# Theory Sigma_Algebra

```(*  Title:      HOL/Analysis/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.
*)

chapter ‹Measure and Integration Theory›

theory Sigma_Algebra
imports
Complex_Main
"HOL-Library.Countable_Set"
"HOL-Library.FuncSet"
"HOL-Library.Indicator_Function"
"HOL-Library.Extended_Nonnegative_Real"
"HOL-Library.Disjoint_Sets"
begin

section ‹Sigma Algebra›

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✐‹tag important› subset_class =
fixes Ω :: "'a set" and M :: "'a set set"
assumes space_closed: "M ⊆ Pow Ω"

lemma (in subset_class) sets_into_space: "x ∈ M ⟹ x ⊆ Ω"
by (metis PowD contra_subsetD space_closed)

subsubsection ‹Semiring of sets›

locale✐‹tag important› semiring_of_sets = subset_class +
assumes empty_sets[iff]: "{} ∈ M"
assumes Int[intro]: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ a ∩ b ∈ M"
assumes Diff_cover:
"⋀a b. a ∈ M ⟹ b ∈ M ⟹ ∃C⊆M. finite C ∧ disjoint C ∧ a - b = ⋃C"

lemma (in semiring_of_sets) finite_INT[intro]:
assumes "finite I" "I ≠ {}" "⋀i. i ∈ I ⟹ A i ∈ M"
shows "(⋂i∈I. A i) ∈ M"
using assms by (induct rule: finite_ne_induct) auto

lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x ∈ M ⟹ Ω ∩ x = x"
by (metis Int_absorb1 sets_into_space)

lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x ∈ M ⟹ x ∩ Ω = x"
by (metis Int_absorb2 sets_into_space)

lemma (in semiring_of_sets) sets_Collect_conj:
assumes "{x∈Ω. P x} ∈ M" "{x∈Ω. Q x} ∈ M"
shows "{x∈Ω. Q x ∧ P x} ∈ M"
proof -
have "{x∈Ω. Q x ∧ P x} = {x∈Ω. Q x} ∩ {x∈Ω. P x}"
by auto
with assms show ?thesis by auto
qed

lemma (in semiring_of_sets) sets_Collect_finite_All':
assumes "⋀i. i ∈ S ⟹ {x∈Ω. P i x} ∈ M" "finite S" "S ≠ {}"
shows "{x∈Ω. ∀i∈S. P i x} ∈ M"
proof -
have "{x∈Ω. ∀i∈S. P i x} = (⋂i∈S. {x∈Ω. P i x})"
using ‹S ≠ {}› by auto
with assms show ?thesis by auto
qed

subsubsection ‹Ring of sets›

locale✐‹tag important› ring_of_sets = semiring_of_sets +
assumes Un [intro]: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ a ∪ b ∈ M"

lemma (in ring_of_sets) finite_Union [intro]:
"finite X ⟹ X ⊆ M ⟹ ⋃X ∈ M"
by (induct set: finite) (auto simp add: Un)

lemma (in ring_of_sets) finite_UN[intro]:
assumes "finite I" and "⋀i. i ∈ I ⟹ A i ∈ M"
shows "(⋃i∈I. A i) ∈ M"
using assms by induct auto

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

lemma ring_of_setsI:
assumes space_closed: "M ⊆ Pow Ω"
assumes empty_sets[iff]: "{} ∈ M"
assumes Un[intro]: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ a ∪ b ∈ M"
assumes Diff[intro]: "⋀a b. a ∈ M ⟹ b ∈ M ⟹ a - b ∈ M"
shows "ring_of_sets Ω M"
proof
fix a b assume ab: "a ∈ M" "b ∈ M"
from ab show "∃C⊆M. finite C ∧ disjoint C ∧ a - b = ⋃C"
by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def)
have "a ∩ b = a - (a - b)" by auto
also have "… ∈ M" using ab by auto
finally show "a ∩ b ∈ M" .
qed fact+

lemma ring_of_sets_iff: "ring_of_sets Ω M ⟷ M ⊆ Pow Ω ∧ {} ∈ M ∧ (∀a∈M. ∀b∈M. a ∪ b ∈ M) ∧ (∀a∈M. ∀b∈M. a - b ∈ M)"
proof
assume "ring_of_sets Ω M"
then interpret ring_of_sets Ω M .
show "M ⊆ Pow Ω ∧ {} ∈ M ∧ (∀a∈M. ∀b∈M. a ∪ b ∈ M) ∧ (∀a∈M. ∀b∈M. a - b ∈ M)"
using space_closed by auto
qed (auto intro!: ring_of_setsI)

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

lemma (in ring_of_sets) sets_Collect_disj:
assumes "{x∈Ω. P x} ∈ M" "{x∈Ω. Q x} ∈ M"
shows "{x∈Ω. Q x ∨ P x} ∈ M"
proof -
have "{x∈Ω. Q x ∨ P x} = {x∈Ω. Q x} ∪ {x∈Ω. P x}"
by auto
with assms show ?thesis by auto
qed

lemma (in ring_of_sets) sets_Collect_finite_Ex:
assumes "⋀i. i ∈ S ⟹ {x∈Ω. P i x} ∈ M" "finite S"
shows "{x∈Ω. ∃i∈S. P i x} ∈ M"
proof -
have "{x∈Ω. ∃i∈S. P i x} = (⋃i∈S. {x∈Ω. P i x})"
by auto
with assms show ?thesis by auto
qed

subsubsection ‹Algebra of sets›

locale✐‹tag important› algebra = ring_of_sets +
assumes top [iff]: "Ω ∈ M"

lemma (in algebra) compl_sets [intro]:
"a ∈ M ⟹ Ω - a ∈ M"
by auto

proposition algebra_iff_Un:
"algebra Ω M ⟷
M ⊆ Pow Ω ∧
{} ∈ M ∧
(∀a ∈ M. Ω - a ∈ M) ∧
(∀a ∈ M. ∀ b ∈ M. a ∪ b ∈ M)" (is "_ ⟷ ?Un")
proof
assume "algebra Ω M"
then interpret algebra Ω M .
show ?Un using sets_into_space by auto
next
assume ?Un
then have "Ω ∈ M" by auto
interpret ring_of_sets Ω M
proof (rule ring_of_setsI)
show Ω: "M ⊆ Pow Ω" "{} ∈ M"
using ‹?Un› by auto
fix a b assume a: "a ∈ M" and b: "b ∈ M"
then show "a ∪ b ∈ M" using ‹?Un› by auto
have "a - b = Ω - ((Ω - a) ∪ b)"
using Ω a b by auto
then show "a - b ∈ M"
using a b  ‹?Un› by auto
qed
show "algebra Ω M" proof qed fact
qed

proposition algebra_iff_Int:
"algebra Ω M ⟷
M ⊆ Pow Ω & {} ∈ M &
(∀a ∈ M. Ω - a ∈ M) &
(∀a ∈ M. ∀ b ∈ M. a ∩ b ∈ M)" (is "_ ⟷ ?Int")
proof
assume "algebra Ω M"
then interpret algebra Ω M .
show ?Int using sets_into_space by auto
next
assume ?Int
show "algebra Ω M"
proof (unfold algebra_iff_Un, intro conjI ballI)
show Ω: "M ⊆ Pow Ω" "{} ∈ M"
using ‹?Int› by auto
from ‹?Int› show "⋀a. a ∈ M ⟹ Ω - a ∈ M" by auto
fix a b assume M: "a ∈ M" "b ∈ M"
hence "a ∪ b = Ω - ((Ω - a) ∩ (Ω - b))"
using Ω by blast
also have "... ∈ M"
using M ‹?Int› by auto
finally show "a ∪ b ∈ M" .
