# Theory Sigma_Algebra

theory Sigma_Algebra
imports FuncSet Extended_Nonnegative_Real
(*  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.
*)

section ‹Sigma Algebra›

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

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%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%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%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%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%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%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 **: "range ?A' = range A"
using surj_from_nat
by (auto simp: image_comp [symmetric] intro!: imageI)
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 intro: from_nat_into) (metis from_nat_into_surj)
also have "… ∈ M" using assms
by (auto intro!: countable_nat_UN) (metis ‹X ≠ {}› from_nat_into set_mp)
finally show ?thesis .
qed simp

lemma (in sigma_algebra) countable_UN[intro]:
fixes A :: "'i::countable ⇒ 'a set"
assumes "AX ⊆ 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: "AX ⊆ 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: "AX ⊆ 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: "AX ⊆ 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%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: strong_SUP_cong)

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

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)"
by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def
algebra_iff_Un Un_range_binary)

subsubsection ‹Initial Sigma Algebra›

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

inductive_set%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 sigma_sets_Un:
"a ∈ sigma_sets sp A ⟹ b ∈ sigma_sets sp A ⟹ a ∪ b ∈ sigma_sets sp A"
apply (rule Union, simp add: binary_def)
done

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"
apply (cases "B = {}")
using from_nat_into [of B] range_from_nat_into [of B] sigma_sets.Union [of "from_nat_into B" X A]
apply simp
apply auto
apply (metis Sup_bot_conv(1) Union_empty ‹⟦B ≠ {}; countable B⟧ ⟹ range (from_nat_into B) = B›)
done

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)
from choice[OF this] guess f ..
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
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

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%unimportant ‹Ring generated by a semiring›

definition (in semiring_of_sets)
"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 guess Ca .. note Ca = this
from b guess Cb .. note Cb = this
show ?thesis
proof
show "disjoint (Ca ∪ Cb)"
using ‹a ∩ b = {}› Ca Cb by (auto intro!: disjoint_union)
qed (insert Ca Cb, 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) ⟹ UNION I A ∈ 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 guess Ca .. note Ca = this
from b guess Cb .. note Cb = this
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 (insert Ca Cb, 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) ⟹ INTER I A ∈ 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 assume a: "a ∈ ?R" then guess Ca .. note Ca = this
fix b assume b: "b ∈ ?R" then guess Cb .. note Cb = this

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%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: strong_SUP_cong)

subsubsection ‹Closed CDI›

definition%important closed_cdi 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%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 by auto
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%important "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%important dynkin 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 ⊆ Ω ∧ (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%important ‹The reason to introduce Dynkin-systems is the following induction rules for $\sigma$-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%important positive :: "'a set set ⇒ ('a set ⇒ ennreal) ⇒ bool" where
"positive M μ ⟷ μ {} = 0"

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

definition%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%important 'a measure = "{(Ω::'a set, A, μ). (∀a∈-A. μ a = 0) ∧ measure_space Ω A μ }"
proof%unimportant
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%important space :: "'a measure ⇒ 'a set" where
"space M = fst (Rep_measure M)"

definition%important sets :: "'a measure ⇒ 'a set set" where
"sets M = fst (snd (Rep_measure M))"

definition%important emeasure :: "'a measure ⇒ 'a set ⇒ ennreal" where
"emeasure M = snd (snd (Rep_measure M))"

definition%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%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 (sigma_sets Ω A) μ'"
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
shows space_measure_of_conv: "space (measure_of Ω A μ) = Ω" (is ?space)
and sets_measure_of_conv:
"sets (measure_of Ω A μ) = (if A ⊆ Pow Ω then sigma_sets Ω A else {{}, Ω})" (is ?sets)
and emeasure_measure_of_conv:
"emeasure (measure_of Ω A μ) =
(λB. if B ∈ sigma_sets Ω A ∧ measure_space Ω (sigma_sets Ω A) μ then μ B else 0)" (is ?emeasure)
proof -
have "?space ∧ ?sets ∧ ?emeasure"
proof(cases "measure_space Ω (sigma_sets Ω A) μ")
case True
from measure_space_closed[OF this] sigma_sets_superset_generator[of A Ω]
have "A ⊆ Pow Ω" by simp
hence "measure_space Ω (sigma_sets Ω A) μ = measure_space Ω (sigma_sets Ω A)
(λa. if a ∈ sigma_sets Ω A then μ a else 0)"
by(rule measure_space_eq) auto
with True ‹A ⊆ Pow Ω› show ?thesis
by(simp add: measure_of_def space_def sets_def emeasure_def Abs_measure_inverse)
next
case False thus ?thesis
by(cases "A ⊆ Pow Ω")(simp_all add: Abs_measure_inverse measure_of_def sets_def space_def emeasure_def measure_space_0 measure_space_0')
qed
thus ?space ?sets ?emeasure by simp_all
qed

