src/HOL/Probability/Sigma_Algebra.thy
 author wenzelm Thu, 18 Apr 2013 17:07:01 +0200 changeset 51717 9e7d1c139569 parent 51683 baefa3b461c2 child 53374 a14d2a854c02 permissions -rw-r--r--
simplifier uses proper Proof.context instead of historic type simpset;
```
(*  Title:      HOL/Probability/Sigma_Algebra.thy
Author:     Stefan Richter, Markus Wenzel, TU München
Author:     Johannes Hölzl, TU München
Plus material from the Hurd/Coble measure theory development,
translated by Lawrence Paulson.
*)

theory Sigma_Algebra
imports
Complex_Main
"~~/src/HOL/Library/Countable_Set"
"~~/src/HOL/Library/FuncSet"
"~~/src/HOL/Library/Indicator_Function"
"~~/src/HOL/Library/Extended_Real"
begin

text {* Sigma algebras are an elementary concept in measure
theory. To measure --- that is to integrate --- functions, we first have
to measure sets. Unfortunately, when dealing with a large universe,
it is often not possible to consistently assign a measure to every
subset. Therefore it is necessary to define the set of measurable
subsets of the universe. A sigma algebra is such a set that has
three very natural and desirable properties. *}

subsection {* Families of sets *}

locale subset_class =
fixes \<Omega> :: "'a set" and M :: "'a set set"
assumes space_closed: "M \<subseteq> Pow \<Omega>"

lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>"
by (metis PowD contra_subsetD space_closed)

subsection {* Semiring of sets *}

subsubsection {* Disjoint sets *}

definition "disjoint A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"

lemma disjointI:
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}) \<Longrightarrow> disjoint A"
unfolding disjoint_def by auto

lemma disjointD:
"disjoint A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}"
unfolding disjoint_def by auto

lemma disjoint_empty[iff]: "disjoint {}"
by (auto simp: disjoint_def)

lemma disjoint_union:
assumes C: "disjoint C" and B: "disjoint B" and disj: "\<Union>C \<inter> \<Union>B = {}"
shows "disjoint (C \<union> B)"
proof (rule disjointI)
fix c d assume sets: "c \<in> C \<union> B" "d \<in> C \<union> B" and "c \<noteq> d"
show "c \<inter> d = {}"
proof cases
assume "(c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B)"
then show ?thesis
proof
assume "c \<in> C \<and> d \<in> C" with `c \<noteq> d` C show "c \<inter> d = {}"
by (auto simp: disjoint_def)
next
assume "c \<in> B \<and> d \<in> B" with `c \<noteq> d` B show "c \<inter> d = {}"
by (auto simp: disjoint_def)
qed
next
assume "\<not> ((c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B))"
with sets have "(c \<subseteq> \<Union>C \<and> d \<subseteq> \<Union>B) \<or> (c \<subseteq> \<Union>B \<and> d \<subseteq> \<Union>C)"
by auto
with disj show "c \<inter> d = {}" by auto
qed
qed

locale semiring_of_sets = subset_class +
assumes empty_sets[iff]: "{} \<in> M"
assumes Int[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
assumes Diff_cover:
"\<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"

lemma (in semiring_of_sets) finite_INT[intro]:
assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
shows "(\<Inter>i\<in>I. A i) \<in> M"
using assms by (induct rule: finite_ne_induct) auto

lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x"
by (metis Int_absorb1 sets_into_space)

lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x"
by (metis Int_absorb2 sets_into_space)

lemma (in semiring_of_sets) sets_Collect_conj:
assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M"
proof -
have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}"
by auto
with assms show ?thesis by auto
qed

lemma (in semiring_of_sets) sets_Collect_finite_All':
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}"
shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})"
using `S \<noteq> {}` by auto
with assms show ?thesis by auto
qed

locale ring_of_sets = semiring_of_sets +
assumes Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"

lemma (in ring_of_sets) finite_Union [intro]:
"finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> Union X \<in> M"
by (induct set: finite) (auto simp add: Un)

lemma (in ring_of_sets) finite_UN[intro]:
assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"
shows "(\<Union>i\<in>I. A i) \<in> M"
using assms by induct auto

lemma (in ring_of_sets) Diff [intro]:
assumes "a \<in> M" "b \<in> M" shows "a - b \<in> M"
using Diff_cover[OF assms] by auto

lemma ring_of_setsI:
assumes space_closed: "M \<subseteq> Pow \<Omega>"
assumes empty_sets[iff]: "{} \<in> M"
assumes Un[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M"
assumes Diff[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M"
shows "ring_of_sets \<Omega> M"
proof
fix a b assume ab: "a \<in> M" "b \<in> M"
from ab show "\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C"
by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def)
have "a \<inter> b = a - (a - b)" by auto
also have "\<dots> \<in> M" using ab by auto
finally show "a \<inter> b \<in> M" .
qed fact+

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)"
proof
assume "ring_of_sets \<Omega> M"
then interpret ring_of_sets \<Omega> M .
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)"
using space_closed by auto
qed (auto intro!: ring_of_setsI)

lemma (in ring_of_sets) insert_in_sets:
assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M"
proof -
have "{x} \<union> A \<in> M" using assms by (rule Un)
thus ?thesis by auto
qed

lemma (in ring_of_sets) sets_Collect_disj:
assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M"
shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M"
proof -
have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}"
by auto
with assms show ?thesis by auto
qed

lemma (in ring_of_sets) sets_Collect_finite_Ex:
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S"
shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})"
by auto
with assms show ?thesis by auto
qed

locale algebra = ring_of_sets +
assumes top [iff]: "\<Omega> \<in> M"

lemma (in algebra) compl_sets [intro]:
"a \<in> M \<Longrightarrow> \<Omega> - a \<in> M"
by auto

lemma algebra_iff_Un:
"algebra \<Omega> M \<longleftrightarrow>
M \<subseteq> Pow \<Omega> \<and>
{} \<in> M \<and>
(\<forall>a \<in> M. \<Omega> - a \<in> M) \<and>
(\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un")
proof
assume "algebra \<Omega> M"
then interpret algebra \<Omega> M .
show ?Un using sets_into_space by auto
next
assume ?Un
then have "\<Omega> \<in> M" by auto
interpret ring_of_sets \<Omega> M
proof (rule ring_of_setsI)
show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
using `?Un` by auto
fix a b assume a: "a \<in> M" and b: "b \<in> M"
then show "a \<union> b \<in> M" using `?Un` by auto
have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)"
using \<Omega> a b by auto
then show "a - b \<in> M"
using a b  `?Un` by auto
qed
show "algebra \<Omega> M" proof qed fact
qed

lemma algebra_iff_Int:
"algebra \<Omega> M \<longleftrightarrow>
M \<subseteq> Pow \<Omega> & {} \<in> M &
(\<forall>a \<in> M. \<Omega> - a \<in> M) &
(\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int")
proof
assume "algebra \<Omega> M"
then interpret algebra \<Omega> M .
show ?Int using sets_into_space by auto
next
assume ?Int
show "algebra \<Omega> M"
proof (unfold algebra_iff_Un, intro conjI ballI)
show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M"
using `?Int` by auto
from `?Int` show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto
fix a b assume M: "a \<in> M" "b \<in> M"
hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))"
using \<Omega> by blast
also have "... \<in> M"
using M `?Int` by auto
finally show "a \<union> b \<in> M" .