qed
qed

lemma (in algebra) sets_Collect_neg:
assumes "{x∈Ω. P x} ∈ M"
shows "{x∈Ω. ¬ P x} ∈ M"
proof -
have "{x∈Ω. ¬ P x} = Ω - {x∈Ω. P x}" by auto
with assms show ?thesis by auto
qed

lemma (in algebra) sets_Collect_imp:
"{x∈Ω. P x} ∈ M ⟹ {x∈Ω. Q x} ∈ M ⟹ {x∈Ω. Q x ⟶ P x} ∈ M"
unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg)

lemma (in algebra) sets_Collect_const:
"{x∈Ω. P} ∈ M"
by (cases P) auto

lemma algebra_single_set:
"X ⊆ S ⟹ algebra S { {}, X, S - X, S }"
by (auto simp: algebra_iff_Int)

subsubsection✐‹tag unimportant› ‹Restricted algebras›

abbreviation (in algebra)
"restricted_space A ≡ ((∩) A) ` M"

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

subsubsection ‹Sigma Algebras›

locale✐‹tag important› sigma_algebra = algebra +
assumes countable_nat_UN [intro]: "⋀A. range A ⊆ M ⟹ (⋃i::nat. A i) ∈ M"

lemma (in algebra) is_sigma_algebra:
assumes "finite M"
shows "sigma_algebra Ω M"
proof
fix A :: "nat ⇒ 'a set" assume "range A ⊆ M"
then have "(⋃i. A i) = (⋃s∈M ∩ range A. s)"
by auto
also have "(⋃s∈M ∩ range A. s) ∈ M"
using ‹finite M› by auto
finally show "(⋃i. A i) ∈ M" .
qed

lemma countable_UN_eq:
fixes A :: "'i::countable ⇒ 'a set"
shows "(range A ⊆ M ⟶ (⋃i. A i) ∈ M) ⟷
(range (A ∘ from_nat) ⊆ M ⟶ (⋃i. (A ∘ from_nat) i) ∈ M)"
proof -
let ?A' = "A ∘ from_nat"
have *: "(⋃i. ?A' i) = (⋃i. A i)" (is "?l = ?r")
proof safe
fix x i assume "x ∈ A i" thus "x ∈ ?l"
by (auto intro!: exI[of _ "to_nat i"])
next
fix x i assume "x ∈ ?A' i" thus "x ∈ ?r"
by (auto intro!: exI[of _ "from_nat i"])
qed
have "A ` range from_nat = range A"
using surj_from_nat by simp
then have **: "range ?A' = range A"
by (simp only: image_comp [symmetric])
show ?thesis unfolding * ** ..
qed

lemma (in sigma_algebra) countable_Union [intro]:
assumes "countable X" "X ⊆ M" shows "⋃X ∈ M"
proof cases
assume "X ≠ {}"
hence "⋃X = (⋃n. from_nat_into X n)"
using assms by (auto cong del: SUP_cong)
also have "… ∈ M" using assms
by (auto intro!: countable_nat_UN) (metis ‹X ≠ {}› from_nat_into subsetD)
finally show ?thesis .
qed simp

lemma (in sigma_algebra) countable_UN[intro]:
fixes A :: "'i::countable ⇒ 'a set"
assumes "A`X ⊆ M"
shows  "(⋃x∈X. A x) ∈ M"
proof -
let ?A = "λi. if i ∈ X then A i else {}"
from assms have "range ?A ⊆ M" by auto
with countable_nat_UN[of "?A ∘ from_nat"] countable_UN_eq[of ?A M]
have "(⋃x. ?A x) ∈ M" by auto
moreover have "(⋃x. ?A x) = (⋃x∈X. A x)" by (auto split: if_split_asm)
ultimately show ?thesis by simp
qed

lemma (in sigma_algebra) countable_UN':
fixes A :: "'i ⇒ 'a set"
assumes X: "countable X"
assumes A: "A`X ⊆ M"
shows  "(⋃x∈X. A x) ∈ M"
proof -
have "(⋃x∈X. A x) = (⋃i∈to_nat_on X ` X. A (from_nat_into X i))"
using X by auto
also have "… ∈ M"
using A X
by (intro countable_UN) auto
finally show ?thesis .
qed

lemma (in sigma_algebra) countable_UN'':
"⟦ countable X; ⋀x y. x ∈ X ⟹ A x ∈ M ⟧ ⟹ (⋃x∈X. A x) ∈ M"
by(erule countable_UN')(auto)

lemma (in sigma_algebra) countable_INT [intro]:
fixes A :: "'i::countable ⇒ 'a set"
assumes A: "A`X ⊆ M" "X ≠ {}"
shows "(⋂i∈X. A i) ∈ M"
proof -
from A have "∀i∈X. A i ∈ M" by fast
hence "Ω - (⋃i∈X. Ω - A i) ∈ M" by blast
moreover
have "(⋂i∈X. A i) = Ω - (⋃i∈X. Ω - A i)" using space_closed A
by blast
ultimately show ?thesis by metis
qed

lemma (in sigma_algebra) countable_INT':
fixes A :: "'i ⇒ 'a set"
assumes X: "countable X" "X ≠ {}"
assumes A: "A`X ⊆ M"
shows  "(⋂x∈X. A x) ∈ M"
proof -
have "(⋂x∈X. A x) = (⋂i∈to_nat_on X ` X. A (from_nat_into X i))"
using X by auto
also have "… ∈ M"
using A X
by (intro countable_INT) auto
finally show ?thesis .
