by auto
2200   finally show ?thesis .
2201 qed
2203 lemma dynkin_systemI:
2204   assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"
2205   assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
2206   assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
2207           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
2208   shows "dynkin_system \<Omega> M"
2209   using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)
2211 lemma dynkin_systemI':
2212   assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"
2213   assumes empty: "{} \<in> M"
2214   assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
2215   assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
2216           \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
2217   shows "dynkin_system \<Omega> M"
2218 proof -
2219   from Diff[OF empty] have "\<Omega> \<in> M" by auto
2220   from 1 this Diff 2 show ?thesis
2221     by (intro dynkin_systemI) auto
2222 qed
2224 lemma dynkin_system_trivial:
2225   shows "dynkin_system A (Pow A)"
2226   by (rule dynkin_systemI) auto
2228 lemma sigma_algebra_imp_dynkin_system:
2229   assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"
2230 proof -
2231   interpret sigma_algebra \<Omega> M by fact
2232   show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
2233 qed
2235 subsection "Intersection stable algebras"
2237 definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"
2239 lemma (in algebra) Int_stable: "Int_stable M"
2240   unfolding Int_stable_def by auto
2242 lemma Int_stableI:
2243   "(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"
2244   unfolding Int_stable_def by auto
2246 lemma Int_stableD:
2247   "Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
2248   unfolding Int_stable_def by auto
2250 lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
2251   "sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"
2252 proof
2253   assume "sigma_algebra \<Omega> M" then show "Int_stable M"
2254     unfolding sigma_algebra_def using algebra.Int_stable by auto
2255 next
2256   assume "Int_stable M"
2257   show "sigma_algebra \<Omega> M"
2258     unfolding sigma_algebra_disjoint_iff algebra_iff_Un
2259   proof (intro conjI ballI allI impI)
2260     show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto
2261   next
2262     fix A B assume "A \<in> M" "B \<in> M"
2263     then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"
2264               "\<Omega> - A \<in> M" "\<Omega> - B \<in> M"
2265       using sets_into_space by auto
2266     then show "A \<union> B \<in> M"
2267       using `Int_stable M` unfolding Int_stable_def by auto
2268   qed auto
2269 qed
2271 subsection "Smallest Dynkin systems"
2273 definition dynkin where
2274   "dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"
2276 lemma dynkin_system_dynkin:
2277   assumes "M \<subseteq> Pow (\<Omega>)"
2278   shows "dynkin_system \<Omega> (dynkin \<Omega> M)"
2279 proof (rule dynkin_systemI)
2280   fix A assume "A \<in> dynkin \<Omega> M"
2281   moreover
2282   { fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"
2283     then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }
2284   moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
2285     using assms dynkin_system_trivial by fastforce
2286   ultimately show "A \<subseteq> \<Omega>"
2287     unfolding dynkin_def using assms
2288     by auto
2289 next
2290   show "\<Omega> \<in> dynkin \<Omega> M"
2291     unfolding dynkin_def using dynkin_system.space by fastforce
2292 next
2293   fix A assume "A \<in> dynkin \<Omega> M"
2294   then show "\<Omega> - A \<in> dynkin \<Omega> M"
2295     unfolding dynkin_def using dynkin_system.compl by force
2296 next
2297   fix A :: "nat \<Rightarrow> 'a set"
2298   assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"
2299   show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def
2300   proof (simp, safe)
2301     fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"
2302     with A have "(\<Union>i. A i) \<in> D"
2303       by (intro dynkin_system.UN) (auto simp: dynkin_def)
2304     then show "(\<Union>i. A i) \<in> D" by auto
2305   qed
2306 qed
2308 lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"
2309   unfolding dynkin_def by auto
2311 lemma (in dynkin_system) restricted_dynkin_system:
2312   assumes "D \<in> M"
2313   shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
2314 proof (rule dynkin_systemI, simp_all)
2315   have "\<Omega> \<inter> D = D"
2316     using `D \<in> M` sets_into_space by auto
2317   then show "\<Omega> \<inter> D \<in> M"
2318     using `D \<in> M` by auto
2319 next
2320   fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"
2321   moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"
2322     by auto
2323   ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"
2324     using  `D \<in> M` by (auto intro: diff)
2325 next
2326   fix A :: "nat \<Rightarrow> 'a set"
2327   assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
2328   then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"
2329     "range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
2330     by ((fastforce simp: disjoint_family_on_def)+)
2331   then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"
2332     by (auto simp del: UN_simps)
2333 qed
2335 lemma (in dynkin_system) dynkin_subset:
2336   assumes "N \<subseteq> M"
2337   shows "dynkin \<Omega> N \<subseteq> M"
2338 proof -
2339   have "dynkin_system \<Omega> M" by default
2340   then have "dynkin_system \<Omega> M"
2341     using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
2342   with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def)
2343 qed
2345 lemma sigma_eq_dynkin:
2346   assumes sets: "M \<subseteq> Pow \<Omega>"
2347   assumes "Int_stable M"
2348   shows "sigma_sets \<Omega> M = dynkin \<Omega> M"
2349 proof -
2350   have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
2351     using sigma_algebra_imp_dynkin_system
2352     unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
2353   moreover
2354   interpret dynkin_system \<Omega> "dynkin \<Omega> M"
2355     using dynkin_system_dynkin[OF sets] .