qed
qed

lemma (in algebra) sets_Collect_neg:
assumes "{x\<in>\<Omega>. P x} \<in> M"
shows "{x\<in>\<Omega>. \<not> P x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto
with assms show ?thesis by auto
qed

lemma (in algebra) sets_Collect_imp:
"{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"
unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)

lemma (in algebra) sets_Collect_const:
"{x\<in>\<Omega>. P} \<in> M"
by (cases P) auto

lemma algebra_single_set:
"X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }"
by (auto simp: algebra_iff_Int)

subsection {* Restricted algebras *}

abbreviation (in algebra)
"restricted_space A \<equiv> (op \<inter> A) ` M"

lemma (in algebra) restricted_algebra:
assumes "A \<in> M" shows "algebra A (restricted_space A)"
using assms by (auto simp: algebra_iff_Int)

subsection {* Sigma Algebras *}

locale sigma_algebra = algebra +
assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

lemma (in algebra) is_sigma_algebra:
assumes "finite M"
shows "sigma_algebra \<Omega> M"
proof
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M"
then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)"
by auto
also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M"
using `finite M` by auto
finally show "(\<Union>i. A i) \<in> M" .
qed

lemma countable_UN_eq:
fixes A :: "'i::countable \<Rightarrow> 'a set"
shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow>
(range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)"
proof -
let ?A' = "A \<circ> from_nat"
have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r")
proof safe
fix x i assume "x \<in> A i" thus "x \<in> ?l"
by (auto intro!: exI[of _ "to_nat i"])
next
fix x i assume "x \<in> ?A' i" thus "x \<in> ?r"
by (auto intro!: exI[of _ "from_nat i"])
qed
have **: "range ?A' = range A"
using surj_from_nat
by (auto simp: image_compose intro!: imageI)
show ?thesis unfolding * ** ..
qed

lemma (in sigma_algebra) countable_Union [intro]:
assumes "countable X" "X \<subseteq> M" shows "Union X \<in> M"
proof cases
assume "X \<noteq> {}"
hence "\<Union>X = (\<Union>n. from_nat_into X n)"
using assms by (auto intro: from_nat_into) (metis from_nat_into_surj)
also have "\<dots> \<in> M" using assms
by (auto intro!: countable_nat_UN) (metis `X \<noteq> {}` from_nat_into set_mp)
finally show ?thesis .
qed simp

lemma (in sigma_algebra) countable_UN[intro]:
fixes A :: "'i::countable \<Rightarrow> 'a set"
assumes "A`X \<subseteq> M"
shows  "(\<Union>x\<in>X. A x) \<in> M"
proof -
let ?A = "\<lambda>i. if i \<in> X then A i else {}"
from assms have "range ?A \<subseteq> M" by auto
with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M]
have "(\<Union>x. ?A x) \<in> M" by auto
moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm)
ultimately show ?thesis by simp
qed

lemma (in sigma_algebra) countable_UN':
fixes A :: "'i \<Rightarrow> 'a set"
assumes X: "countable X"
assumes A: "A`X \<subseteq> M"
shows  "(\<Union>x\<in>X. A x) \<in> M"
proof -
have "(\<Union>x\<in>X. A x) = (\<Union>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
using X by auto
also have "\<dots> \<in> M"
using A X
by (intro countable_UN) auto
finally show ?thesis .
qed

lemma (in sigma_algebra) countable_INT [intro]:
fixes A :: "'i::countable \<Rightarrow> 'a set"
assumes A: "A`X \<subseteq> M" "X \<noteq> {}"
shows "(\<Inter>i\<in>X. A i) \<in> M"
proof -
from A have "\<forall>i\<in>X. A i \<in> M" by fast
hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast
moreover
have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A
by blast
ultimately show ?thesis by metis
qed

lemma (in sigma_algebra) countable_INT':
fixes A :: "'i \<Rightarrow> 'a set"
assumes X: "countable X" "X \<noteq> {}"
assumes A: "A`X \<subseteq> M"
shows  "(\<Inter>x\<in>X. A x) \<in> M"
proof -
have "(\<Inter>x\<in>X. A x) = (\<Inter>i\<in>to_nat_on X ` X. A (from_nat_into X i))"
using X by auto
also have "\<dots> \<in> M"
using A X
by (intro countable_INT) auto
finally show ?thesis .
qed

lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)"
by (auto simp: ring_of_sets_iff)

lemma algebra_Pow: "algebra sp (Pow sp)"
by (auto simp: algebra_iff_Un)

lemma sigma_algebra_iff:
"sigma_algebra \<Omega> M \<longleftrightarrow>
algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"

lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)"
by (auto simp: sigma_algebra_iff algebra_iff_Int)

lemma (in sigma_algebra) sets_Collect_countable_All:
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto
with assms show ?thesis by auto
qed

lemma (in sigma_algebra) sets_Collect_countable_Ex:
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto
with assms show ?thesis by auto
qed

lemma (in sigma_algebra) sets_Collect_countable_Ex':
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
assumes "countable I"
shows "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} \<in> M"
proof -
have "{x\<in>\<Omega>. \<exists>i\<in>I. P i x} = (\<Union>i\<in>I. {x\<in>\<Omega>. P i x})" by auto
with assms show ?thesis
by (auto intro!: countable_UN')
qed

lemmas (in sigma_algebra) sets_Collect =
sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const
sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All

lemma (in sigma_algebra) sets_Collect_countable_Ball:
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M"
unfolding Ball_def by (intro sets_Collect assms)

lemma (in sigma_algebra) sets_Collect_countable_Bex:
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M"
shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M"
unfolding Bex_def by (intro sets_Collect assms)

lemma sigma_algebra_single_set:
assumes "X \<subseteq> S"
shows "sigma_algebra S { {}, X, S - X, S }"
using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp

subsection {* Binary Unions *}

definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"
where "binary a b =  (\<lambda>x. b)(0 := a)"

lemma range_binary_eq: "range(binary a b) = {a,b}"

lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"

lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"

lemma sigma_algebra_iff2:
"sigma_algebra \<Omega> M \<longleftrightarrow>
M \<subseteq> Pow \<Omega> \<and>
{} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and>
(\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
algebra_iff_Un Un_range_binary)

subsection {* Initial Sigma Algebra *}

text {*Sigma algebras can naturally be created as the closure of any set of
M with regard to the properties just postulated.  *}

inductive_set sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
for sp :: "'a set" and A :: "'a set set"
where
Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A"
| Empty: "{} \<in> sigma_sets sp A"
| Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A"
| Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A"

lemma (in sigma_algebra) sigma_sets_subset:
assumes a: "a \<subseteq> M"
shows "sigma_sets \<Omega> a \<subseteq> M"
proof
fix x
assume "x \<in> sigma_sets \<Omega> a"
from this show "x \<in> M"
by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
qed

lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp"
by (erule sigma_sets.induct, auto)

lemma sigma_algebra_sigma_sets:
"a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp
intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl)

lemma sigma_sets_least_sigma_algebra:
assumes "A \<subseteq> Pow S"
shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
proof safe
fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B"
and X: "X \<in> sigma_sets S A"
from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X
show "X \<in> B" by auto
next
fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}"
then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B"
by simp
have "A \<subseteq> sigma_sets S A" using assms by auto
moreover have "sigma_algebra S (sigma_sets S A)"
using assms by (intro sigma_algebra_sigma_sets[of A]) auto
ultimately show "X \<in> sigma_sets S A" by auto
qed

lemma sigma_sets_top: "sp \<in> sigma_sets sp A"
by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)

lemma sigma_sets_Un:
"a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A"
apply (rule Union, simp add: binary_def)
done

lemma sigma_sets_Inter:
assumes Asb: "A \<subseteq> Pow sp"
shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A"
proof -
assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A"
hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A"
by (rule sigma_sets.Compl)
hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
by (rule sigma_sets.Union)
hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A"
by (rule sigma_sets.Compl)
also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)"
by auto
also have "... = (\<Inter>i. a i)" using ai
by (blast dest: sigma_sets_into_sp [OF Asb])
finally show ?thesis .