qed

lemma (in sigma_algebra) countable_INT'':
"UNIV ∈ M ⟹ countable I ⟹ (⋀i. i ∈ I ⟹ F i ∈ M) ⟹ (⋂i∈I. F i) ∈ M"
by (cases "I = {}") (auto intro: countable_INT')

lemma (in sigma_algebra) countable:
assumes "⋀a. a ∈ A ⟹ {a} ∈ M" "countable A"
shows "A ∈ M"
proof -
have "(⋃a∈A. {a}) ∈ M"
using assms by (intro countable_UN') auto
also have "(⋃a∈A. {a}) = A" by auto
finally show ?thesis by auto
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 Ω M ⟷
algebra Ω M ∧ (∀A. range A ⊆ M ⟶ (⋃i::nat. A i) ∈ 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 "⋀i. {x∈Ω. P i x} ∈ M"
shows "{x∈Ω. ∀i::'i::countable. P i x} ∈ M"
proof -
have "{x∈Ω. ∀i::'i::countable. P i x} = (⋂i. {x∈Ω. P i x})" by auto
with assms show ?thesis by auto
qed

lemma (in sigma_algebra) sets_Collect_countable_Ex:
assumes "⋀i. {x∈Ω. P i x} ∈ M"
shows "{x∈Ω. ∃i::'i::countable. P i x} ∈ M"
proof -
have "{x∈Ω. ∃i::'i::countable. P i x} = (⋃i. {x∈Ω. P i x})" by auto
with assms show ?thesis by auto
qed

lemma (in sigma_algebra) sets_Collect_countable_Ex':
assumes "⋀i. i ∈ I ⟹ {x∈Ω. P i x} ∈ M"
assumes "countable I"
shows "{x∈Ω. ∃i∈I. P i x} ∈ M"
proof -
have "{x∈Ω. ∃i∈I. P i x} = (⋃i∈I. {x∈Ω. P i x})" by auto
with assms show ?thesis
by (auto intro!: countable_UN')
qed

lemma (in sigma_algebra) sets_Collect_countable_All':
assumes "⋀i. i ∈ I ⟹ {x∈Ω. P i x} ∈ M"
assumes "countable I"
shows "{x∈Ω. ∀i∈I. P i x} ∈ M"
proof -
have "{x∈Ω. ∀i∈I. P i x} = (⋂i∈I. {x∈Ω. P i x}) ∩ Ω" by auto
with assms show ?thesis
by (cases "I = {}") (auto intro!: countable_INT')
qed

lemma (in sigma_algebra) sets_Collect_countable_Ex1':
assumes "⋀i. i ∈ I ⟹ {x∈Ω. P i x} ∈ M"
assumes "countable I"
shows "{x∈Ω. ∃!i∈I. P i x} ∈ M"
proof -
have "{x∈Ω. ∃!i∈I. P i x} = {x∈Ω. ∃i∈I. P i x ∧ (∀j∈I. P j x ⟶ i = j)}"
by auto
with assms show ?thesis
by (auto intro!: sets_Collect_countable_All' sets_Collect_countable_Ex' sets_Collect_conj sets_Collect_imp sets_Collect_const)
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 "⋀i. {x∈Ω. P i x} ∈ M"
shows "{x∈Ω. ∀i::'i::countable∈X. P i x} ∈ M"
unfolding Ball_def by (intro sets_Collect assms)

lemma (in sigma_algebra) sets_Collect_countable_Bex:
assumes "⋀i. {x∈Ω. P i x} ∈ M"
shows "{x∈Ω. ∃i::'i::countable∈X. P i x} ∈ M"
unfolding Bex_def by (intro sets_Collect assms)

lemma sigma_algebra_single_set:
assumes "X ⊆ S"
shows "sigma_algebra S { {}, X, S - X, S }"
using algebra.is_sigma_algebra[OF algebra_single_set[OF ‹X ⊆ S›]] by simp

subsubsection✐‹tag unimportant› ‹Binary Unions›

definition binary :: "'a ⇒ 'a ⇒ nat ⇒ 'a"
where "binary a b =  (λx. b)(0 := a)"

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

lemma Un_range_binary: "a ∪ b = (⋃i::nat. binary a b i)"
by (simp add: range_binary_eq cong del: SUP_cong_simp)

lemma Int_range_binary: "a ∩ b = (⋂i::nat. binary a b i)"
by (simp add: range_binary_eq cong del: INF_cong_simp)

lemma sigma_algebra_iff2:
"sigma_algebra Ω M ⟷
M ⊆ Pow Ω ∧ {} ∈ M ∧ (∀s ∈ M. Ω - s ∈ M)
∧ (∀A. range A ⊆ M ⟶(⋃ i::nat. A i) ∈ M)" (is "?P ⟷ ?R ∧ ?S ∧ ?V ∧ ?W")
proof
assume ?P
then interpret sigma_algebra Ω M .
from space_closed show "?R ∧ ?S ∧ ?V ∧ ?W"
by auto
next
assume "?R ∧ ?S ∧ ?V ∧ ?W"
then have ?R ?S ?V ?W
by simp_all
show ?P
proof (rule sigma_algebra.intro)
show "sigma_algebra_axioms M"
by standard (use ‹?W› in simp)
from ‹?W› have *: "range (binary a b) ⊆ M ⟹ ⋃ (range (binary a b)) ∈ M" for a b
by auto
show "algebra Ω M"
unfolding algebra_iff_Un using ‹?R› ‹?S› ‹?V› *
qed
qed

subsubsection ‹Initial Sigma Algebra›

text✐‹tag important› ‹Sigma algebras can naturally be created as the closure of any set of
M with regard to the properties just postulated.›

inductive_set✐‹tag important› sigma_sets :: "'a set ⇒ 'a set set ⇒ 'a set set"
for sp :: "'a set" and A :: "'a set set"
where
Basic[intro, simp]: "a ∈ A ⟹ a ∈ sigma_sets sp A"
| Empty: "{} ∈ sigma_sets sp A"
| Compl: "a ∈ sigma_sets sp A ⟹ sp - a ∈ sigma_sets sp A"
| Union: "(⋀i::nat. a i ∈ sigma_sets sp A) ⟹ (⋃i. a i) ∈ sigma_sets sp A"

lemma (in sigma_algebra) sigma_sets_subset:
assumes a: "a ⊆ M"
shows "sigma_sets Ω a ⊆ M"
proof
fix x
assume "x ∈ sigma_sets Ω a"
from this show "x ∈ M"
by (induct rule: sigma_sets.induct, auto) (metis a subsetD)
qed

lemma sigma_sets_into_sp: "A ⊆ Pow sp ⟹ x ∈ sigma_sets sp A ⟹ x ⊆ sp"
by (erule sigma_sets.induct, auto)

lemma sigma_algebra_sigma_sets:
"a ⊆ Pow Ω ⟹ sigma_algebra Ω (sigma_sets Ω 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 ⊆ Pow S"
shows "sigma_sets S A = ⋂{B. A ⊆ B ∧ sigma_algebra S B}"
proof safe
fix B X assume "A ⊆ B" and sa: "sigma_algebra S B"
and X: "X ∈ sigma_sets S A"
from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF ‹A ⊆ B›] X
show "X ∈ B" by auto
next
fix X assume "X ∈ ⋂{B. A ⊆ B ∧ sigma_algebra S B}"
then have [intro!]: "⋀B. A ⊆ B ⟹ sigma_algebra S B ⟹ X ∈ B"
by simp
have "A ⊆ 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 ∈ sigma_sets S A" by auto
qed

lemma sigma_sets_top: "sp ∈ sigma_sets sp A"
by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty)

lemma binary_in_sigma_sets:
"binary a b i ∈ sigma_sets sp A" if "a ∈ sigma_sets sp A" and "b ∈ sigma_sets sp A"
using that by (simp add: binary_def)

lemma sigma_sets_Un:
"a ∪ b ∈ sigma_sets sp A" if "a ∈ sigma_sets sp A" and "b ∈ sigma_sets sp A"
using that by (simp add: Un_range_binary binary_in_sigma_sets Union)

lemma sigma_sets_Inter:
assumes Asb: "A ⊆ Pow sp"
shows "(⋀i::nat. a i ∈ sigma_sets sp A) ⟹ (⋂i. a i) ∈ sigma_sets sp A"
proof -
assume ai: "⋀i::nat. a i ∈ sigma_sets sp A"
hence "⋀i::nat. sp-(a i) ∈ sigma_sets sp A"
by (rule sigma_sets.Compl)
hence "(⋃i. sp-(a i)) ∈ sigma_sets sp A"
by (rule sigma_sets.Union)
hence "sp-(⋃i. sp-(a i)) ∈ sigma_sets sp A"
by (rule sigma_sets.Compl)
also have "sp-(⋃i. sp-(a i)) = sp Int (⋂i. a i)"