2356   have "sigma_algebra \<Omega> (dynkin \<Omega> M)"
2357     unfolding sigma_algebra_eq_Int_stable Int_stable_def
2358   proof (intro ballI)
2359     fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"
2360     let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"
2361     have "M \<subseteq> ?D B"
2362     proof
2363       fix E assume "E \<in> M"
2364       then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
2365         using sets_into_space `Int_stable M` by (auto simp: Int_stable_def)
2366       then have "dynkin \<Omega> M \<subseteq> ?D E"
2367         using restricted_dynkin_system `E \<in> dynkin \<Omega> M`
2368         by (intro dynkin_system.dynkin_subset) simp_all
2369       then have "B \<in> ?D E"
2370         using `B \<in> dynkin \<Omega> M` by auto
2371       then have "E \<inter> B \<in> dynkin \<Omega> M"
2372         by (subst Int_commute) simp
2373       then show "E \<in> ?D B"
2374         using sets `E \<in> M` by auto
2375     qed
2376     then have "dynkin \<Omega> M \<subseteq> ?D B"
2377       using restricted_dynkin_system `B \<in> dynkin \<Omega> M`
2378       by (intro dynkin_system.dynkin_subset) simp_all
2379     then show "A \<inter> B \<in> dynkin \<Omega> M"
2380       using `A \<in> dynkin \<Omega> M` sets_into_space by auto
2381   qed
2382   from sigma_algebra.sigma_sets_subset[OF this, of "M"]
2383   have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
2384   ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto
2385   then show ?thesis
2386     by (auto simp: dynkin_def)
2387 qed
2389 lemma (in dynkin_system) dynkin_idem:
2390   "dynkin \<Omega> M = M"
2391 proof -
2392   have "dynkin \<Omega> M = M"
2393   proof
2394     show "M \<subseteq> dynkin \<Omega> M"
2395       using dynkin_Basic by auto
2396     show "dynkin \<Omega> M \<subseteq> M"
2397       by (intro dynkin_subset) auto
2398   qed
2399   then show ?thesis
2400     by (auto simp: dynkin_def)
2401 qed
2403 lemma (in dynkin_system) dynkin_lemma:
2404   assumes "Int_stable E"
2405   and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"
2406   shows "sigma_sets \<Omega> E = M"
2407 proof -
2408   have "E \<subseteq> Pow \<Omega>"
2409     using E sets_into_space by force
2410   then have "sigma_sets \<Omega> E = dynkin \<Omega> E"
2411     using `Int_stable E` by (rule sigma_eq_dynkin)
2412   moreover then have "dynkin \<Omega> E = M"
2413     using assms dynkin_subset[OF E(1)] by simp
2414   ultimately show ?thesis
2415     using assms by (auto simp: dynkin_def)
2416 qed
2418 lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
2419   assumes "Int_stable G"
2420     and closed: "G \<subseteq> Pow \<Omega>"
2421     and A: "A \<in> sigma_sets \<Omega> G"
2422   assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"
2423     and empty: "P {}"
2424     and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"
2425     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)"
2426   shows "P A"
2427 proof -
2428   let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"
2429   interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"
2430     using closed by (rule sigma_algebra_sigma_sets)
2431   from compl[OF _ empty] closed have space: "P \<Omega>" by simp
2432   interpret dynkin_system \<Omega> ?D
2433     by default (auto dest: sets_into_space intro!: space compl union)
2434   have "sigma_sets \<Omega> G = ?D"
2435     by (rule dynkin_lemma) (auto simp: basic `Int_stable G`)
2436   with A show ?thesis by auto
2437 qed
2439 end