qed

lemma sigma_sets_INTER:
assumes Asb: "A \<subseteq> Pow sp"
and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}"
shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A"
proof -
from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A"
hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A"
by (rule sigma_sets_Inter [OF Asb])
also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)"
by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+
finally show ?thesis .
qed

lemma sigma_sets_UNION: "countable B \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A) \<Longrightarrow> (\<Union>B) \<in> sigma_sets X A"
using from_nat_into[of B] range_from_nat_into[of B] sigma_sets.Union[of "from_nat_into B" X A]
apply (cases "B = {}")
apply (simp del: Union_image_eq add: Union_image_eq[symmetric])
done

lemma (in sigma_algebra) sigma_sets_eq:
"sigma_sets \<Omega> M = M"
proof
show "M \<subseteq> sigma_sets \<Omega> M"
by (metis Set.subsetI sigma_sets.Basic)
next
show "sigma_sets \<Omega> M \<subseteq> M"
by (metis sigma_sets_subset subset_refl)
qed

lemma sigma_sets_eqI:
assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"
assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A"
shows "sigma_sets M A = sigma_sets M B"
proof (intro set_eqI iffI)
fix a assume "a \<in> sigma_sets M A"
from this A show "a \<in> sigma_sets M B"
by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
next
fix b assume "b \<in> sigma_sets M B"
from this B show "b \<in> sigma_sets M A"
by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic)
qed

lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
proof
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
qed

lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
proof
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros(2-))
qed

lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
proof
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-))
qed

lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
by (auto intro: sigma_sets.Basic)

lemma (in sigma_algebra) restriction_in_sets:
fixes A :: "nat \<Rightarrow> 'a set"
assumes "S \<in> M"
and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r")
shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
proof -
{ fix i have "A i \<in> ?r" using * by auto
hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto
hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto }
thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M"
by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"])
qed

lemma (in sigma_algebra) restricted_sigma_algebra:
assumes "S \<in> M"
shows "sigma_algebra S (restricted_space S)"
unfolding sigma_algebra_def sigma_algebra_axioms_def
proof safe
show "algebra S (restricted_space S)" using restricted_algebra[OF assms] .
next
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S"
from restriction_in_sets[OF assms this[simplified]]
show "(\<Union>i. A i) \<in> restricted_space S" by simp
qed

lemma sigma_sets_Int:
assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp"
shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)"
proof (intro equalityI subsetI)
fix x assume "x \<in> op \<inter> A ` sigma_sets sp st"
then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto
then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)"
proof (induct arbitrary: x)
case (Compl a)
then show ?case
by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps)
next
case (Union a)
then show ?case
by (auto intro!: sigma_sets.Union
simp add: UN_extend_simps simp del: UN_simps)
qed (auto intro!: sigma_sets.intros(2-))
then show "x \<in> sigma_sets A (op \<inter> A ` st)"
using `A \<subseteq> sp` by (simp add: Int_absorb2)
next
fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)"
then show "x \<in> op \<inter> A ` sigma_sets sp st"
proof induct
case (Compl a)
then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto
then show ?case using `A \<subseteq> sp`
by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)
next
case (Union a)
then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x"
by (auto simp: image_iff Bex_def)
from choice[OF this] guess f ..
then show ?case
by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union
qed (auto intro!: sigma_sets.intros(2-))
qed

lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}"
proof (intro set_eqI iffI)
fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}"
by induct blast+
qed (auto intro: sigma_sets.Empty sigma_sets_top)

lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}"
proof (intro set_eqI iffI)
fix x assume "x \<in> sigma_sets A {A}"
then show "x \<in> {{}, A}"
by induct blast+
next
fix x assume "x \<in> {{}, A}"
then show "x \<in> sigma_sets A {A}"
by (auto intro: sigma_sets.Empty sigma_sets_top)
qed

lemma sigma_sets_sigma_sets_eq:
"M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M"
by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto

lemma sigma_sets_singleton:
assumes "X \<subseteq> S"
shows "sigma_sets S { X } = { {}, X, S - X, S }"
proof -
interpret sigma_algebra S "{ {}, X, S - X, S }"
by (rule sigma_algebra_single_set) fact
have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }"
by (rule sigma_sets_subseteq) simp
moreover have "\<dots> = { {}, X, S - X, S }"
using sigma_sets_eq by simp
moreover
{ fix A assume "A \<in> { {}, X, S - X, S }"
then have "A \<in> sigma_sets S { X }"
by (auto intro: sigma_sets.intros(2-) sigma_sets_top) }
ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }"
by (intro antisym) auto
with sigma_sets_eq show ?thesis by simp
qed

lemma restricted_sigma:
assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>"
shows "algebra.restricted_space (sigma_sets \<Omega> M) S =
sigma_sets S (algebra.restricted_space M S)"
proof -
from S sigma_sets_into_sp[OF M]
have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto
from sigma_sets_Int[OF this]
show ?thesis by simp
qed

lemma sigma_sets_vimage_commute:
assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'"
shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'}
= sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R")
proof
show "?L \<subseteq> ?R"
proof clarify
fix A assume "A \<in> sigma_sets \<Omega>' M'"
then show "X -` A \<inter> \<Omega> \<in> ?R"
proof induct
case Empty then show ?case
by (auto intro!: sigma_sets.Empty)
next
case (Compl B)
have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)"
by (auto simp add: funcset_mem [OF X])
with Compl show ?case
by (auto intro!: sigma_sets.Compl)
next
case (Union F)
then show ?case
by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps
intro!: sigma_sets.Union)
qed auto
qed
show "?R \<subseteq> ?L"
proof clarify
fix A assume "A \<in> ?R"
then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'"
proof induct
case (Basic B) then show ?case by auto
next
case Empty then show ?case
by (auto intro!: sigma_sets.Empty exI[of _ "{}"])
next
case (Compl B)
then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto
then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>"
by (auto simp add: funcset_mem [OF X])
with A(2) show ?case
by (auto intro: sigma_sets.Compl)
next
case (Union F)
then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto
from choice[OF this] guess A .. note A = this
with A show ?case
by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)
qed
qed
qed

subsection "Disjoint families"

definition
disjoint_family_on  where
"disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"

abbreviation
"disjoint_family A \<equiv> disjoint_family_on A UNIV"

lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B"
by blast

lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
by blast

lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
by blast

lemma disjoint_family_subset:
"disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"

lemma disjoint_family_on_bisimulation:
assumes "disjoint_family_on f S"
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 = {}"
shows "disjoint_family_on g S"
using assms unfolding disjoint_family_on_def by auto

lemma disjoint_family_on_mono:
"A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"
unfolding disjoint_family_on_def by auto

lemma disjoint_family_Suc:
assumes Suc: "!!n. A n \<subseteq> A (Suc n)"
shows "disjoint_family (\<lambda>i. A (Suc i) - A i)"
proof -
{
fix m
have "!!n. A n \<subseteq> A (m+n)"
proof (induct m)
case 0 show ?case by simp
next
case (Suc m) thus ?case
qed
}
hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n"
thus ?thesis
(metis insert_absorb insert_subset le_SucE le_antisym not_leE)
qed

lemma setsum_indicator_disjoint_family:
fixes f :: "'d \<Rightarrow> 'e::semiring_1"
assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
proof -
have "P \<inter> {i. x \<in> A i} = {j}"
using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
by auto
thus ?thesis
unfolding indicator_def
by (simp add: if_distrib setsum_cases[OF `finite P`])
qed

definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"

lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n)
thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
qed

lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
apply (rule UN_finite2_eq [where k=0])
done

lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"

lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
(metis neq_iff Int_commute less_disjoint_disjointed)

lemma disjointed_subset: "disjointed A n \<subseteq> A n"

lemma (in ring_of_sets) UNION_in_sets:
fixes A:: "nat \<Rightarrow> 'a set"
assumes A: "range A \<subseteq> M"
shows  "(\<Union>i\<in>{0..<n}. A i) \<in> M"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n)
thus ?case
by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
qed

lemma (in ring_of_sets) range_disjointed_sets:
assumes A: "range A \<subseteq> M"
shows  "range (disjointed A) \<subseteq> M"
fix n
show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets
by (metis A Diff UNIV_I image_subset_iff)
qed

lemma (in algebra) range_disjointed_sets':
"range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M"
using range_disjointed_sets .

