# Theory Binary_Product_Measure

```(*  Title:      HOL/Analysis/Binary_Product_Measure.thy
Author:     Johannes Hölzl, TU München
*)

section ‹Binary Product Measure›

theory Binary_Product_Measure
imports Nonnegative_Lebesgue_Integration
begin

lemma Pair_vimage_times[simp]: "Pair x -` (A × B) = (if x ∈ A then B else {})"
by auto

lemma rev_Pair_vimage_times[simp]: "(λx. (x, y)) -` (A × B) = (if y ∈ B then A else {})"
by auto

subsection "Binary products"

definition✐‹tag important› pair_measure (infixr "⨂⇩M" 80) where
"A ⨂⇩M B = measure_of (space A × space B)
{a × b | a b. a ∈ sets A ∧ b ∈ sets B}
(λX. ∫⇧+x. (∫⇧+y. indicator X (x,y) ∂B) ∂A)"

lemma pair_measure_closed: "{a × b | a b. a ∈ sets A ∧ b ∈ sets B} ⊆ Pow (space A × space B)"
using sets.space_closed[of A] sets.space_closed[of B] by auto

lemma space_pair_measure:
"space (A ⨂⇩M B) = space A × space B"
unfolding pair_measure_def using pair_measure_closed[of A B]
by (rule space_measure_of)

lemma SIGMA_Collect_eq: "(SIGMA x:space M. {y∈space N. P x y}) = {x∈space (M ⨂⇩M N). P (fst x) (snd x)}"
by (auto simp: space_pair_measure)

lemma sets_pair_measure:
"sets (A ⨂⇩M B) = sigma_sets (space A × space B) {a × b | a b. a ∈ sets A ∧ b ∈ sets B}"
unfolding pair_measure_def using pair_measure_closed[of A B]
by (rule sets_measure_of)

lemma sets_pair_measure_cong[measurable_cong, cong]:
"sets M1 = sets M1' ⟹ sets M2 = sets M2' ⟹ sets (M1 ⨂⇩M M2) = sets (M1' ⨂⇩M M2')"
unfolding sets_pair_measure by (simp cong: sets_eq_imp_space_eq)

lemma pair_measureI[intro, simp, measurable]:
"x ∈ sets A ⟹ y ∈ sets B ⟹ x × y ∈ sets (A ⨂⇩M B)"
by (auto simp: sets_pair_measure)

lemma sets_Pair: "{x} ∈ sets M1 ⟹ {y} ∈ sets M2 ⟹ {(x, y)} ∈ sets (M1 ⨂⇩M M2)"
using pair_measureI[of "{x}" M1 "{y}" M2] by simp

lemma measurable_pair_measureI:
assumes 1: "f ∈ space M → space M1 × space M2"
assumes 2: "⋀A B. A ∈ sets M1 ⟹ B ∈ sets M2 ⟹ f -` (A × B) ∩ space M ∈ sets M"
shows "f ∈ measurable M (M1 ⨂⇩M M2)"
unfolding pair_measure_def using 1 2
by (intro measurable_measure_of) (auto dest: sets.sets_into_space)

lemma measurable_split_replace[measurable (raw)]:
"(λx. f x (fst (g x)) (snd (g x))) ∈ measurable M N ⟹ (λx. case_prod (f x) (g x)) ∈ measurable M N"
unfolding split_beta' .

lemma measurable_Pair[measurable (raw)]:
assumes f: "f ∈ measurable M M1" and g: "g ∈ measurable M M2"
shows "(λx. (f x, g x)) ∈ measurable M (M1 ⨂⇩M M2)"
proof (rule measurable_pair_measureI)
show "(λx. (f x, g x)) ∈ space M → space M1 × space M2"
using f g by (auto simp: measurable_def)
fix A B assume *: "A ∈ sets M1" "B ∈ sets M2"
have "(λx. (f x, g x)) -` (A × B) ∩ space M = (f -` A ∩ space M) ∩ (g -` B ∩ space M)"
by auto
also have "… ∈ sets M"
by (rule sets.Int) (auto intro!: measurable_sets * f g)
finally show "(λx. (f x, g x)) -` (A × B) ∩ space M ∈ sets M" .