by auto
also have "... = (⋂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 ⊆ Pow sp"
and ai: "⋀i::nat. i ∈ S ⟹ a i ∈ sigma_sets sp A" and non: "S ≠ {}"
shows "(⋂i∈S. a i) ∈ sigma_sets sp A"
proof -
from ai have "⋀i. (if i∈S then a i else sp) ∈ sigma_sets sp A"
hence "(⋂i. (if i∈S then a i else sp)) ∈ sigma_sets sp A"
by (rule sigma_sets_Inter [OF Asb])
also have "(⋂i. (if i∈S then a i else sp)) = (⋂i∈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 ⟹ (⋀b. b ∈ B ⟹ b ∈ sigma_sets X A) ⟹ ⋃ B ∈ 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]
by (cases "B = {}") (simp_all add: sigma_sets.Empty cong del: SUP_cong)

lemma (in sigma_algebra) sigma_sets_eq:
"sigma_sets Ω M = M"
proof
show "M ⊆ sigma_sets Ω M"
by (metis Set.subsetI sigma_sets.Basic)
next
show "sigma_sets Ω M ⊆ M"
by (metis sigma_sets_subset subset_refl)
qed

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

lemma sigma_sets_subseteq: assumes "A ⊆ B" shows "sigma_sets X A ⊆ sigma_sets X B"
proof
fix x assume "x ∈ sigma_sets X A" then show "x ∈ sigma_sets X B"
by induct (insert ‹A ⊆ B›, auto intro: sigma_sets.intros(2-))
qed

lemma sigma_sets_mono: assumes "A ⊆ sigma_sets X B" shows "sigma_sets X A ⊆ sigma_sets X B"
proof
fix x assume "x ∈ sigma_sets X A" then show "x ∈ sigma_sets X B"
by induct (insert ‹A ⊆ sigma_sets X B›, auto intro: sigma_sets.intros(2-))
qed

lemma sigma_sets_mono': assumes "A ⊆ B" shows "sigma_sets X A ⊆ sigma_sets X B"
proof
fix x assume "x ∈ sigma_sets X A" then show "x ∈ sigma_sets X B"
by induct (insert ‹A ⊆ B›, auto intro: sigma_sets.intros(2-))
qed

lemma sigma_sets_superset_generator: "A ⊆ sigma_sets X A"
by (auto intro: sigma_sets.Basic)

lemma (in sigma_algebra) restriction_in_sets:
fixes A :: "nat ⇒ 'a set"
assumes "S ∈ M"
and *: "range A ⊆ (λA. S ∩ A) ` M" (is "_ ⊆ ?r")
shows "range A ⊆ M" "(⋃i. A i) ∈ (λA. S ∩ A) ` M"
proof -
{ fix i have "A i ∈ ?r" using * by auto
hence "∃B. A i = B ∩ S ∧ B ∈ M" by auto
hence "A i ⊆ S" "A i ∈ M" using ‹S ∈ M› by auto }
thus "range A ⊆ M" "(⋃i. A i) ∈ (λA. S ∩ A) ` M"
by (auto intro!: image_eqI[of _ _ "(⋃i. A i)"])
qed

lemma (in sigma_algebra) restricted_sigma_algebra:
assumes "S ∈ 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 ⇒ 'a set" assume "range A ⊆ restricted_space S"
from restriction_in_sets[OF assms this[simplified]]
show "(⋃i. A i) ∈ restricted_space S" by simp
qed

lemma sigma_sets_Int:
assumes "A ∈ sigma_sets sp st" "A ⊆ sp"
shows "(∩) A ` sigma_sets sp st = sigma_sets A ((∩) A ` st)"
proof (intro equalityI subsetI)
fix x assume "x ∈ (∩) A ` sigma_sets sp st"
then obtain y where "y ∈ sigma_sets sp st" "x = y ∩ A" by auto
then have "x ∈ sigma_sets (A ∩ sp) ((∩) 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 ∈ sigma_sets A ((∩) A ` st)"
using ‹A ⊆ sp› by (simp add: Int_absorb2)
next
fix x assume "x ∈ sigma_sets A ((∩) A ` st)"
then show "x ∈ (∩) A ` sigma_sets sp st"
proof induct
case (Compl a)
then obtain x where "a = A ∩ x" "x ∈ sigma_sets sp st" by auto
then show ?case using ‹A ⊆ sp›
by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl)
next
case (Union a)
then have "∀i. ∃x. x ∈ sigma_sets sp st ∧ a i = A ∩ x"
by (auto simp: image_iff Bex_def)
then obtain f where "∀x. f x ∈ sigma_sets sp st ∧ a x = A ∩ f x"
by metis
then show ?case
by (auto intro!: bexI[of _ "(⋃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 ∈ sigma_sets A {}" then show "a ∈ {{}, 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 ∈ sigma_sets A {A}"
then show "x ∈ {{}, A}"
by induct blast+
next
fix x assume "x ∈ {{}, A}"
then show "x ∈ sigma_sets A {A}"
by (auto intro: sigma_sets.Empty sigma_sets_top)
qed

lemma sigma_sets_sigma_sets_eq:
"M ⊆ Pow S ⟹ 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 ⊆ 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 } ⊆ sigma_sets S { {}, X, S - X, S }"
by (rule sigma_sets_subseteq) simp
moreover have "… = { {}, X, S - X, S }"
using sigma_sets_eq by simp
moreover
{ fix A assume "A ∈ { {}, X, S - X, S }"
then have "A ∈ 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 ∈ sigma_sets Ω M" and M: "M ⊆ Pow Ω"
shows "algebra.restricted_space (sigma_sets Ω M) S =
sigma_sets S (algebra.restricted_space M S)"
proof -
from S sigma_sets_into_sp[OF M]
have "S ∈ sigma_sets Ω M" "S ⊆ Ω" by auto
from sigma_sets_Int[OF this]
show ?thesis by simp
qed

lemma sigma_sets_vimage_commute:
assumes X: "X ∈ Ω → Ω'"
shows "{X -` A ∩ Ω |A. A ∈ sigma_sets Ω' M'}
= sigma_sets Ω {X -` A ∩ Ω |A. A ∈ M'}" (is "?L = ?R")
proof
show "?L ⊆ ?R"
proof clarify
fix A assume "A ∈ sigma_sets Ω' M'"
then show "X -` A ∩ Ω ∈ ?R"
proof induct
case Empty then show ?case
by (auto intro!: sigma_sets.Empty)
next
case (Compl B)
have [simp]: "X -` (Ω' - B) ∩ Ω = Ω - (X -` B ∩ Ω)"
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 ⊆ ?L"
proof clarify
fix A assume "A ∈ ?R"
then show "∃B. A = X -` B ∩ Ω ∧ B ∈ sigma_sets Ω' 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 ∩ Ω" "A ∈ sigma_sets Ω' M'" by auto
then have [simp]: "Ω - B = X -` (Ω' - A) ∩ Ω"
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 "∀i. ∃B. F i = X -` B ∩ Ω ∧ B ∈ sigma_sets Ω' M'" by auto
then obtain A where "∀x. F x = X -` A x ∩ Ω ∧ A x ∈ sigma_sets Ω' M'"
by metis
then show ?case
by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union)
qed
qed
qed

lemma (in ring_of_sets) UNION_in_sets:
fixes A:: "nat ⇒ 'a set"
assumes A: "range A ⊆ M"
shows  "(⋃i∈{0..<n}. A i) ∈ 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 ⊆ M"
shows  "range (disjointed A) ⊆ M"
fix n
show "A n - (⋃i∈{0..<n}. A i) ∈ M" using UNION_in_sets
by (metis A Diff UNIV_I image_subset_iff)
qed

lemma (in algebra) range_disjointed_sets':
"range A ⊆ M ⟹ range (disjointed A) ⊆ M"
using range_disjointed_sets .