lemma disjointed_0[simp]: "disjointed A 0 = A 0"

lemma incseq_Un:
"incseq A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n"
unfolding incseq_def by auto

lemma disjointed_incseq:
"incseq A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n"
using incseq_Un[of A]
by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)

lemma sigma_algebra_disjoint_iff:
"sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and>
(\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"
fix A :: "nat \<Rightarrow> 'a set"
assume M: "algebra \<Omega> M"
and A: "range A \<subseteq> M"
and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M"
hence "range (disjointed A) \<subseteq> M \<longrightarrow>
disjoint_family (disjointed A) \<longrightarrow>
(\<Union>i. disjointed A i) \<in> M" by blast
hence "(\<Union>i. disjointed A i) \<in> M"
by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed)
thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq)
qed

lemma disjoint_family_on_disjoint_image:
"disjoint_family_on A I \<Longrightarrow> disjoint (A ` I)"
unfolding disjoint_family_on_def disjoint_def by force

lemma disjoint_image_disjoint_family_on:
assumes d: "disjoint (A ` I)" and i: "inj_on A I"
shows "disjoint_family_on A I"
unfolding disjoint_family_on_def
proof (intro ballI impI)
fix n m assume nm: "m \<in> I" "n \<in> I" and "n \<noteq> m"
with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}"
by (intro disjointD[OF d]) auto
qed

subsection {* Ring generated by a semiring *}

definition (in semiring_of_sets)
"generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }"

lemma (in semiring_of_sets) generated_ringE[elim?]:
assumes "a \<in> generated_ring"
obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
using assms unfolding generated_ring_def by auto

lemma (in semiring_of_sets) generated_ringI[intro?]:
assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C"
shows "a \<in> generated_ring"
using assms unfolding generated_ring_def by auto

lemma (in semiring_of_sets) generated_ringI_Basic:
"A \<in> M \<Longrightarrow> A \<in> generated_ring"
by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def)

lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]:
assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
and "a \<inter> b = {}"
shows "a \<union> b \<in> generated_ring"
proof -
from a guess Ca .. note Ca = this
from b guess Cb .. note Cb = this
show ?thesis
proof
show "disjoint (Ca \<union> Cb)"
using `a \<inter> b = {}` Ca Cb by (auto intro!: disjoint_union)
qed (insert Ca Cb, auto)
qed

lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring"
by (auto simp: generated_ring_def disjoint_def)

lemma (in semiring_of_sets) generated_ring_disjoint_Union:
assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring"
using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty)

lemma (in semiring_of_sets) generated_ring_disjoint_UNION:
"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"
unfolding SUP_def by (intro generated_ring_disjoint_Union) auto

lemma (in semiring_of_sets) generated_ring_Int:
assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring"
shows "a \<inter> b \<in> generated_ring"
proof -
from a guess Ca .. note Ca = this
from b guess Cb .. note Cb = this
def C \<equiv> "(\<lambda>(a,b). a \<inter> b)` (Ca\<times>Cb)"
show ?thesis
proof
show "disjoint C"
proof (simp add: disjoint_def C_def, intro ballI impI)
fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb"
assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2"
then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto
then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}"
proof
assume "a1 \<noteq> a2"
with sets Ca have "a1 \<inter> a2 = {}"
by (auto simp: disjoint_def)
then show ?thesis by auto
next
assume "b1 \<noteq> b2"
with sets Cb have "b1 \<inter> b2 = {}"
by (auto simp: disjoint_def)
then show ?thesis by auto
qed
qed
qed (insert Ca Cb, auto simp: C_def)
qed

lemma (in semiring_of_sets) generated_ring_Inter:
assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring"
using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int)

lemma (in semiring_of_sets) generated_ring_INTER:
"finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring"
unfolding INF_def by (intro generated_ring_Inter) auto

lemma (in semiring_of_sets) generating_ring:
"ring_of_sets \<Omega> generated_ring"
proof (rule ring_of_setsI)
let ?R = generated_ring
show "?R \<subseteq> Pow \<Omega>"
using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)
show "{} \<in> ?R" by (rule generated_ring_empty)

{ fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this
fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this

show "a - b \<in> ?R"
proof cases
assume "Cb = {}" with Cb `a \<in> ?R` show ?thesis
by simp
next
assume "Cb \<noteq> {}"
with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto
also have "\<dots> \<in> ?R"
proof (intro generated_ring_INTER generated_ring_disjoint_UNION)
fix a b assume "a \<in> Ca" "b \<in> Cb"
with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R"
next
show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)"
using Ca by (auto simp add: disjoint_def `Cb \<noteq> {}`)
next
show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+
qed
finally show "a - b \<in> ?R" .
qed }
note Diff = this

fix a b assume sets: "a \<in> ?R" "b \<in> ?R"
have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto
also have "\<dots> \<in> ?R"
by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto
finally show "a \<union> b \<in> ?R" .
qed

lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M"
proof
interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M"
using space_closed by (rule sigma_algebra_sigma_sets)
show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M"
by (blast intro!: sigma_sets_mono elim: generated_ringE)
qed (auto intro!: generated_ringI_Basic sigma_sets_mono)

subsection {* Measure type *}

definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
"positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)"

definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
"countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow>
(\<Sum>i. f (A i)) = f (\<Union>i. A i))"

definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where
"measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>"

typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }"
proof
have "sigma_algebra UNIV {{}, UNIV}"
by (auto simp: sigma_algebra_iff2)
then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} "
by (auto simp: measure_space_def positive_def countably_additive_def)
qed

definition space :: "'a measure \<Rightarrow> 'a set" where
"space M = fst (Rep_measure M)"

definition sets :: "'a measure \<Rightarrow> 'a set set" where
"sets M = fst (snd (Rep_measure M))"

definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where
"emeasure M = snd (snd (Rep_measure M))"

definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
"measure M A = real (emeasure M A)"

declare [[coercion sets]]

declare [[coercion measure]]

declare [[coercion emeasure]]

lemma measure_space: "measure_space (space M) (sets M) (emeasure M)"
by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse)

interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure"
using measure_space[of M] by (auto simp: measure_space_def)

definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
"measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, sigma_sets \<Omega> A,
\<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"

abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)"

lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)"
unfolding measure_space_def
by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)

lemma (in ring_of_sets) positive_cong_eq:
"(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>"

"(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>"

lemma measure_space_eq:
assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a"
shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
proof -
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets)
from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis
by (auto simp: measure_space_def)
qed

lemma measure_of_eq:
assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)"
shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
proof -
have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
using assms by (rule measure_space_eq)
with eq show ?thesis
by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure])
qed

lemma
assumes A: "A \<subseteq> Pow \<Omega>"
shows sets_measure_of[simp]: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A" (is ?sets)
and space_measure_of[simp]: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space)
proof -
have "?sets \<and> ?space"
proof cases
assume "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A)
(\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)"
using A by (rule measure_space_eq) auto
ultimately show "?sets \<and> ?space"
by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def)
next
assume "\<not> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
with A show "?sets \<and> ?space"
by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def measure_space_0)
qed
then show ?sets ?space by auto
qed

lemma (in sigma_algebra) sets_measure_of_eq[simp]:
"sets (measure_of \<Omega> M \<mu>) = M"
using space_closed by (auto intro!: sigma_sets_eq)

lemma (in sigma_algebra) space_measure_of_eq[simp]:
"space (measure_of \<Omega> M \<mu>) = \<Omega>"
using space_closed by (auto intro!: sigma_sets_eq)

lemma measure_of_subset:
"M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
by (auto intro!: sigma_sets_subseteq)

lemma sigma_sets_mono'':
assumes "A \<in> sigma_sets C D"
assumes "B \<subseteq> D"
assumes "D \<subseteq> Pow C"
shows "sigma_sets A B \<subseteq> sigma_sets C D"
proof
fix x assume "x \<in> sigma_sets A B"
thus "x \<in> sigma_sets C D"
proof induct
case (Basic a) with assms have "a \<in> D" by auto
thus ?case ..