qed

lemma measurable_fst[intro!, simp, measurable]: "fst ∈ measurable (M1 ⨂⇩M M2) M1"
by (auto simp: fst_vimage_eq_Times space_pair_measure sets.sets_into_space Times_Int_Times
measurable_def)

lemma measurable_snd[intro!, simp, measurable]: "snd ∈ measurable (M1 ⨂⇩M M2) M2"
by (auto simp: snd_vimage_eq_Times space_pair_measure sets.sets_into_space Times_Int_Times
measurable_def)

lemma measurable_Pair_compose_split[measurable_dest]:
assumes f: "case_prod f ∈ measurable (M1 ⨂⇩M M2) N"
assumes g: "g ∈ measurable M M1" and h: "h ∈ measurable M M2"
shows "(λx. f (g x) (h x)) ∈ measurable M N"
using measurable_compose[OF measurable_Pair f, OF g h] by simp

lemma measurable_Pair1_compose[measurable_dest]:
assumes f: "(λx. (f x, g x)) ∈ measurable M (M1 ⨂⇩M M2)"
assumes [measurable]: "h ∈ measurable N M"
shows "(λx. f (h x)) ∈ measurable N M1"
using measurable_compose[OF f measurable_fst] by simp

lemma measurable_Pair2_compose[measurable_dest]:
assumes f: "(λx. (f x, g x)) ∈ measurable M (M1 ⨂⇩M M2)"
assumes [measurable]: "h ∈ measurable N M"
shows "(λx. g (h x)) ∈ measurable N M2"
using measurable_compose[OF f measurable_snd] by simp

lemma measurable_pair:
assumes "(fst ∘ f) ∈ measurable M M1" "(snd ∘ f) ∈ measurable M M2"
shows "f ∈ measurable M (M1 ⨂⇩M M2)"
using measurable_Pair[OF assms] by simp

lemma
assumes f[measurable]: "f ∈ measurable M (N ⨂⇩M P)"
shows measurable_fst': "(λx. fst (f x)) ∈ measurable M N"
and measurable_snd': "(λx. snd (f x)) ∈ measurable M P"
by simp_all

lemma
assumes f[measurable]: "f ∈ measurable M N"
shows measurable_fst'': "(λx. f (fst x)) ∈ measurable (M ⨂⇩M P) N"
and measurable_snd'': "(λx. f (snd x)) ∈ measurable (P ⨂⇩M M) N"
by simp_all

lemma sets_pair_in_sets:
assumes "⋀a b. a ∈ sets A ⟹ b ∈ sets B ⟹ a × b ∈ sets N"
shows "sets (A ⨂⇩M B) ⊆ sets N"
unfolding sets_pair_measure
by (intro sets.sigma_sets_subset') (auto intro!: assms)

lemma  sets_pair_eq_sets_fst_snd:
"sets (A ⨂⇩M B) = sets (Sup {vimage_algebra (space A × space B) fst A, vimage_algebra (space A × space B) snd B})"
(is "?P = sets (Sup {?fst, ?snd})")
proof -
{ fix a b assume ab: "a ∈ sets A" "b ∈ sets B"
then have "a × b = (fst -` a ∩ (space A × space B)) ∩ (snd -` b ∩ (space A × space B))"
by (auto dest: sets.sets_into_space)
also have "… ∈ sets (Sup {?fst, ?snd})"
apply (rule sets.Int)
apply (rule in_sets_Sup)
apply auto []
apply (rule insertI1)
apply (auto intro: ab in_vimage_algebra) []
apply (rule in_sets_Sup)
apply auto []
apply (rule insertI2)
apply (auto intro: ab in_vimage_algebra)
done
finally have "a × b ∈ sets (Sup {?fst, ?snd})" . }
moreover have "sets ?fst ⊆ sets (A ⨂⇩M B)"
by (rule sets_image_in_sets) (auto simp: space_pair_measure[symmetric])
moreover have "sets ?snd ⊆ sets (A ⨂⇩M B)"
by (rule sets_image_in_sets) (auto simp: space_pair_measure)
ultimately show ?thesis
apply (intro antisym[of "sets A" for A] sets_Sup_in_sets sets_pair_in_sets)
apply simp
apply simp
apply simp
apply (elim disjE)
apply (simp add: space_pair_measure)
apply (simp add: space_pair_measure)
apply (auto simp add: space_pair_measure)
done
qed

lemma measurable_pair_iff:
"f ∈ measurable M (M1 ⨂⇩M M2) ⟷ (fst ∘ f) ∈ measurable M M1 ∧ (snd ∘ f) ∈ measurable M M2"
by (auto intro: measurable_pair[of f M M1 M2])

lemma  measurable_split_conv:
"(λ(x, y). f x y) ∈ measurable A B ⟷ (λx. f (fst x) (snd x)) ∈ measurable A B"
by (intro arg_cong2[where f="(∈)"]) auto

lemma measurable_pair_swap': "(λ(x,y). (y, x)) ∈ measurable (M1 ⨂⇩M M2) (M2 ⨂⇩M M1)"
by (auto intro!: measurable_Pair simp: measurable_split_conv)

lemma  measurable_pair_swap:
assumes f: "f ∈ measurable (M1 ⨂⇩M M2) M" shows "(λ(x,y). f (y, x)) ∈ measurable (M2 ⨂⇩M M1) M"
using measurable_comp[OF measurable_Pair f] by (auto simp: measurable_split_conv comp_def)

lemma measurable_pair_swap_iff:
"f ∈ measurable (M2 ⨂⇩M M1) M ⟷ (λ(x,y). f (y,x)) ∈ measurable (M1 ⨂⇩M M2) M"
by (auto dest: measurable_pair_swap)

lemma measurable_Pair1': "x ∈ space M1 ⟹ Pair x ∈ measurable M2 (M1 ⨂⇩M M2)"
by simp

lemma sets_Pair1[measurable (raw)]:
assumes A: "A ∈ sets (M1 ⨂⇩M M2)" shows "Pair x -` A ∈ sets M2"
proof -
have "Pair x -` A = (if x ∈ space M1 then Pair x -` A ∩ space M2 else {})"
using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
also have "… ∈ sets M2"
using A by (auto simp add: measurable_Pair1' intro!: measurable_sets split: if_split_asm)
finally show ?thesis .