lemma [simp]:
assumes A: "A ⊆ Pow Ω"
shows sets_measure_of: "sets (measure_of Ω A μ) = sigma_sets Ω A"
and space_measure_of: "space (measure_of Ω A μ) = Ω"
using assms

lemma space_in_measure_of[simp]: "Ω ∈ sets (measure_of Ω M μ)"
by (subst sets_measure_of_conv) (auto simp: sigma_sets_top)

lemma (in sigma_algebra) sets_measure_of_eq[simp]: "sets (measure_of Ω M μ) = M"
using space_closed by (auto intro!: sigma_sets_eq)

lemma (in sigma_algebra) space_measure_of_eq[simp]: "space (measure_of Ω M μ) = Ω"
by (rule space_measure_of_conv)

lemma measure_of_subset: "M ⊆ Pow Ω ⟹ M' ⊆ M ⟹ sets (measure_of Ω M' μ) ⊆ sets (measure_of Ω M μ')"
by (auto intro!: sigma_sets_subseteq)

lemma emeasure_sigma: "emeasure (sigma Ω A) = (λx. 0)"
unfolding measure_of_def emeasure_def
by (subst Abs_measure_inverse)
intro!: sigma_algebra_sigma_sets sigma_algebra_trivial)

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

lemma in_measure_of[intro, simp]: "M ⊆ Pow Ω ⟹ A ∈ M ⟹ A ∈ sets (measure_of Ω M μ)"
by auto

lemma space_empty_iff: "space N = {} ⟷ sets N = {{}}"
by (metis Pow_empty Sup_bot_conv(1) cSup_singleton empty_iff
sets.sigma_sets_eq sets.space_closed sigma_sets_top subset_singletonD)

subsubsection ‹Constructing simple @{typ "'a measure"}›

proposition emeasure_measure_of:
assumes M: "M = measure_of Ω A μ"
assumes ms: "A ⊆ Pow Ω" "positive (sets M) μ" "countably_additive (sets M) μ"
assumes X: "X ∈ sets M"
shows "emeasure M X = μ X"
proof -
interpret sigma_algebra Ω "sigma_sets Ω A" by (rule sigma_algebra_sigma_sets) fact
have "measure_space Ω (sigma_sets Ω A) μ"
using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets)
thus ?thesis using X ms
qed

lemma emeasure_measure_of_sigma:
assumes ms: "sigma_algebra Ω M" "positive M μ" "countably_additive M μ"
assumes A: "A ∈ M"
shows "emeasure (measure_of Ω M μ) A = μ A"
proof -
interpret sigma_algebra Ω M by fact
have "measure_space Ω (sigma_sets Ω M) μ"
using ms sigma_sets_eq by (simp add: measure_space_def)
thus ?thesis by(simp add: emeasure_measure_of_conv A)
qed

lemma measure_cases[cases type: measure]:
obtains (measure) Ω A μ where "x = Abs_measure (Ω, A, μ)" "∀a∈-A. μ a = 0" "measure_space Ω A μ"
by atomize_elim (cases x, auto)

lemma sets_le_imp_space_le: "sets A ⊆ sets B ⟹ space A ⊆ space B"
by (auto dest: sets.sets_into_space)

lemma sets_eq_imp_space_eq: "sets M = sets M' ⟹ space M = space M'"
by (auto intro!: antisym sets_le_imp_space_le)

lemma emeasure_notin_sets: "A ∉ sets M ⟹ emeasure M A = 0"
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def)

lemma emeasure_neq_0_sets: "emeasure M A ≠ 0 ⟹ A ∈ sets M"
using emeasure_notin_sets[of A M] by blast

lemma measure_notin_sets: "A ∉ sets M ⟹ measure M A = 0"
by (simp add: measure_def emeasure_notin_sets zero_ennreal.rep_eq)