next
case Empty show ?case by (rule sigma_sets.Empty)
next
from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`])
ultimately have "A - a \<in> sets (sigma C D)" ..
thus ?case by (subst (asm) sets_measure_of[OF `D \<subseteq> Pow C`])
next
case (Union a)
thus ?case by (intro sigma_sets.Union)
qed
qed

lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)"
by auto

subsection {* Constructing simple @{typ "'a measure"} *}

lemma emeasure_measure_of:
assumes M: "M = measure_of \<Omega> A \<mu>"
assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>"
assumes X: "X \<in> sets M"
shows "emeasure M X = \<mu> X"
proof -
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact
have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)
= measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>"
using ms(1) by (rule measure_space_eq) auto
moreover have "X \<in> sigma_sets \<Omega> A"
using X M ms by simp
ultimately show ?thesis
unfolding emeasure_def measure_of_def M
by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq)
qed

lemma emeasure_measure_of_sigma:
assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>"
assumes A: "A \<in> M"
shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
proof -
interpret sigma_algebra \<Omega> M by fact
have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
using ms sigma_sets_eq by (simp add: measure_space_def)
moreover have "measure_space \<Omega> (sigma_sets \<Omega> M) (\<lambda>a. if a \<in> sigma_sets \<Omega> M then \<mu> a else 0)
= measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
using space_closed by (rule measure_space_eq) auto
ultimately show ?thesis using A
unfolding emeasure_def measure_of_def
by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq)
qed

lemma measure_cases[cases type: measure]:
obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>"
by atomize_elim (cases x, auto)

lemma sets_eq_imp_space_eq:
"sets M = sets M' \<Longrightarrow> space M = space M'"
using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M']
by blast

lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0"
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0"

lemma measure_eqI:
fixes M N :: "'a measure"
assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A"
shows "M = N"
proof (cases M N rule: measure_cases[case_product measure_cases])
case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>')
interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def)
interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def)
have "A = sets M" "A' = sets N"
using measure_measure by (simp_all add: sets_def Abs_measure_inverse)
with `sets M = sets N` have "A = A'" by simp
moreover with M.top N.top M.space_closed N.space_closed have "\<Omega> = \<Omega>'" by auto
moreover { fix B have "\<mu> B = \<mu>' B"
proof cases
assume "B \<in> A"
with eq `A = sets M` have "emeasure M B = emeasure N B" by simp
with measure_measure show "\<mu> B = \<mu>' B"
next
assume "B \<notin> A"
with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N"
by auto
then have "emeasure M B = 0" "emeasure N B = 0"
with measure_measure show "\<mu> B = \<mu>' B"
qed }
then have "\<mu> = \<mu>'" by auto
ultimately show "M = N"
qed

lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)"
using measure_space_0[of A \<Omega>]
by (simp add: measure_of_def emeasure_def Abs_measure_inverse)

lemma sigma_eqI:
assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N"
shows "sigma \<Omega> M = sigma \<Omega> N"
by (rule measure_eqI) (simp_all add: emeasure_sigma)

subsection {* Measurable functions *}

definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where
"measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}"

lemma measurable_space:
"f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A"
unfolding measurable_def by auto

lemma measurable_sets:
"f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
unfolding measurable_def by auto

lemma measurable_sets_Collect:
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"
proof -
have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}"
using measurable_space[OF f] by auto
with measurable_sets[OF f P] show ?thesis
by simp
qed

lemma measurable_sigma_sets:
assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>"
and f: "f \<in> space M \<rightarrow> \<Omega>"
and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
shows "f \<in> measurable M N"
proof -
interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets)
from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force

{ fix X assume "X \<in> sigma_sets \<Omega> A"
then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>"
proof induct
case (Basic a) then show ?case
by (auto simp add: ba) (metis B(2) subsetD PowD)
next
case (Compl a)
have [simp]: "f -` \<Omega> \<inter> space M = space M"
by (auto simp add: funcset_mem [OF f])
then show ?case
by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl)
next
case (Union a)
then show ?case
by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast
qed auto }
with f show ?thesis
by (auto simp add: measurable_def B \<Omega>)
qed

lemma measurable_measure_of:
assumes B: "N \<subseteq> Pow \<Omega>"
and f: "f \<in> space M \<rightarrow> \<Omega>"
and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M"
shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)"
proof -
have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N"
using B by (rule sets_measure_of)
from this assms show ?thesis by (rule measurable_sigma_sets)
qed

lemma measurable_iff_measure_of:
assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>"
shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)"
by (metis assms in_measure_of measurable_measure_of assms measurable_sets)

lemma measurable_cong_sets:
assumes sets: "sets M = sets M'" "sets N = sets N'"
shows "measurable M N = measurable M' N'"
using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def)

lemma measurable_cong:
assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w"
shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'"
unfolding measurable_def using assms
by (simp cong: vimage_inter_cong Pi_cong)

lemma measurable_eqI:
"\<lbrakk> space m1 = space m1' ; space m2 = space m2' ;
sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk>
\<Longrightarrow> measurable m1 m2 = measurable m1' m2'"

lemma measurable_compose:
assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L"
shows "(\<lambda>x. g (f x)) \<in> measurable M L"
proof -
have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M"
using measurable_space[OF f] by auto
with measurable_space[OF f] measurable_space[OF g] show ?thesis
by (auto intro: measurable_sets[OF f] measurable_sets[OF g]
simp del: vimage_Int simp add: measurable_def)
qed

lemma measurable_comp:
"f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L"
using measurable_compose[of f M N g L] by (simp add: comp_def)

lemma measurable_const:
"c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'"

lemma measurable_If:
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
assumes P: "{x\<in>space M. P x} \<in> sets M"
shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'"
unfolding measurable_def
proof safe
fix x assume "x \<in> space M"
thus "(if P x then f x else g x) \<in> space M'"
using measure unfolding measurable_def by auto
next
fix A assume "A \<in> sets M'"
hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M =
((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union>
((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))"
using measure unfolding measurable_def by (auto split: split_if_asm)
show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M"
using `A \<in> sets M'` measure P unfolding * measurable_def
by (auto intro!: sets.Un)
qed

lemma measurable_If_set:
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'"
assumes P: "A \<inter> space M \<in> sets M"
shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'"
proof (rule measurable_If[OF measure])
have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto
thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto
qed

lemma measurable_ident: "id \<in> measurable M M"

lemma measurable_ident_sets:
assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'"
using measurable_ident[of M]
unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] .