lemma sigma_algebra_disjoint_iff:
"sigma_algebra Ω M ⟷ algebra Ω M ∧
(∀A. range A ⊆ M ⟶ disjoint_family A ⟶ (⋃i::nat. A i) ∈ M)"
fix A :: "nat ⇒ 'a set"
assume M: "algebra Ω M"
and A: "range A ⊆ M"
and UnA: "∀A. range A ⊆ M ⟶ disjoint_family A ⟶ (⋃i::nat. A i) ∈ M"
hence "range (disjointed A) ⊆ M ⟶
disjoint_family (disjointed A) ⟶
(⋃i. disjointed A i) ∈ M" by blast
hence "(⋃i. disjointed A i) ∈ M"
by (simp add: algebra.range_disjointed_sets'[of Ω] M A disjoint_family_disjointed)
thus "(⋃i::nat. A i) ∈ M" by (simp add: UN_disjointed_eq)
qed

subsubsection✐‹tag unimportant› ‹Ring generated by a semiring›

definition (in semiring_of_sets) generated_ring :: "'a set set" where
"generated_ring = { ⋃C | C. C ⊆ M ∧ finite C ∧ disjoint C }"

lemma (in semiring_of_sets) generated_ringE[elim?]:
assumes "a ∈ generated_ring"
obtains C where "finite C" "disjoint C" "C ⊆ M" "a = ⋃C"
using assms unfolding generated_ring_def by auto

lemma (in semiring_of_sets) generated_ringI[intro?]:
assumes "finite C" "disjoint C" "C ⊆ M" "a = ⋃C"
shows "a ∈ generated_ring"
using assms unfolding generated_ring_def by auto

lemma (in semiring_of_sets) generated_ringI_Basic:
"A ∈ M ⟹ A ∈ 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 ∈ generated_ring" and b: "b ∈ generated_ring"
and "a ∩ b = {}"
shows "a ∪ b ∈ generated_ring"
proof -
from a b obtain Ca Cb
where Ca: "finite Ca" "disjoint Ca" "Ca ⊆ M" "a = ⋃ Ca"
and Cb: "finite Cb" "disjoint Cb" "Cb ⊆ M" "b = ⋃ Cb"
using generated_ringE by metis
show ?thesis
proof
from ‹a ∩ b = {}› Ca Cb show "disjoint (Ca ∪ Cb)"
by (auto intro!: disjoint_union)
qed (use Ca Cb in auto)
qed

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

lemma (in semiring_of_sets) generated_ring_disjoint_Union:
assumes "finite A" shows "A ⊆ generated_ring ⟹ disjoint A ⟹ ⋃A ∈ 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 ⟹ disjoint (A ` I) ⟹ (⋀i. i ∈ I ⟹ A i ∈ generated_ring) ⟹ ⋃(A ` I) ∈ generated_ring"
by (intro generated_ring_disjoint_Union) auto

lemma (in semiring_of_sets) generated_ring_Int:
assumes a: "a ∈ generated_ring" and b: "b ∈ generated_ring"
shows "a ∩ b ∈ generated_ring"
proof -
from a b obtain Ca Cb
where Ca: "finite Ca" "disjoint Ca" "Ca ⊆ M" "a = ⋃ Ca"
and Cb: "finite Cb" "disjoint Cb" "Cb ⊆ M" "b = ⋃ Cb"
using generated_ringE by metis
define C where "C = (λ(a,b). a ∩ b)` (Ca×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 ∈ Ca" "b1 ∈ Cb" "a2 ∈ Ca" "b2 ∈ Cb"
assume "a1 ∩ b1 ≠ a2 ∩ b2"
then have "a1 ≠ a2 ∨ b1 ≠ b2" by auto
then show "(a1 ∩ b1) ∩ (a2 ∩ b2) = {}"
proof
assume "a1 ≠ a2"
with sets Ca have "a1 ∩ a2 = {}"
by (auto simp: disjoint_def)
then show ?thesis by auto
next
assume "b1 ≠ b2"
with sets Cb have "b1 ∩ b2 = {}"
by (auto simp: disjoint_def)
then show ?thesis by auto
qed
qed
qed (use Ca Cb in ‹auto simp: C_def›)
qed

lemma (in semiring_of_sets) generated_ring_Inter:
assumes "finite A" "A ≠ {}" shows "A ⊆ generated_ring ⟹ ⋂A ∈ 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 ⟹ I ≠ {} ⟹ (⋀i. i ∈ I ⟹ A i ∈ generated_ring) ⟹ ⋂(A ` I) ∈ generated_ring"
by (intro generated_ring_Inter) auto

lemma (in semiring_of_sets) generating_ring:
"ring_of_sets Ω generated_ring"
proof (rule ring_of_setsI)
let ?R = generated_ring
show "?R ⊆ Pow Ω"
using sets_into_space by (auto simp: generated_ring_def generated_ring_empty)
show "{} ∈ ?R" by (rule generated_ring_empty)

{
fix a b assume "a ∈ ?R" "b ∈ ?R"
then obtain Ca Cb
where Ca: "finite Ca" "disjoint Ca" "Ca ⊆ M" "a = ⋃ Ca"
and Cb: "finite Cb" "disjoint Cb" "Cb ⊆ M" "b = ⋃ Cb"
using generated_ringE by metis
show "a - b ∈ ?R"
proof cases
assume "Cb = {}" with Cb ‹a ∈ ?R› show ?thesis
by simp
next
assume "Cb ≠ {}"
with Ca Cb have "a - b = (⋃a'∈Ca. ⋂b'∈Cb. a' - b')" by auto
also have "… ∈ ?R"
proof (intro generated_ring_INTER generated_ring_disjoint_UNION)
fix a b assume "a ∈ Ca" "b ∈ Cb"
with Ca Cb Diff_cover[of a b] show "a - b ∈ ?R"
(metis DiffI Diff_eq_empty_iff empty_iff)
next
show "disjoint ((λa'. ⋂b'∈Cb. a' - b')`Ca)"
using Ca by (auto simp add: disjoint_def ‹Cb ≠ {}›)
next
show "finite Ca" "finite Cb" "Cb ≠ {}" by fact+
qed
finally show "a - b ∈ ?R" .
qed
}
note Diff = this

fix a b assume sets: "a ∈ ?R" "b ∈ ?R"
have "a ∪ b = (a - b) ∪ (a ∩ b) ∪ (b - a)" by auto
also have "… ∈ ?R"
by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto
finally show "a ∪ b ∈ ?R" .