lemma measure_eqI:
fixes M N :: "'a measure"
assumes "sets M = sets N" and eq: "⋀A. A ∈ sets M ⟹ emeasure M A = emeasure N A"
shows "M = N"
proof (cases M N rule: measure_cases[case_product measure_cases])
case (measure_measure Ω A μ Ω' A' μ')
interpret M: sigma_algebra Ω A using measure_measure by (auto simp: measure_space_def)
interpret N: sigma_algebra Ω' 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 AA': "A = A'" by simp
moreover from M.top N.top M.space_closed N.space_closed AA' have "Ω = Ω'" by auto
moreover { fix B have "μ B = μ' B"
proof cases
assume "B ∈ A"
with eq ‹A = sets M› have "emeasure M B = emeasure N B" by simp
with measure_measure show "μ B = μ' B"
next
assume "B ∉ A"
with ‹A = sets M› ‹A' = sets N› ‹A = A'› have "B ∉ sets M" "B ∉ sets N"
by auto
then have "emeasure M B = 0" "emeasure N B = 0"
with measure_measure show "μ B = μ' B"
qed }
then have "μ = μ'" by auto
ultimately show "M = N"
qed

lemma sigma_eqI:
assumes [simp]: "M ⊆ Pow Ω" "N ⊆ Pow Ω" "sigma_sets Ω M = sigma_sets Ω N"
shows "sigma Ω M = sigma Ω N"
by (rule measure_eqI) (simp_all add: emeasure_sigma)

subsubsection ‹Measurable functions›

definition%important measurable :: "'a measure ⇒ 'b measure ⇒ ('a ⇒ 'b) set" (infixr "→⇩M" 60) where
"measurable A B = {f ∈ space A → space B. ∀y ∈ sets B. f - y ∩ space A ∈ sets A}"

lemma measurableI:
"(⋀x. x ∈ space M ⟹ f x ∈ space N) ⟹ (⋀A. A ∈ sets N ⟹ f - A ∩ space M ∈ sets M) ⟹
f ∈ measurable M N"
by (auto simp: measurable_def)

lemma measurable_space:
"f ∈ measurable M A ⟹ x ∈ space M ⟹ f x ∈ space A"
unfolding measurable_def by auto

lemma measurable_sets:
"f ∈ measurable M A ⟹ S ∈ sets A ⟹ f - S ∩ space M ∈ sets M"
unfolding measurable_def by auto

lemma measurable_sets_Collect:
assumes f: "f ∈ measurable M N" and P: "{x∈space N. P x} ∈ sets N" shows "{x∈space M. P (f x)} ∈ sets M"
proof -
have "f - {x ∈ space N. P x} ∩ space M = {x∈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 Ω A" "A ⊆ Pow Ω"
and f: "f ∈ space M → Ω"
and ba: "⋀y. y ∈ A ⟹ (f - y) ∩ space M ∈ sets M"
shows "f ∈ measurable M N"
proof -
interpret A: sigma_algebra Ω "sigma_sets Ω 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 Ω: "Ω = space N" by force

{ fix X assume "X ∈ sigma_sets Ω A"
then have "f - X ∩ space M ∈ sets M ∧ X ⊆ Ω"
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 - Ω ∩ 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 Ω)
qed

lemma measurable_measure_of:
assumes B: "N ⊆ Pow Ω"
and f: "f ∈ space M → Ω"
and ba: "⋀y. y ∈ N ⟹ (f - y) ∩ space M ∈ sets M"
shows "f ∈ measurable M (measure_of Ω N μ)"
proof -
have "sets (measure_of Ω N μ) = sigma_sets Ω 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 ⊆ Pow Ω" "f ∈ space M → Ω"
shows "f ∈ measurable M (measure_of Ω N μ) ⟷ (∀A∈N. f - A ∩ space M ∈ 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 "⋀w. w ∈ space M ⟹ f w = g w"
shows "f ∈ measurable M M' ⟷ g ∈ measurable M M'"
unfolding measurable_def using assms
by (simp cong: vimage_inter_cong Pi_cong)

lemma measurable_cong':
assumes "⋀w. w ∈ space M =simp=> f w = g w"
shows "f ∈ measurable M M' ⟷ g ∈ measurable M M'"
unfolding measurable_def using assms
by (simp cong: vimage_inter_cong Pi_cong add: simp_implies_def)

lemma measurable_cong_strong:
"M = N ⟹ M' = N' ⟹ (⋀w. w ∈ space M ⟹ f w = g w) ⟹
f ∈ measurable M M' ⟷ g ∈ measurable N N'"
by (metis measurable_cong)

lemma measurable_compose:
assumes f: "f ∈ measurable M N" and g: "g ∈ measurable N L"
shows "(λx. g (f x)) ∈ measurable M L"
proof -
have "⋀A. (λx. g (f x)) - A ∩ space M = f - (g - A ∩ space N) ∩ 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 ∈ measurable M N ⟹ g ∈ measurable N L ⟹ g ∘ f ∈ measurable M L"
using measurable_compose[of f M N g L] by (simp add: comp_def)