qed

lemma measurable_Pair2': "y ∈ space M2 ⟹ (λx. (x, y)) ∈ measurable M1 (M1 ⨂⇩M M2)"
by (auto intro!: measurable_Pair)

lemma sets_Pair2: assumes A: "A ∈ sets (M1 ⨂⇩M M2)" shows "(λx. (x, y)) -` A ∈ sets M1"
proof -
have "(λx. (x, y)) -` A = (if y ∈ space M2 then (λx. (x, y)) -` A ∩ space M1 else {})"
using A[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
also have "… ∈ sets M1"
using A by (auto simp add: measurable_Pair2' intro!: measurable_sets split: if_split_asm)
finally show ?thesis .
qed

lemma measurable_Pair2:
assumes f: "f ∈ measurable (M1 ⨂⇩M M2) M" and x: "x ∈ space M1"
shows "(λy. f (x, y)) ∈ measurable M2 M"
using measurable_comp[OF measurable_Pair1' f, OF x]
by (simp add: comp_def)

lemma measurable_Pair1:
assumes f: "f ∈ measurable (M1 ⨂⇩M M2) M" and y: "y ∈ space M2"
shows "(λx. f (x, y)) ∈ measurable M1 M"
using measurable_comp[OF measurable_Pair2' f, OF y]
by (simp add: comp_def)

lemma Int_stable_pair_measure_generator: "Int_stable {a × b | a b. a ∈ sets A ∧ b ∈ sets B}"
unfolding Int_stable_def
by safe (auto simp add: Times_Int_Times)

lemma (in finite_measure) finite_measure_cut_measurable:
assumes [measurable]: "Q ∈ sets (N ⨂⇩M M)"
shows "(λx. emeasure M (Pair x -` Q)) ∈ borel_measurable N"
(is "?s Q ∈ _")
using Int_stable_pair_measure_generator pair_measure_closed assms
unfolding sets_pair_measure
proof (induct rule: sigma_sets_induct_disjoint)
case (compl A)
with sets.sets_into_space have "⋀x. emeasure M (Pair x -` ((space N × space M) - A)) =
(if x ∈ space N then emeasure M (space M) - ?s A x else 0)"
unfolding sets_pair_measure[symmetric]
by (auto intro!: emeasure_compl simp: vimage_Diff sets_Pair1)
with compl sets.top show ?case
by (auto intro!: measurable_If simp: space_pair_measure)
next
case (union F)
then have "⋀x. emeasure M (Pair x -` (⋃i. F i)) = (∑i. ?s (F i) x)"
by (simp add: suminf_emeasure disjoint_family_on_vimageI subset_eq vimage_UN sets_pair_measure[symmetric])
with union show ?case
unfolding sets_pair_measure[symmetric] by simp
qed (auto simp add: if_distrib Int_def[symmetric] intro!: measurable_If)

lemma (in sigma_finite_measure) measurable_emeasure_Pair:
assumes Q: "Q ∈ sets (N ⨂⇩M M)" shows "(λx. emeasure M (Pair x -` Q)) ∈ borel_measurable N" (is "?s Q ∈ _")
proof -
obtain F :: "nat ⇒ 'a set" where F:
"range F ⊆ sets M"
"⋃ (range F) = space M"
"⋀i. emeasure M (F i) ≠ ∞"
"disjoint_family F" by (blast intro: sigma_finite_disjoint)
then have F_sets: "⋀i. F i ∈ sets M" by auto
let ?C = "λx i. F i ∩ Pair x -` Q"
{ fix i
have [simp]: "space N × F i ∩ space N × space M = space N × F i"
using F sets.sets_into_space by auto
let ?R = "density M (indicator (F i))"
have "finite_measure ?R"
using F by (intro finite_measureI) (auto simp: emeasure_restricted subset_eq)
then have "(λx. emeasure ?R (Pair x -` (space N × space ?R ∩ Q))) ∈ borel_measurable N"
by (rule finite_measure.finite_measure_cut_measurable) (auto intro: Q)