qed

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

subsubsection✐‹tag unimportant› ‹A Two-Element Series›

definition binaryset :: "'a set ⇒ 'a set ⇒ nat ⇒ 'a set"
where "binaryset A B = (λx. {})(0 := A, Suc 0 := B)"

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

lemma UN_binaryset_eq: "(⋃i. binaryset A B i) = A ∪ B"
by (simp add: range_binaryset_eq cong del: SUP_cong_simp)

subsubsection ‹Closed CDI›

definition✐‹tag important› closed_cdi :: "'a set ⇒ 'a set set ⇒ bool" where
"closed_cdi Ω M ⟷
M ⊆ Pow Ω &
(∀s ∈ M. Ω - s ∈ M) &
(∀A. (range A ⊆ M) & (A 0 = {}) & (∀n. A n ⊆ A (Suc n)) ⟶
(⋃i. A i) ∈ M) &
(∀A. (range A ⊆ M) & disjoint_family A ⟶ (⋃i::nat. A i) ∈ M)"

inductive_set
smallest_ccdi_sets :: "'a set ⇒ 'a set set ⇒ 'a set set"
for Ω M
where
Basic [intro]:
"a ∈ M ⟹ a ∈ smallest_ccdi_sets Ω M"
| Compl [intro]:
"a ∈ smallest_ccdi_sets Ω M ⟹ Ω - a ∈ smallest_ccdi_sets Ω M"
| Inc:
"range A ∈ Pow(smallest_ccdi_sets Ω M) ⟹ A 0 = {} ⟹ (⋀n. A n ⊆ A (Suc n))
⟹ (⋃i. A i) ∈ smallest_ccdi_sets Ω M"
| Disj:
"range A ∈ Pow(smallest_ccdi_sets Ω M) ⟹ disjoint_family A
⟹ (⋃i::nat. A i) ∈ smallest_ccdi_sets Ω M"

lemma (in subset_class) smallest_closed_cdi1: "M ⊆ smallest_ccdi_sets Ω M"
by auto

lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets Ω M ⊆ Pow Ω"
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 Ω (smallest_ccdi_sets Ω 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 Ω M ⟹ M ⊆ Pow Ω"

lemma closed_cdi_Compl: "closed_cdi Ω M ⟹ s ∈ M ⟹ Ω - s ∈ M"

lemma closed_cdi_Inc:
"closed_cdi Ω M ⟹ range A ⊆ M ⟹ A 0 = {} ⟹ (!!n. A n ⊆ A (Suc n)) ⟹ (⋃i. A i) ∈ M"

lemma closed_cdi_Disj:
"closed_cdi Ω M ⟹ range A ⊆ M ⟹ disjoint_family A ⟹ (⋃i::nat. A i) ∈ M"

lemma closed_cdi_Un:
assumes cdi: "closed_cdi Ω M" and empty: "{} ∈ M"
and A: "A ∈ M" and B: "B ∈ M"
and disj: "A ∩ B = {}"
shows "A ∪ B ∈ M"
proof -
have ra: "range (binaryset A B) ⊆ 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 ∈ smallest_ccdi_sets Ω M" and B: "B ∈ smallest_ccdi_sets Ω M"
and disj: "A ∩ B = {}"
shows "A ∪ B ∈ smallest_ccdi_sets Ω M"
proof -
have ra: "range (binaryset A B) ∈ Pow (smallest_ccdi_sets Ω 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 ∈ M"
shows "b ∈ smallest_ccdi_sets Ω M ⟹ a ∩ b ∈ smallest_ccdi_sets Ω 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 ∩ (Ω - x) = Ω - ((Ω - a) ∪ (a ∩ x))"
by blast
also have "... ∈ smallest_ccdi_sets Ω 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: "(⋃i. (λi. a ∩ A i) i) = a ∩ (⋃i. A i)"
by blast
have "range (λi. a ∩ A i) ∈ Pow(smallest_ccdi_sets Ω M)" using Inc
by blast
moreover have "(λi. a ∩ A i) 0 = {}"
moreover have "!!n. (λi. a ∩ A i) n ⊆ (λi. a ∩ A i) (Suc n)" using Inc
by blast
ultimately have 2: "(⋃i. (λi. a ∩ A i) i) ∈ smallest_ccdi_sets Ω M"
by (rule smallest_ccdi_sets.Inc)
show ?case
by (metis 1 2)
next
case (Disj A)
have 1: "(⋃i. (λi. a ∩ A i) i) = a ∩ (⋃i. A i)"
by blast
have "range (λi. a ∩ A i) ∈ Pow(smallest_ccdi_sets Ω M)" using Disj
by blast
moreover have "disjoint_family (λi. a ∩ A i)" using Disj
ultimately have 2: "(⋃i. (λi. a ∩ A i) i) ∈ smallest_ccdi_sets Ω M"
by (rule smallest_ccdi_sets.Disj)
show ?case
by (metis 1 2)
qed

lemma (in algebra) smallest_ccdi_sets_Int:
assumes b: "b ∈ smallest_ccdi_sets Ω M"
shows "a ∈ smallest_ccdi_sets Ω M ⟹ a ∩ b ∈ smallest_ccdi_sets Ω 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 "(Ω - x) ∩ b = Ω - (x ∩ b ∪ (Ω - b))"
by blast
also have "... ∈ smallest_ccdi_sets Ω 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: "(⋃i. (λi. A i ∩ b) i) = (⋃i. A i) ∩ b"
by blast
have "range (λi. A i ∩ b) ∈ Pow(smallest_ccdi_sets Ω M)" using Inc
by blast
moreover have "(λi. A i ∩ b) 0 = {}"
moreover have "!!n. (λi. A i ∩ b) n ⊆ (λi. A i ∩ b) (Suc n)" using Inc
by blast
ultimately have 2: "(⋃i. (λi. A i ∩ b) i) ∈ smallest_ccdi_sets Ω M"
by (rule smallest_ccdi_sets.Inc)
show ?case
by (metis 1 2)
next
case (Disj A)
have 1: "(⋃i. (λi. A i ∩ b) i) = (⋃i. A i) ∩ b"
by blast
have "range (λi. A i ∩ b) ∈ Pow(smallest_ccdi_sets Ω M)" using Disj
by blast
moreover have "disjoint_family (λi. A i ∩ b)" using Disj
ultimately have 2: "(⋃i. (λi. A i ∩ b) i) ∈ smallest_ccdi_sets Ω M"
by (rule smallest_ccdi_sets.Disj)
show ?case
by (metis 1 2)
qed

lemma (in algebra) sigma_property_disjoint_lemma:
assumes sbC: "M ⊆ C"
and ccdi: "closed_cdi Ω C"
shows "sigma_sets Ω M ⊆ C"
proof -
have "smallest_ccdi_sets Ω M ∈ {B . M ⊆ B ∧ sigma_algebra Ω 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 (Ω) (M) ⊆ smallest_ccdi_sets Ω M"
by clarsimp
(drule sigma_algebra.sigma_sets_subset [where a="M"], auto)
also have "...  ⊆ C"
proof
fix x
assume x: "x ∈ smallest_ccdi_sets Ω M"
thus "x ∈ 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 ⊆ C"
and compl: "!!s. s ∈ C ∩ sigma_sets (Ω) (M) ⟹ Ω - s ∈ C"
and inc: "!!A. range A ⊆ C ∩ sigma_sets (Ω) (M)
⟹ A 0 = {} ⟹ (!!n. A n ⊆ A (Suc n))
⟹ (⋃i. A i) ∈ C"
and disj: "!!A. range A ⊆ C ∩ sigma_sets (Ω) (M)
⟹ disjoint_family A ⟹ (⋃i::nat. A i) ∈ C"
shows "sigma_sets (Ω) (M) ⊆ C"
proof -
have "sigma_sets (Ω) (M) ⊆ C ∩ sigma_sets (Ω) (M)"
proof (rule sigma_property_disjoint_lemma)
show "M ⊆ C ∩ sigma_sets (Ω) (M)"
by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic)
next
show "closed_cdi Ω (C ∩ sigma_sets (Ω) (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

subsubsection ‹Dynkin systems›

locale✐‹tag important› Dynkin_system = subset_class +
assumes space: "Ω ∈ M"
and   compl[intro!]: "⋀A. A ∈ M ⟹ Ω - A ∈ M"
and   UN[intro!]: "⋀A. disjoint_family A ⟹ range A ⊆ M
⟹ (⋃i::nat. A i) ∈ M"

lemma (in Dynkin_system) empty[intro, simp]: "{} ∈ M"
using space compl[of "Ω"] by simp

lemma (in Dynkin_system) diff:
assumes sets: "D ∈ M" "E ∈ M" and "D ⊆ E"
shows "E - D ∈ M"
proof -
let ?f = "λx. if x = 0 then D else if x = Suc 0 then Ω - E else {}"
have "range ?f = {D, Ω - E, {}}"
by (auto simp: image_iff)
moreover have "D ∪ (Ω - E) = (⋃i. ?f i)"
by (auto simp: image_iff split: if_split_asm)
moreover
have "disjoint_family ?f" unfolding disjoint_family_on_def
using ‹D ∈ M›[THEN sets_into_space] ‹D ⊆ E› by auto
ultimately have "Ω - (D ∪ (Ω - E)) ∈ M"
using sets UN by auto fastforce
also have "Ω - (D ∪ (Ω - E)) = E - D"
using assms sets_into_space by auto
finally show ?thesis .