lemma sets_Least:
assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M"
shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
proof -
{ fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M"
proof cases
assume i: "(LEAST j. False) = i"
have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
{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}))"
(insert i, auto dest: Least_le intro: LeastI intro!: Least_equality)
with meas show ?thesis
by (auto intro!: sets.Int)
next
assume i: "(LEAST j. False) \<noteq> i"
then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M =
{x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))"
fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)"
have "\<exists>j. P j x"
by (rule ccontr) (insert neq, auto)
then show "P (LEAST j. P j x) x" by (rule LeastI_ex)
qed (auto dest: Least_le intro!: Least_equality)
with meas show ?thesis
by auto
qed }
then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M"
by (intro sets.countable_UN) auto
moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) =
(\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto
ultimately show ?thesis by auto
qed

lemma measurable_strong:
fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c"
assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c"
and t: "f ` (space a) \<subseteq> t"
and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b"
shows "(g o f) \<in> measurable a c"
proof -
have fab: "f \<in> (space a -> space b)"
and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f
have eq: "\<And>y. f -` g -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t
by force
show ?thesis
apply (auto simp add: measurable_def vimage_compose)
apply (metis funcset_mem fab g)
apply (subst eq, metis ba cb)
done
qed

lemma measurable_mono1:
"M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow>
measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N"
using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def)

subsection {* Counting space *}

definition count_space :: "'a set \<Rightarrow> 'a measure" where
"count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)"

lemma
shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>"
and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>"
using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>]
by (auto simp: count_space_def)

lemma measurable_count_space_eq1[simp]:
"f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M"
unfolding measurable_def by simp

lemma measurable_count_space_eq2:
assumes "finite A"
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))"
proof -
{ fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A"
with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X"
by (auto dest: finite_subset)
moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M"
ultimately have "f -` X \<inter> space M \<in> sets M"
using `X \<subseteq> A` by (auto intro!: sets.finite_UN simp del: UN_simps) }
then show ?thesis
unfolding measurable_def by auto
qed

lemma measurable_compose_countable:
assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)"
shows "(\<lambda>x. f (g x) x) \<in> measurable M N"
unfolding measurable_def
proof safe
fix x assume "x \<in> space M" then show "f (g x) x \<in> space N"
using f[THEN measurable_space] g[THEN measurable_space] by auto
next
fix A assume A: "A \<in> sets N"
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))"
by auto
also have "\<dots> \<in> sets M" using f[THEN measurable_sets, OF A] g[THEN measurable_sets]
by (auto intro!: sets.countable_UN measurable_sets)
finally show "(\<lambda>x. f (g x) x) -` A \<inter> space M \<in> sets M" .
qed

lemma measurable_count_space_const:
"(\<lambda>x. c) \<in> measurable M (count_space UNIV)"

lemma measurable_count_space:
"f \<in> measurable (count_space A) (count_space UNIV)"
by simp

lemma measurable_compose_rev:
assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L"
shows "(\<lambda>x. f (g x)) \<in> measurable M N"
using measurable_compose[OF g f] .

subsection {* Extend measure *}

definition "extend_measure \<Omega> I G \<mu> =
(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)
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>')
else measure_of \<Omega> (G`I) (\<lambda>_. 0))"

lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
unfolding extend_measure_def by simp

lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)"
unfolding extend_measure_def by simp

lemma emeasure_extend_measure:
assumes M: "M = extend_measure \<Omega> I G \<mu>"
and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"
and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
and "i \<in> I"
shows "emeasure M (G i) = \<mu> i"
proof cases
assume *: "(\<forall>i\<in>I. \<mu> i = 0)"
with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)"
from measure_space_0[OF ms(1)] ms `i\<in>I`
have "emeasure M (G i) = 0"
by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure)
with `i\<in>I` * show ?thesis
by simp
next
def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'"
assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)"
moreover
have "measure_space (space M) (sets M) \<mu>'"
using ms unfolding measure_space_def by auto default
with ms eq have "\<exists>\<mu>'. P \<mu>'"
unfolding P_def
by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure)
ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)"
by (simp add: M extend_measure_def P_def[symmetric])

from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex)
show "emeasure M (G i) = \<mu> i"
proof (subst emeasure_measure_of[OF M_eq])
have sets_M: "sets M = sigma_sets \<Omega> (G`I)"
using M_eq ms by (auto simp: sets_extend_measure)
then show "G i \<in> sets M" using `i \<in> I` by auto
show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i"
using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def)
qed fact
qed

lemma emeasure_extend_measure_Pair:
assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"
and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"
and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'"
and "I i j"
shows "emeasure M (G i j) = \<mu> i j"
using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j`
by (auto simp: subset_eq)

subsection {* Sigma algebra generated by function preimages *}

definition
"vimage_algebra M S f = sigma S ((\<lambda>A. f -` A \<inter> S) ` sets M)"

lemma sigma_algebra_preimages:
fixes f :: "'x \<Rightarrow> 'a"
assumes "f \<in> S \<rightarrow> space M"
shows "sigma_algebra S ((\<lambda>A. f -` A \<inter> S) ` sets M)"
(is "sigma_algebra _ (?F ` sets M)")
show "{} \<in> ?F ` sets M" by blast
next
fix A assume "A \<in> sets M"
moreover have "S - ?F A = ?F (space M - A)"
using assms by auto
ultimately show "S - ?F A \<in> ?F ` sets M"
by blast
next
fix A :: "nat \<Rightarrow> 'x set" assume *: "range A \<subseteq> ?F ` M"
have "\<forall>i. \<exists>b. b \<in> M \<and> A i = ?F b"
proof safe
fix i
have "A i \<in> ?F ` M" using * by auto
then show "\<exists>b. b \<in> M \<and> A i = ?F b" by auto
qed
from choice[OF this] obtain b where b: "range b \<subseteq> M" "\<And>i. A i = ?F (b i)"
by auto
then have "(\<Union>i. A i) = ?F (\<Union>i. b i)" by auto
then show "(\<Union>i. A i) \<in> ?F ` M" using b(1) by blast
qed

lemma sets_vimage_algebra[simp]:
"f \<in> S \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra M S f) = (\<lambda>A. f -` A \<inter> S) ` sets M"
using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_preimages, of f S M]

lemma space_vimage_algebra[simp]:
"f \<in> S \<rightarrow> space M \<Longrightarrow> space (vimage_algebra M S f) = S"
using sigma_algebra.space_measure_of_eq[OF sigma_algebra_preimages, of f S M]

lemma in_vimage_algebra[simp]:
"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)"

lemma measurable_vimage_algebra:
fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M"
shows "f \<in> measurable (vimage_algebra M S f) M"
unfolding measurable_def using assms by force

lemma measurable_vimage:
fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a"
assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M"
shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra M S f) M2"
proof -
note measurable_vimage_algebra[OF assms(2)]
from measurable_comp[OF this assms(1)]
show ?thesis by (simp add: comp_def)
qed

lemma sigma_sets_vimage:
assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S"
shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A"
proof (intro set_eqI iffI)
let ?F = "\<lambda>X. f -` X \<inter> S'"
fix X assume "X \<in> sigma_sets S' (?F ` A)"
then show "X \<in> ?F ` sigma_sets S A"
proof induct
case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A"
by auto
then show ?case by auto
next
case Empty then show ?case
by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty)
next
case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A"
by auto
then have "S - X' \<in> sigma_sets S A"
by (auto intro!: sigma_sets.Compl)
then show ?case
using X assms by (auto intro!: image_eqI[where x="S - X'"])
next
case (Union F)
then have "\<forall>i. \<exists>F'.  F' \<in> sigma_sets S A \<and> F i = f -` F' \<inter> S'"
by (auto simp: image_iff Bex_def)
from choice[OF this] obtain F' where
"\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f -` F' i \<inter> S'"
by auto
then show ?case
by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"])
qed
next
let ?F = "\<lambda>X. f -` X \<inter> S'"
fix X assume "X \<in> ?F ` sigma_sets S A"
then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto
then show "X \<in> sigma_sets S' (?F ` A)"
proof (induct arbitrary: X)
case Empty then show ?case by (auto intro: sigma_sets.Empty)
next
case (Compl X')
have "S' - (S' - X) \<in> sigma_sets S' (?F ` A)"
apply (rule sigma_sets.Compl)
using assms by (auto intro!: Compl.hyps simp: Compl.prems)
also have "S' - (S' - X) = X"
using assms Compl by auto
finally show ?case .