lemma measurable_const:
"c ∈ space M' ⟹ (λx. c) ∈ measurable M M'"

lemma measurable_ident: "id ∈ measurable M M"

lemma measurable_id: "(λx. x) ∈ measurable M M"

lemma measurable_ident_sets:
assumes eq: "sets M = sets M'" shows "(λx. x) ∈ 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: "⋀i::nat. {x∈space M. P i x} ∈ M"
shows "(λx. LEAST j. P j x) - A ∩ space M ∈ sets M"
proof -
{ fix i have "(λx. LEAST j. P j x) - {i} ∩ space M ∈ sets M"
proof cases
assume i: "(LEAST j. False) = i"
have "(λx. LEAST j. P j x) - {i} ∩ space M =
{x∈space M. P i x} ∩ (space M - (⋃j<i. {x∈space M. P j x})) ∪ (space M - (⋃i. {x∈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) ≠ i"
then have "(λx. LEAST j. P j x) - {i} ∩ space M =
{x∈space M. P i x} ∩ (space M - (⋃j<i. {x∈space M. P j x}))"
fix x assume neq: "(LEAST j. False) ≠ (LEAST j. P j x)"
have "∃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 "(⋃i∈A. (λx. LEAST j. P j x) - {i} ∩ space M) ∈ sets M"
by (intro sets.countable_UN) auto
moreover have "(⋃i∈A. (λx. LEAST j. P j x) - {i} ∩ space M) =
(λx. LEAST j. P j x) - A ∩ space M" by auto
ultimately show ?thesis by auto
qed

lemma measurable_mono1:
"M' ⊆ Pow Ω ⟹ M ⊆ M' ⟹
measurable (measure_of Ω M μ) N ⊆ measurable (measure_of Ω M' μ') N"
using measure_of_subset[of M' Ω M] by (auto simp add: measurable_def)

subsubsection ‹Counting space›

definition%important count_space :: "'a set ⇒ 'a measure" where
"count_space Ω = measure_of Ω (Pow Ω) (λA. if finite A then of_nat (card A) else ∞)"

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

lemma measurable_count_space_eq1[simp]:
"f ∈ measurable (count_space A) M ⟷ f ∈ A → space M"
unfolding measurable_def by simp

lemma measurable_compose_countable':
assumes f: "⋀i. i ∈ I ⟹ (λx. f i x) ∈ measurable M N"
and g: "g ∈ measurable M (count_space I)" and I: "countable I"
shows "(λx. f (g x) x) ∈ measurable M N"
unfolding measurable_def
proof safe
fix x assume "x ∈ space M" then show "f (g x) x ∈ space N"
using measurable_space[OF f] g[THEN measurable_space] by auto
next
fix A assume A: "A ∈ sets N"
have "(λx. f (g x) x) - A ∩ space M = (⋃i∈I. (g - {i} ∩ space M) ∩ (f i - A ∩ space M))"
using measurable_space[OF g] by auto
also have "… ∈ sets M"
using f[THEN measurable_sets, OF _ A] g[THEN measurable_sets]
by (auto intro!: sets.countable_UN' I intro: sets.Int[OF measurable_sets measurable_sets])
finally show "(λx. f (g x) x) - A ∩ space M ∈ sets M" .
qed

lemma measurable_count_space_eq_countable:
assumes "countable A"
shows "f ∈ measurable M (count_space A) ⟷ (f ∈ space M → A ∧ (∀a∈A. f - {a} ∩ space M ∈ sets M))"
proof -
{ fix X assume "X ⊆ A" "f ∈ space M → A"
with ‹countable A› have "f - X ∩ space M = (⋃a∈X. f - {a} ∩ space M)" "countable X"
by (auto dest: countable_subset)
moreover assume "∀a∈A. f - {a} ∩ space M ∈ sets M"
ultimately have "f - X ∩ space M ∈ sets M"
using ‹X ⊆ A› by (auto intro!: sets.countable_UN' simp del: UN_simps) }
then show ?thesis
unfolding measurable_def by auto
qed

lemma measurable_count_space_eq2:
"finite A ⟹ f ∈ measurable M (count_space A) ⟷ (f ∈ space M → A ∧ (∀a∈A. f - {a} ∩ space M ∈ sets M))"
by (intro measurable_count_space_eq_countable countable_finite)

lemma measurable_count_space_eq2_countable:
fixes f :: "'a => 'c::countable"
shows "f ∈ measurable M (count_space A) ⟷ (f ∈ space M → A ∧ (∀a∈A. f - {a} ∩ space M ∈ sets M))"
by (intro measurable_count_space_eq_countable countableI_type)