qed

lemma Dynkin_systemI:
assumes "⋀ A. A ∈ M ⟹ A ⊆ Ω" "Ω ∈ M"
assumes "⋀ A. A ∈ M ⟹ Ω - A ∈ M"
assumes "⋀ A. disjoint_family A ⟹ range A ⊆ M
⟹ (⋃i::nat. A i) ∈ M"
shows "Dynkin_system Ω M"
using assms by (auto simp: Dynkin_system_def Dynkin_system_axioms_def subset_class_def)

lemma Dynkin_systemI':
assumes 1: "⋀ A. A ∈ M ⟹ A ⊆ Ω"
assumes empty: "{} ∈ M"
assumes Diff: "⋀ A. A ∈ M ⟹ Ω - A ∈ M"
assumes 2: "⋀ A. disjoint_family A ⟹ range A ⊆ M
⟹ (⋃i::nat. A i) ∈ M"
shows "Dynkin_system Ω M"
proof -
from Diff[OF empty] have "Ω ∈ 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 Ω M" shows "Dynkin_system Ω M"
proof -
interpret sigma_algebra Ω M by fact
show ?thesis using sets_into_space by (fastforce intro!: Dynkin_systemI)
qed

subsubsection "Intersection sets systems"

definition✐‹tag important› Int_stable :: "'a set set ⇒ bool" where
"Int_stable M ⟷ (∀ a ∈ M. ∀ b ∈ M. a ∩ b ∈ M)"

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

lemma Int_stableI_image:
"(⋀i j. i ∈ I ⟹ j ∈ I ⟹ ∃k∈I. A i ∩ A j = A k) ⟹ Int_stable (A ` I)"
by (auto simp: Int_stable_def image_def)

lemma Int_stableI:
"(⋀a b. a ∈ A ⟹ b ∈ A ⟹ a ∩ b ∈ A) ⟹ Int_stable A"
unfolding Int_stable_def by auto

lemma Int_stableD:
"Int_stable M ⟹ a ∈ M ⟹ b ∈ M ⟹ a ∩ b ∈ M"
unfolding Int_stable_def by auto

lemma (in Dynkin_system) sigma_algebra_eq_Int_stable:
"sigma_algebra Ω M ⟷ Int_stable M"
proof
assume "sigma_algebra Ω M" then show "Int_stable M"
unfolding sigma_algebra_def using algebra.Int_stable by auto
next
assume "Int_stable M"
show "sigma_algebra Ω M"
unfolding sigma_algebra_disjoint_iff algebra_iff_Un
proof (intro conjI ballI allI impI)
show "M ⊆ Pow (Ω)" using sets_into_space by auto
next
fix A B assume "A ∈ M" "B ∈ M"
then have "A ∪ B = Ω - ((Ω - A) ∩ (Ω - B))"
"Ω - A ∈ M" "Ω - B ∈ M"
using sets_into_space by auto
then show "A ∪ B ∈ M"
using ‹Int_stable M› unfolding Int_stable_def by auto
qed auto
qed

subsubsection "Smallest Dynkin systems"

definition✐‹tag important› Dynkin :: "'a set ⇒ 'a set set ⇒ 'a set set" where
"Dynkin Ω M =  (⋂{D. Dynkin_system Ω D ∧ M ⊆ D})"

lemma Dynkin_system_Dynkin:
assumes "M ⊆ Pow (Ω)"
shows "Dynkin_system Ω (Dynkin Ω M)"
proof (rule Dynkin_systemI)
fix A assume "A ∈ Dynkin Ω M"
moreover
{ fix D assume "A ∈ D" and d: "Dynkin_system Ω D"
then have "A ⊆ Ω" by (auto simp: Dynkin_system_def subset_class_def) }
moreover have "{D. Dynkin_system Ω D ∧ M ⊆ D} ≠ {}"
using assms Dynkin_system_trivial by fastforce
ultimately show "A ⊆ Ω"
unfolding Dynkin_def using assms
by auto
next
show "Ω ∈ Dynkin Ω M"
unfolding Dynkin_def using Dynkin_system.space by fastforce
next
fix A assume "A ∈ Dynkin Ω M"
then show "Ω - A ∈ Dynkin Ω M"
unfolding Dynkin_def using Dynkin_system.compl by force
next
fix A :: "nat ⇒ 'a set"
assume A: "disjoint_family A" "range A ⊆ Dynkin Ω M"
show "(⋃i. A i) ∈ Dynkin Ω M" unfolding Dynkin_def
proof (simp, safe)
fix D assume "Dynkin_system Ω D" "M ⊆ D"
with A have "(⋃i. A i) ∈ D"
by (intro Dynkin_system.UN) (auto simp: Dynkin_def)
then show "(⋃i. A i) ∈ D" by auto
qed
qed

lemma Dynkin_Basic[intro]: "A ∈ M ⟹ A ∈ Dynkin Ω M"
unfolding Dynkin_def by auto

lemma (in Dynkin_system) restricted_Dynkin_system:
assumes "D ∈ M"
shows "Dynkin_system Ω {Q. Q ⊆ Ω ∧ Q ∩ D ∈ M}"
proof (rule Dynkin_systemI, simp_all)
have "Ω ∩ D = D"
using ‹D ∈ M› sets_into_space by auto
then show "Ω ∩ D ∈ M"
using ‹D ∈ M› by auto
next
fix A assume "A ⊆ Ω ∧ A ∩ D ∈ M"
moreover have "(Ω - A) ∩ D = (Ω - (A ∩ D)) - (Ω - D)"
by auto
ultimately show "(Ω - A) ∩ D ∈ M"
using  ‹D ∈ M› by (auto intro: diff)
next
fix A :: "nat ⇒ 'a set"
assume "disjoint_family A" "range A ⊆ {Q. Q ⊆ Ω ∧ Q ∩ D ∈ M}"
then have "⋀i. A i ⊆ Ω" "disjoint_family (λi. A i ∩ D)"
"range (λi. A i ∩ D) ⊆ M" "(⋃x. A x) ∩ D = (⋃x. A x ∩ D)"
by ((fastforce simp: disjoint_family_on_def)+)
then show "(⋃x. A x) ⊆ Ω ∧ (⋃x. A x) ∩ D ∈ M"
by (auto simp del: UN_simps)
qed

lemma (in Dynkin_system) Dynkin_subset:
assumes "N ⊆ M"
shows "Dynkin Ω N ⊆ M"
proof -
have "Dynkin_system Ω M" ..
then have "Dynkin_system Ω M"
using assms unfolding Dynkin_system_def Dynkin_system_axioms_def subset_class_def by simp
with ‹N ⊆ M› show ?thesis by (auto simp add: Dynkin_def)
qed

lemma sigma_eq_Dynkin:
assumes sets: "M ⊆ Pow Ω"
assumes "Int_stable M"
shows "sigma_sets Ω M = Dynkin Ω M"
proof -
have "Dynkin Ω M ⊆ sigma_sets (Ω) (M)"
using sigma_algebra_imp_Dynkin_system
unfolding Dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto
moreover
interpret Dynkin_system Ω "Dynkin Ω M"
using Dynkin_system_Dynkin[OF sets] .