next
case (Union F)
have "(\<Union>i. f -` F i \<inter> S') \<in> sigma_sets S' (?F ` A)"
by (intro sigma_sets.Union Union.hyps) simp
also have "(\<Union>i. f -` F i \<inter> S') = X"
using assms Union by auto
finally show ?case .
qed auto
qed

subsection {* A Two-Element Series *}

definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"

lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
apply (rule set_eqI)
done

lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"

section {* Closed CDI *}

definition closed_cdi where
"closed_cdi \<Omega> M \<longleftrightarrow>
M \<subseteq> Pow \<Omega> &
(\<forall>s \<in> M. \<Omega> - s \<in> M) &
(\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow>
(\<Union>i. A i) \<in> M) &
(\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)"

inductive_set
smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set"
for \<Omega> M
where
Basic [intro]:
"a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M"
| Compl [intro]:
"a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"
| Inc:
"range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n))
\<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M"
| Disj:
"range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A
\<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M"

lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M"
by auto

lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>"
apply (rule subsetI)
apply (erule smallest_ccdi_sets.induct)
apply (auto intro: range_subsetD dest: sets_into_space)
done

lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)"
apply (auto simp add: closed_cdi_def smallest_ccdi_sets)
apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) +
done

lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>"

lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M"

lemma closed_cdi_Inc:
"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"

lemma closed_cdi_Disj:
"closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

lemma closed_cdi_Un:
assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M"
and A: "A \<in> M" and B: "B \<in> M"
and disj: "A \<inter> B = {}"
shows "A \<union> B \<in> M"
proof -
have ra: "range (binaryset A B) \<subseteq> M"
by (simp add: range_binaryset_eq empty A B)
have di:  "disjoint_family (binaryset A B)" using disj
by (simp add: disjoint_family_on_def binaryset_def Int_commute)
from closed_cdi_Disj [OF cdi ra di]
show ?thesis
qed

lemma (in algebra) smallest_ccdi_sets_Un:
assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M"
and disj: "A \<inter> B = {}"
shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"
proof -
have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"
by (simp add: range_binaryset_eq  A B smallest_ccdi_sets.Basic)
have di:  "disjoint_family (binaryset A B)" using disj
by (simp add: disjoint_family_on_def binaryset_def Int_commute)
from Disj [OF ra di]
show ?thesis
qed

lemma (in algebra) smallest_ccdi_sets_Int1:
assumes a: "a \<in> M"
shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
proof (induct rule: smallest_ccdi_sets.induct)
case (Basic x)
thus ?case
by (metis a Int smallest_ccdi_sets.Basic)
next
case (Compl x)
have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))"
by blast
also have "... \<in> smallest_ccdi_sets \<Omega> M"
by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2
Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un
smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl)
finally show ?case .
next
case (Inc A)
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
by blast
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
by blast
moreover have "(\<lambda>i. a \<inter> A i) 0 = {}"
moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc
by blast
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Inc)
show ?case
by (metis 1 2)
next
case (Disj A)
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)"
by blast
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
by blast
moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Disj)
show ?case
by (metis 1 2)
qed

lemma (in algebra) smallest_ccdi_sets_Int:
assumes b: "b \<in> smallest_ccdi_sets \<Omega> M"
shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
proof (induct rule: smallest_ccdi_sets.induct)
case (Basic x)
thus ?case
by (metis b smallest_ccdi_sets_Int1)
next
case (Compl x)
have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))"
by blast
also have "... \<in> smallest_ccdi_sets \<Omega> M"
by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b
smallest_ccdi_sets.Compl smallest_ccdi_sets_Un)
finally show ?case .
next
case (Inc A)
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
by blast
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc
by blast
moreover have "(\<lambda>i. A i \<inter> b) 0 = {}"
moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc
by blast
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Inc)
show ?case
by (metis 1 2)
next
case (Disj A)
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b"
by blast
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj
by blast
moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M"
by (rule smallest_ccdi_sets.Disj)
show ?case
by (metis 1 2)
qed

lemma (in algebra) sigma_property_disjoint_lemma:
assumes sbC: "M \<subseteq> C"
and ccdi: "closed_cdi \<Omega> C"
shows "sigma_sets \<Omega> M \<subseteq> C"
proof -
have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}"
apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int
smallest_ccdi_sets_Int)
apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets)
apply (blast intro: smallest_ccdi_sets.Disj)
done
hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M"
by clarsimp
(drule sigma_algebra.sigma_sets_subset [where a="M"], auto)
also have "...  \<subseteq> C"
proof
fix x
assume x: "x \<in> smallest_ccdi_sets \<Omega> M"
thus "x \<in> C"
proof (induct rule: smallest_ccdi_sets.induct)
case (Basic x)
thus ?case
by (metis Basic subsetD sbC)
next
case (Compl x)
thus ?case
by (blast intro: closed_cdi_Compl [OF ccdi, simplified])
next
case (Inc A)
thus ?case
by (auto intro: closed_cdi_Inc [OF ccdi, simplified])
next
case (Disj A)
thus ?case
by (auto intro: closed_cdi_Disj [OF ccdi, simplified])
qed
qed
finally show ?thesis .
qed

lemma (in algebra) sigma_property_disjoint:
assumes sbC: "M \<subseteq> C"
and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C"
and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
\<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n))
\<Longrightarrow> (\<Union>i. A i) \<in> C"
and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)
\<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C"
shows "sigma_sets (\<Omega>) (M) \<subseteq> C"
proof -
have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
proof (rule sigma_property_disjoint_lemma)
show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)"
by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
next
show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))"
by (simp add: closed_cdi_def compl inc disj)
(metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed
IntE sigma_sets.Compl range_subsetD sigma_sets.Union)
qed
thus ?thesis
by blast
qed

subsection {* Dynkin systems *}

locale dynkin_system = subset_class +
assumes space: "\<Omega> \<in> M"
and   compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
and   UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M"

lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M"
using space compl[of "\<Omega>"] by simp

lemma (in dynkin_system) diff:
assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E"
shows "E - D \<in> M"
proof -
let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}"
have "range ?f = {D, \<Omega> - E, {}}"
by (auto simp: image_iff)
moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)"
by (auto simp: image_iff split: split_if_asm)
moreover
then have "disjoint_family ?f" unfolding disjoint_family_on_def
using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto
ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M"
using sets by auto
also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D"
using assms sets_into_space by auto
finally show ?thesis .