lemma measurable_compose_countable:
assumes f: "⋀i::'i::countable. (λx. f i x) ∈ measurable M N" and g: "g ∈ measurable M (count_space UNIV)"
shows "(λx. f (g x) x) ∈ measurable M N"
by (rule measurable_compose_countable'[OF assms]) auto

lemma measurable_count_space_const:
"(λx. c) ∈ measurable M (count_space UNIV)"

lemma measurable_count_space:
"f ∈ measurable (count_space A) (count_space UNIV)"
by simp

lemma measurable_compose_rev:
assumes f: "f ∈ measurable L N" and g: "g ∈ measurable M L"
shows "(λx. f (g x)) ∈ measurable M N"
using measurable_compose[OF g f] .

lemma measurable_empty_iff:
"space N = {} ⟹ f ∈ measurable M N ⟷ space M = {}"
by (auto simp add: measurable_def Pi_iff)

subsubsection%unimportant ‹Extend measure›

definition "extend_measure Ω I G μ =
(if (∃μ'. (∀i∈I. μ' (G i) = μ i) ∧ measure_space Ω (sigma_sets Ω (GI)) μ') ∧ ¬ (∀i∈I. μ i = 0)
then measure_of Ω (GI) (SOME μ'. (∀i∈I. μ' (G i) = μ i) ∧ measure_space Ω (sigma_sets Ω (GI)) μ')
else measure_of Ω (GI) (λ_. 0))"

lemma space_extend_measure: "G  I ⊆ Pow Ω ⟹ space (extend_measure Ω I G μ) = Ω"
unfolding extend_measure_def by simp

lemma sets_extend_measure: "G  I ⊆ Pow Ω ⟹ sets (extend_measure Ω I G μ) = sigma_sets Ω (GI)"
unfolding extend_measure_def by simp

lemma emeasure_extend_measure:
assumes M: "M = extend_measure Ω I G μ"
and eq: "⋀i. i ∈ I ⟹ μ' (G i) = μ i"
and ms: "G  I ⊆ Pow Ω" "positive (sets M) μ'" "countably_additive (sets M) μ'"
and "i ∈ I"
shows "emeasure M (G i) = μ i"
proof cases
assume *: "(∀i∈I. μ i = 0)"
with M have M_eq: "M = measure_of Ω (GI) (λ_. 0)"
from measure_space_0[OF ms(1)] ms ‹i∈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∈I› * show ?thesis
by simp
next
define P where "P μ' ⟷ (∀i∈I. μ' (G i) = μ i) ∧ measure_space Ω (sigma_sets Ω (GI)) μ'" for μ'
assume "¬ (∀i∈I. μ i = 0)"
moreover
have "measure_space (space M) (sets M) μ'"
using ms unfolding measure_space_def by auto standard
with ms eq have "∃μ'. P μ'"
unfolding P_def
by (intro exI[of _ μ']) (auto simp add: M space_extend_measure sets_extend_measure)
ultimately have M_eq: "M = measure_of Ω (GI) (Eps P)"
by (simp add: M extend_measure_def P_def[symmetric])

from ‹∃μ'. P μ'› have P: "P (Eps P)" by (rule someI_ex)
show "emeasure M (G i) = μ i"
proof (subst emeasure_measure_of[OF M_eq])
have sets_M: "sets M = sigma_sets Ω (GI)"
using M_eq ms by (auto simp: sets_extend_measure)
then show "G i ∈ sets M" using ‹i ∈ I› by auto
show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = μ i"
using P ‹i∈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 Ω {(i, j). I i j} (λ(i, j). G i j) (λ(i, j). μ i j)"
and eq: "⋀i j. I i j ⟹ μ' (G i j) = μ i j"
and ms: "⋀i j. I i j ⟹ G i j ∈ Pow Ω" "positive (sets M) μ'" "countably_additive (sets M) μ'"
and "I i j"
shows "emeasure M (G i j) = μ 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 ‹The smallest $\sigma$-algebra regarding a function›

definition%important
"vimage_algebra X f M = sigma X {f - A ∩ X | A. A ∈ sets M}"

lemma space_vimage_algebra[simp]: "space (vimage_algebra X f M) = X"
unfolding vimage_algebra_def by (rule space_measure_of) auto

lemma sets_vimage_algebra: "sets (vimage_algebra X f M) = sigma_sets X {f - A ∩ X | A. A ∈ sets M}"
unfolding vimage_algebra_def by (rule sets_measure_of) auto

lemma sets_vimage_algebra2:
"f ∈ X → space M ⟹ sets (vimage_algebra X f M) = {f - A ∩ X | A. A ∈ sets M}"
using sigma_sets_vimage_commute[of f X "space M" "sets M"]
unfolding sets_vimage_algebra sets.sigma_sets_eq by simp