have "sigma_algebra Ω (Dynkin Ω M)"
unfolding sigma_algebra_eq_Int_stable Int_stable_def
proof (intro ballI)
fix A B assume "A ∈ Dynkin Ω M" "B ∈ Dynkin Ω M"
let ?D = "λE. {Q. Q ⊆ Ω ∧ Q ∩ E ∈ Dynkin Ω M}"
have "M ⊆ ?D B"
proof
fix E assume "E ∈ M"
then have "M ⊆ ?D E" "E ∈ Dynkin Ω M"
using sets_into_space ‹Int_stable M› by (auto simp: Int_stable_def)
then have "Dynkin Ω M ⊆ ?D E"
using restricted_Dynkin_system ‹E ∈ Dynkin Ω M›
by (intro Dynkin_system.Dynkin_subset) simp_all
then have "B ∈ ?D E"
using ‹B ∈ Dynkin Ω M› by auto
then have "E ∩ B ∈ Dynkin Ω M"
by (subst Int_commute) simp
then show "E ∈ ?D B"
using sets ‹E ∈ M› by auto
qed
then have "Dynkin Ω M ⊆ ?D B"
using restricted_Dynkin_system ‹B ∈ Dynkin Ω M›
by (intro Dynkin_system.Dynkin_subset) simp_all
then show "A ∩ B ∈ Dynkin Ω M"
using ‹A ∈ Dynkin Ω M› sets_into_space by auto
qed
from sigma_algebra.sigma_sets_subset[OF this, of "M"]
have "sigma_sets (Ω) (M) ⊆ Dynkin Ω M" by auto
ultimately have "sigma_sets (Ω) (M) = Dynkin Ω M" by auto
then show ?thesis
by (auto simp: Dynkin_def)
qed

lemma (in Dynkin_system) Dynkin_idem:
"Dynkin Ω M = M"
proof -
have "Dynkin Ω M = M"
proof
show "M ⊆ Dynkin Ω M"
using Dynkin_Basic by auto
show "Dynkin Ω M ⊆ 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 ⊆ M" "M ⊆ sigma_sets Ω E"
shows "sigma_sets Ω E = M"
proof -
have "E ⊆ Pow Ω"
using E sets_into_space by force
then have *: "sigma_sets Ω E = Dynkin Ω E"
using ‹Int_stable E› by (rule sigma_eq_Dynkin)
then have "Dynkin Ω E = M"
using assms Dynkin_subset[OF E(1)] by simp
with * show ?thesis
using assms by (auto simp: Dynkin_def)
qed

subsubsection ‹Induction rule for intersection-stable generators›

text✐‹tag important› ‹The reason to introduce Dynkin-systems is the following induction rules for ‹σ›-algebras
generated by a generator closed under intersection.›

proposition sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]:
assumes "Int_stable G"
and closed: "G ⊆ Pow Ω"
and A: "A ∈ sigma_sets Ω G"
assumes basic: "⋀A. A ∈ G ⟹ P A"
and empty: "P {}"
and compl: "⋀A. A ∈ sigma_sets Ω G ⟹ P A ⟹ P (Ω - A)"
and union: "⋀A. disjoint_family A ⟹ range A ⊆ sigma_sets Ω G ⟹ (⋀i. P (A i)) ⟹ P (⋃i::nat. A i)"
shows "P A"
proof -
let ?D = "{ A ∈ sigma_sets Ω G. P A }"
interpret sigma_algebra Ω "sigma_sets Ω G"
using closed by (rule sigma_algebra_sigma_sets)
from compl[OF _ empty] closed have space: "P Ω" by simp
interpret Dynkin_system Ω ?D
by standard (auto dest: sets_into_space intro!: space compl union)
have "sigma_sets Ω G = ?D"
by (rule Dynkin_lemma) (auto simp: basic ‹Int_stable G›)
with A show ?thesis by auto
qed

subsection ‹Measure type›

definition✐‹tag important› positive :: "'a set set ⇒ ('a set ⇒ ennreal) ⇒ bool" where
"positive M μ ⟷ μ {} = 0"

definition✐‹tag important› countably_additive :: "'a set set ⇒ ('a set ⇒ ennreal) ⇒ bool" where
(∀A. range A ⊆ M ⟶ disjoint_family A ⟶ (⋃i. A i) ∈ M ⟶
(∑i. f (A i)) = f (⋃i. A i))"

definition✐‹tag important› measure_space :: "'a set ⇒ 'a set set ⇒ ('a set ⇒ ennreal) ⇒ bool" where
"measure_space Ω A μ ⟷
sigma_algebra Ω A ∧ positive A μ ∧ countably_additive A μ"

typedef✐‹tag important› 'a measure =
"{(Ω::'a set, A, μ). (∀a∈-A. μ a = 0) ∧ measure_space Ω A μ }"
proof
have "sigma_algebra UNIV {{}, UNIV}"
by (auto simp: sigma_algebra_iff2)
then show "(UNIV, {{}, UNIV}, λA. 0) ∈ {(Ω, A, μ). (∀a∈-A. μ a = 0) ∧ measure_space Ω A μ} "
by (auto simp: measure_space_def positive_def countably_additive_def)
qed

definition✐‹tag important› space :: "'a measure ⇒ 'a set" where
"space M = fst (Rep_measure M)"

definition✐‹tag important› sets :: "'a measure ⇒ 'a set set" where
"sets M = fst (snd (Rep_measure M))"

definition✐‹tag important› emeasure :: "'a measure ⇒ 'a set ⇒ ennreal" where
"emeasure M = snd (snd (Rep_measure M))"

definition✐‹tag important› measure :: "'a measure ⇒ 'a set ⇒ real" where
"measure M A = enn2real (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✐‹tag important› measure_of :: "'a set ⇒ 'a set set ⇒ ('a set ⇒ ennreal) ⇒ 'a measure"
where
"measure_of Ω A μ =
Abs_measure (Ω, if A ⊆ Pow Ω then sigma_sets Ω A else {{}, Ω},
λa. if a ∈ sigma_sets Ω A ∧ measure_space Ω (sigma_sets Ω A) μ then μ a else 0)"

abbreviation "sigma Ω A ≡ measure_of Ω A (λx. 0)"

lemma measure_space_0: "A ⊆ Pow Ω ⟹ measure_space Ω (sigma_sets Ω A) (λx. 0)"
unfolding measure_space_def
by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def)

lemma sigma_algebra_trivial: "sigma_algebra Ω {{}, Ω}"
by unfold_locales(fastforce intro: exI[where x="{{}}"] exI[where x="{Ω}"])+

lemma measure_space_0': "measure_space Ω {{}, Ω} (λx. 0)"

lemma measure_space_closed:
assumes "measure_space Ω M μ"
shows "M ⊆ Pow Ω"
proof -
interpret sigma_algebra Ω M using assms by(simp add: measure_space_def)
show ?thesis by(rule space_closed)
qed

lemma (in ring_of_sets) positive_cong_eq:
"(⋀a. a ∈ M ⟹ μ' a = μ a) ⟹ positive M μ' = positive M μ"

"(⋀a. a ∈ M ⟹ μ' a = μ a) ⟹ countably_additive M μ' = countably_additive M μ"

lemma measure_space_eq:
assumes closed: "A ⊆ Pow Ω" and eq: "⋀a. a ∈ sigma_sets Ω A ⟹ μ a = μ' a"
shows "measure_space Ω (sigma_sets Ω A) μ = measure_space Ω (sigma_sets Ω A) μ'"
proof -
interpret sigma_algebra Ω "sigma_sets Ω A" using closed by (rule sigma_algebra_sigma_sets)
from positive_cong_eq[OF eq, of "λi. i"] countably_additive_eq[OF eq, of "λi. i"] show ?thesis
by (auto simp: measure_space_def)
qed

lemma measure_of_eq:
assumes closed: "A ⊆ Pow Ω" and eq: "(⋀a. a ∈ sigma_sets Ω A ⟹ μ a = μ' a)"
shows "measure_of Ω A μ = measure_of Ω A μ'"
proof -
have "measure_space Ω (sigma_sets Ω A) μ = measure_space Ω ```