qed

lemma dynkin_systemI:
assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M"
assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
shows "dynkin_system \<Omega> M"
using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def)

lemma dynkin_systemI':
assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"
assumes empty: "{} \<in> M"
assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"
assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M"
shows "dynkin_system \<Omega> M"
proof -
from Diff[OF empty] have "\<Omega> \<in> M" by auto
from 1 this Diff 2 show ?thesis
by (intro dynkin_systemI) auto
qed

lemma dynkin_system_trivial:
shows "dynkin_system A (Pow A)"
by (rule dynkin_systemI) auto

lemma sigma_algebra_imp_dynkin_system:
assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M"
proof -
interpret sigma_algebra \<Omega> M by fact
show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI)
qed

subsection "Intersection stable algebras"

definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)"

lemma (in algebra) Int_stable: "Int_stable M"
unfolding Int_stable_def by auto

lemma Int_stableI:
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A"
unfolding Int_stable_def by auto

lemma Int_stableD:
"Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M"
unfolding Int_stable_def by auto

lemma (in dynkin_system) sigma_algebra_eq_Int_stable:
"sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M"
proof
assume "sigma_algebra \<Omega> M" then show "Int_stable M"
unfolding sigma_algebra_def using algebra.Int_stable by auto
next
assume "Int_stable M"
show "sigma_algebra \<Omega> M"
unfolding sigma_algebra_disjoint_iff algebra_iff_Un
proof (intro conjI ballI allI impI)
show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto
next
fix A B assume "A \<in> M" "B \<in> M"
then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))"
"\<Omega> - A \<in> M" "\<Omega> - B \<in> M"
using sets_into_space by auto
then show "A \<union> B \<in> M"
using `Int_stable M` unfolding Int_stable_def by auto
qed auto
qed

subsection "Smallest Dynkin systems"

definition dynkin where
"dynkin \<Omega> M =  (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})"

lemma dynkin_system_dynkin:
assumes "M \<subseteq> Pow (\<Omega>)"
shows "dynkin_system \<Omega> (dynkin \<Omega> M)"
proof (rule dynkin_systemI)
fix A assume "A \<in> dynkin \<Omega> M"
moreover
{ fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D"
then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) }
moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}"
using assms dynkin_system_trivial by fastforce
ultimately show "A \<subseteq> \<Omega>"
unfolding dynkin_def using assms
by auto
next
show "\<Omega> \<in> dynkin \<Omega> M"
unfolding dynkin_def using dynkin_system.space by fastforce
next
fix A assume "A \<in> dynkin \<Omega> M"
then show "\<Omega> - A \<in> dynkin \<Omega> M"
unfolding dynkin_def using dynkin_system.compl by force
next
fix A :: "nat \<Rightarrow> 'a set"
assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M"
show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def
proof (simp, safe)
fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D"
with A have "(\<Union>i. A i) \<in> D"
by (intro dynkin_system.UN) (auto simp: dynkin_def)
then show "(\<Union>i. A i) \<in> D" by auto
qed
qed

lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M"
unfolding dynkin_def by auto

lemma (in dynkin_system) restricted_dynkin_system:
assumes "D \<in> M"
shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
proof (rule dynkin_systemI, simp_all)
have "\<Omega> \<inter> D = D"
using `D \<in> M` sets_into_space by auto
then show "\<Omega> \<inter> D \<in> M"
using `D \<in> M` by auto
next
fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M"
moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)"
by auto
ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M"
using  `D \<in> M` by (auto intro: diff)
next
fix A :: "nat \<Rightarrow> 'a set"
assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}"
then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)"
"range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)"
by ((fastforce simp: disjoint_family_on_def)+)
then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M"
by (auto simp del: UN_simps)
qed

lemma (in dynkin_system) dynkin_subset:
assumes "N \<subseteq> M"
shows "dynkin \<Omega> N \<subseteq> M"
proof -
have "dynkin_system \<Omega> M" by default
then have "dynkin_system \<Omega> M"
using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp
with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def)
qed

lemma sigma_eq_dynkin:
assumes sets: "M \<subseteq> Pow \<Omega>"
assumes "Int_stable M"
shows "sigma_sets \<Omega> M = dynkin \<Omega> M"
proof -
have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)"
using sigma_algebra_imp_dynkin_system
unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
moreover
interpret dynkin_system \<Omega> "dynkin \<Omega> M"
using dynkin_system_dynkin[OF sets] .
have "sigma_algebra \<Omega> (dynkin \<Omega> M)"
unfolding sigma_algebra_eq_Int_stable Int_stable_def
proof (intro ballI)
fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M"
let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}"
have "M \<subseteq> ?D B"
proof
fix E assume "E \<in> M"
then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M"
using sets_into_space `Int_stable M` by (auto simp: Int_stable_def)
then have "dynkin \<Omega> M \<subseteq> ?D E"
using restricted_dynkin_system `E \<in> dynkin \<Omega> M`
by (intro dynkin_system.dynkin_subset) simp_all
then have "B \<in> ?D E"
using `B \<in> dynkin \<Omega> M` by auto
then have "E \<inter> B \<in> dynkin \<Omega> M"
by (subst Int_commute) simp
then show "E \<in> ?D B"
using sets `E \<in> M` by auto
qed
then have "dynkin \<Omega> M \<subseteq> ?D B"
using restricted_dynkin_system `B \<in> dynkin \<Omega> M`
by (intro dynkin_system.dynkin_subset) simp_all
then show "A \<inter> B \<in> dynkin \<Omega> M"
using `A \<in> dynkin \<Omega> M` sets_into_space by auto
qed
from sigma_algebra.sigma_sets_subset[OF this, of "M"]
have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto
ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto
then show ?thesis
by (auto simp: dynkin_def)
qed

lemma (in dynkin_system) dynkin_idem:
"dynkin \<Omega> M = M"
proof -
have "dynkin \<Omega> M = M"
proof
show "M \<subseteq> dynkin \<Omega> M"
using dynkin_Basic by auto
show "dynkin \<Omega> M \<subseteq> M"
by (intro dynkin_subset) auto
qed
then show ?thesis
by (auto simp: dynkin_def)
qed

lemma (in dynkin_system) dynkin_lemma:
assumes "Int_stable E"
and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E"
shows "sigma_sets \<Omega> E = M"
proof -
have "E \<subseteq> Pow \<Omega>"
using E sets_into_space by force
then have "sigma_sets \<Omega> E = dynkin \<Omega> E"
using `Int_stable E` by (rule sigma_eq_dynkin)
moreover then have "dynkin \<Omega> E = M"
using assms dynkin_subset[OF E(1)] by simp
ultimately show ?thesis
using assms by (auto simp: dynkin_def)
qed

lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
assumes "Int_stable G"
and closed: "G \<subseteq> Pow \<Omega>"
and A: "A \<in> sigma_sets \<Omega> G"
assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A"
and empty: "P {}"
and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)"
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)"
shows "P A"
proof -
let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }"
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G"
using closed by (rule sigma_algebra_sigma_sets)
from compl[OF _ empty] closed have space: "P \<Omega>" by simp
interpret dynkin_system \<Omega> ?D
by default (auto dest: sets_into_space intro!: space compl union)
have "sigma_sets \<Omega> G = ?D"
by (rule dynkin_lemma) (auto simp: basic `Int_stable G`)
with A show ?thesis by auto
qed

end
```