lemma sets_vimage_algebra_cong: "sets M = sets N ⟹ sets (vimage_algebra X f M) = sets (vimage_algebra X f N)"

lemma vimage_algebra_cong:
assumes "X = Y"
assumes "⋀x. x ∈ Y ⟹ f x = g x"
assumes "sets M = sets N"
shows "vimage_algebra X f M = vimage_algebra Y g N"
by (auto simp: vimage_algebra_def assms intro!: arg_cong2[where f=sigma])

lemma in_vimage_algebra: "A ∈ sets M ⟹ f - A ∩ X ∈ sets (vimage_algebra X f M)"
by (auto simp: vimage_algebra_def)

lemma sets_image_in_sets:
assumes N: "space N = X"
assumes f: "f ∈ measurable N M"
shows "sets (vimage_algebra X f M) ⊆ sets N"
unfolding sets_vimage_algebra N[symmetric]
by (rule sets.sigma_sets_subset) (auto intro!: measurable_sets f)

lemma measurable_vimage_algebra1: "f ∈ X → space M ⟹ f ∈ measurable (vimage_algebra X f M) M"
unfolding measurable_def by (auto intro: in_vimage_algebra)

lemma measurable_vimage_algebra2:
assumes g: "g ∈ space N → X" and f: "(λx. f (g x)) ∈ measurable N M"
shows "g ∈ measurable N (vimage_algebra X f M)"
unfolding vimage_algebra_def
proof (rule measurable_measure_of)
fix A assume "A ∈ {f - A ∩ X | A. A ∈ sets M}"
then obtain Y where Y: "Y ∈ sets M" and A: "A = f - Y ∩ X"
by auto
then have "g - A ∩ space N = (λx. f (g x)) - Y ∩ space N"
using g by auto
also have "… ∈ sets N"
using f Y by (rule measurable_sets)
finally show "g - A ∩ space N ∈ sets N" .
qed (insert g, auto)

lemma vimage_algebra_sigma:
assumes X: "X ⊆ Pow Ω'" and f: "f ∈ Ω → Ω'"
shows "vimage_algebra Ω f (sigma Ω' X) = sigma Ω {f - A ∩ Ω | A. A ∈ X }" (is "?V = ?S")
proof (rule measure_eqI)
have Ω: "{f - A ∩ Ω |A. A ∈ X} ⊆ Pow Ω" by auto
show "sets ?V = sets ?S"
using sigma_sets_vimage_commute[OF f, of X]
by (simp add: space_measure_of_conv f sets_vimage_algebra2 Ω X)

lemma vimage_algebra_vimage_algebra_eq:
assumes *: "f ∈ X → Y" "g ∈ Y → space M"
shows "vimage_algebra X f (vimage_algebra Y g M) = vimage_algebra X (λx. g (f x)) M"
(is "?VV = ?V")
proof (rule measure_eqI)
have "(λx. g (f x)) ∈ X → space M" "⋀A. A ∩ f - Y ∩ X = A ∩ X"
using * by auto
with * show "sets ?VV = sets ?V"
by (simp add: sets_vimage_algebra2 vimage_comp comp_def flip: ex_simps)

subsubsection ‹Restricted Space Sigma Algebra›

definition restrict_space where
"restrict_space M Ω = measure_of (Ω ∩ space M) (((∩) Ω)  sets M) (emeasure M)"

lemma space_restrict_space: "space (restrict_space M Ω) = Ω ∩ space M"
using sets.sets_into_space unfolding restrict_space_def by (subst space_measure_of) auto

lemma space_restrict_space2 [simp]: "Ω ∈ sets M ⟹ space (restrict_space M Ω) = Ω"

lemma sets_restrict_space: "sets (restrict_space M Ω) = ((∩) Ω)  sets M"
unfolding restrict_space_def
proof (subst sets_measure_of)
show "(∩) Ω  sets M ⊆ Pow (Ω ∩ space M)"
by (auto dest: sets.sets_into_space)
have "sigma_sets (Ω ∩ space M) {((λx. x) - X) ∩ (Ω ∩ space M) | X. X ∈ sets M} =
(λX. X ∩ (Ω ∩ space M))  sets M"
by (subst sigma_sets_vimage_commute[symmetric, where Ω' = "space M"])
moreover have "{((λx. x) - X) ∩ (Ω ∩ space M) | X. X ∈ sets M} = (λX. X ∩ (Ω ∩ space M))   sets M"
by auto
moreover have "(λX. X ∩ (Ω ∩ space M))   sets M = ((∩) Ω)  sets M"
by (intro image_cong) (auto dest: sets.sets_into_space)
ultimately show "sigma_sets (Ω ∩ space M) ((∩) Ω  sets M) = (∩) Ω  sets M"
by simp
qed

lemma restrict_space_sets_cong:
"A = B ⟹ sets M = sets N ⟹ sets (restrict_space M A) = sets (restrict_space N B)"
by (auto simp: sets_restrict_space)

lemma sets_restrict_space_count_space :
"sets (restrict_space (count_space A) B) = sets (count_space (A ∩ B))"

lemma sets_restrict_UNIV[simp]: "sets (restrict_space M UNIV) = sets M"

lemma sets_restrict_restrict_space:
"sets (restrict_space (restrict_space M A) B) = sets (restrict_space M (A ∩ B))"
unfolding sets_restrict_space image_comp by (intro image_cong) auto

lemma sets_restrict_space_iff:
"Ω ∩ space M ∈ sets M ⟹ A ∈ sets (restrict_space M Ω) ⟷ (A ⊆ Ω ∧ A ∈ sets M)"
proof (subst sets_restrict_space, safe)
fix A assume "Ω ∩ space M ∈ sets M" and A: "A ∈ sets M"
then have "(Ω ∩ space M) ∩ A ∈ sets M"
by rule
also have "(Ω ∩ space M) ∩ A = Ω ∩ A"
using sets.sets_into_space[OF A] by auto
finally show "Ω ∩ A ∈ sets M"
by auto
qed auto

lemma sets_restrict_space_cong: "sets M = sets N ⟹ sets (restrict_space M Ω) = sets (restrict_space N Ω)"

lemma restrict_space_eq_vimage_algebra:
"Ω ⊆ space M ⟹ sets (restrict_space M Ω) = sets (vimage_algebra Ω (λx. x) M)"
unfolding restrict_space_def
apply (subst sets_measure_of)
apply (auto simp add: image_subset_iff dest: sets.sets_into_space) []
apply (auto simp add: sets_vimage_algebra intro!: arg_cong2[where f=sigma_sets])
done

lemma sets_Collect_restrict_space_iff:
assumes "S ∈ sets M"
shows "{x∈space (restrict_space M S). P x} ∈ sets (restrict_space M S) ⟷ {x∈space M. x ∈ S ∧ P x} ∈ sets M"
proof -
have "{x∈S. P x} = {x∈space M. x ∈ S ∧ P x}"
using sets.sets_into_space[OF assms] by auto
then show ?thesis
by (subst sets_restrict_space_iff) (auto simp add: space_restrict_space assms)
qed

lemma measurable_restrict_space1:
assumes f: "f ∈ measurable M N"
shows "f ∈ measurable (restrict_space M Ω) N"
unfolding measurable_def
proof (intro CollectI conjI ballI)
show sp: "f ∈ space (restrict_space M Ω) → space N"
using measurable_space[OF f] by (auto simp: space_restrict_space)

fix A assume "A ∈ sets N"
have "f - A ∩ space (restrict_space M Ω) = (f - A ∩ space M) ∩ (Ω ∩ space M)"
by (auto simp: space_restrict_space)
also have "… ∈ sets (restrict_space M Ω)"
unfolding sets_restrict_space
using measurable_sets[OF f ‹A ∈ sets N›] by blast
finally show "f - A ∩ space (restrict_space M Ω) ∈ sets (restrict_space M Ω)" .
qed

lemma measurable_restrict_space2_iff:
"f ∈ measurable M (restrict_space N Ω) ⟷ (f ∈ measurable M N ∧ f ∈ space M → Ω)"
proof -
have "⋀A. f ∈ space M → Ω ⟹ f - Ω ∩ f - A ∩ space M = f - A ∩ space M"
by auto
then show ?thesis
by (auto simp: measurable_def space_restrict_space Pi_Int[symmetric] sets_restrict_space)
qed

lemma measurable_restrict_space2:
"f ∈ space M → Ω ⟹ f ∈ measurable M N ⟹ f ∈ measurable M (restrict_space N Ω)"

lemma measurable_piecewise_restrict:
assumes I: "countable C"
and X: "⋀Ω. Ω ∈ C ⟹ Ω ∩ space M ∈ sets M" "space M ⊆ ⋃C"
and f: "⋀Ω. Ω ∈ C ⟹ f ∈ measurable (restrict_space M Ω) N"
shows "f ∈ measurable M N"
proof (rule measurableI)
fix x assume "x ∈ space M"
with X obtain Ω where "Ω ∈ C" "x ∈ Ω" "x ∈ space M" by auto
then show "f x ∈ space N"
by (auto simp: space_restrict_space intro: f measurable_space)
next
fix A assume A: "A ∈ sets N"
have "f - A ∩ space M = (⋃Ω∈C. (f - A ∩ (Ω ∩ space M)))"
using X by (auto simp: subset_eq)
also have "… ∈ sets M"
using measurable_sets[OF f A] X I
by (intro sets.countable_UN') (auto simp: sets_restrict_space_iff space_restrict_space)
finally show "f - A ∩ space M ∈ sets M" .
qed

lemma measurable_piecewise_restrict_iff:
"countable C ⟹ (⋀Ω. Ω ∈ C ⟹ Ω ∩ space M ∈ sets M) ⟹ space M ⊆ (⋃C) ⟹
f ∈ measurable M N ⟷ (∀Ω∈C. f ∈ measurable (restrict_space M Ω) N)"
by (auto intro: measurable_piecewise_restrict measurable_restrict_space1)

lemma measurable_If_restrict_space_iff:
"{x∈space M. P x} ∈ sets M ⟹
(λx. if P x then f x else g x) ∈ measurable M N ⟷
(f ∈ measurable (restrict_space M {x. P x}) N ∧ g ∈ measurable (restrict_space M {x. ¬ P x}) N)"
by (subst measurable_piecewise_restrict_iff[where C="{{x. P x}, {x. ¬ P x}}"])
(auto simp: Int_def sets.sets_Collect_neg space_restrict_space conj_commute[of _ "x ∈ space M" for x]
cong: measurable_cong')

lemma measurable_If:
"f ∈ measurable M M' ⟹ g ∈ measurable M M' ⟹ {x∈space M. P x} ∈ sets M ⟹
(λx. if P x then f x else g x) ∈ measurable M M'"
unfolding measurable_If_restrict_space_iff by (auto intro: measurable_restrict_space1)

lemma measurable_If_set:
assumes measure: "f ∈ measurable M M'" "g ∈ measurable M M'"
assumes P: "A ∩ space M ∈ sets M"
shows "(λx. if x ∈ A then f x else g x) ∈ measurable M M'"
proof (rule measurable_If[OF measure])
have "{x ∈ space M. x ∈ A} = A ∩ space M" by auto
thus "{x ∈ space M. x ∈ A} ∈ sets M" using ‹A ∩ space M ∈ sets M› by auto
qed

lemma measurable_restrict_space_iff:
"Ω ∩ space M ∈ sets M ⟹ c ∈ space N ⟹
f ∈ measurable (restrict_space M Ω) N ⟷ (λx. if x ∈ Ω then f x else c) ∈ measurable M N"
by (subst measurable_If_restrict_space_iff)

lemma restrict_space_singleton: "{x} ∈ sets M ⟹ sets (restrict_space M {x}) = sets (count_space {x})"
using sets_restrict_space_iff[of "{x}" M]
by (auto simp add: sets_restrict_space_iff dest!: subset_singletonD)

lemma measurable_restrict_countable:
assumes X[intro]: "countable X"
assumes sets[simp]: "⋀x. x ∈ X ⟹ {x} ∈ sets M"
assumes space[simp]: "⋀x. x ∈ X ⟹ f x ∈ space N"
assumes f: "f ∈ measurable (restrict_space M (- X)) N"
shows "f ∈ measurable M N"
using f sets.countable[OF sets X]
by (intro measurable_piecewise_restrict[where M=M and C="{- X} ∪ ((λx. {x})  X)"])
(auto simp: Diff_Int_distrib2 Compl_eq_Diff_UNIV Int_insert_left sets.Diff restrict_space_singleton
simp del: sets_count_space  cong: measurable_cong_sets)

lemma measurable_discrete_difference:
assumes f: "f ∈ measurable M N"
assumes X: "countable X" "⋀x. x ∈ X ⟹ {x} ∈ sets M" "⋀x. x ∈ X ⟹ g x ∈ space N"
assumes eq: "⋀x. x ∈ space M ⟹ x ∉ X ⟹ f x = g x"
shows "g ∈ measurable M N"
by (rule measurable_restrict_countable[OF X])
(auto simp: eq[symmetric] space_restrict_space cong: measurable_cong' intro: f measurable_restrict_space1)

lemma measurable_count_space_extend: "A ⊆ B ⟹ f ∈ space M → A ⟹ f ∈ M →⇩M count_space B ⟹ f ∈ M →⇩M count_space A"
by (auto simp: measurable_def)

end
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