moreover have "⋀x. emeasure ?R (Pair x -` (space N × space ?R ∩ Q))
= emeasure M (F i ∩ Pair x -` (space N × space ?R ∩ Q))"
using Q F_sets by (intro emeasure_restricted) (auto intro: sets_Pair1)
moreover have "⋀x. F i ∩ Pair x -` (space N × space ?R ∩ Q) = ?C x i"
using sets.sets_into_space[OF Q] by (auto simp: space_pair_measure)
ultimately have "(λx. emeasure M (?C x i)) ∈ borel_measurable N"
by simp }
moreover
{ fix x
have "(∑i. emeasure M (?C x i)) = emeasure M (⋃i. ?C x i)"
proof (intro suminf_emeasure)
show "range (?C x) ⊆ sets M"
using F ‹Q ∈ sets (N ⨂⇩M M)› by (auto intro!: sets_Pair1)
have "disjoint_family F" using F by auto
show "disjoint_family (?C x)"
by (rule disjoint_family_on_bisimulation[OF ‹disjoint_family F›]) auto
qed
also have "(⋃i. ?C x i) = Pair x -` Q"
using F sets.sets_into_space[OF ‹Q ∈ sets (N ⨂⇩M M)›]
by (auto simp: space_pair_measure)
finally have "emeasure M (Pair x -` Q) = (∑i. emeasure M (?C x i))"
by simp }
ultimately show ?thesis using ‹Q ∈ sets (N ⨂⇩M M)› F_sets
by auto
qed

lemma (in sigma_finite_measure) measurable_emeasure[measurable (raw)]:
assumes space: "⋀x. x ∈ space N ⟹ A x ⊆ space M"
assumes A: "{x∈space (N ⨂⇩M M). snd x ∈ A (fst x)} ∈ sets (N ⨂⇩M M)"
shows "(λx. emeasure M (A x)) ∈ borel_measurable N"
proof -
from space have "⋀x. x ∈ space N ⟹ Pair x -` {x ∈ space (N ⨂⇩M M). snd x ∈ A (fst x)} = A x"
by (auto simp: space_pair_measure)
with measurable_emeasure_Pair[OF A] show ?thesis
by (auto cong: measurable_cong)
qed

lemma (in sigma_finite_measure) emeasure_pair_measure:
assumes "X ∈ sets (N ⨂⇩M M)"
shows "emeasure (N ⨂⇩M M) X = (∫⇧+ x. ∫⇧+ y. indicator X (x, y) ∂M ∂N)" (is "_ = ?μ X")
proof (rule emeasure_measure_of[OF pair_measure_def])
show "positive (sets (N ⨂⇩M M)) ?μ"
by (auto simp: positive_def)
have eq[simp]: "⋀A x y. indicator A (x, y) = indicator (Pair x -` A) y"
by (auto simp: indicator_def)
show "countably_additive (sets (N ⨂⇩M M)) ?μ"
proof (rule countably_additiveI)
fix F :: "nat ⇒ ('b × 'a) set" assume F: "range F ⊆ sets (N ⨂⇩M M)" "disjoint_family F"
from F have *: "⋀i. F i ∈ sets (N ⨂⇩M M)" by auto
moreover have "⋀x. disjoint_family (λi. Pair x -` F i)"
by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
moreover have "⋀x. range (λi. Pair x -` F i) ⊆ sets M"
using F by (auto simp: sets_Pair1)
ultimately show "(∑n. ?μ (F n)) = ?μ (⋃i. F i)"
by (auto simp add: nn_integral_suminf[symmetric] vimage_UN suminf_emeasure
intro!: nn_integral_cong nn_integral_indicator[symmetric])
qed
show "{a × b |a b. a ∈ sets N ∧ b ∈ sets M} ⊆ Pow (space N × space M)"
using sets.space_closed[of N] sets.space_closed[of M] by auto
qed fact

lemma (in sigma_finite_measure) emeasure_pair_measure_alt:
assumes X: "X ∈ sets (N ⨂⇩M M)"
shows "emeasure (N  ⨂⇩M M) X = (∫⇧+x. emeasure M (Pair x -` X) ∂N)"
proof -
have [simp]: "⋀x y. indicator X (x, y) = indicator (Pair x -` X) y"
by (auto simp: indicator_def)
show ?thesis
using X by (auto intro!: nn_integral_cong simp: emeasure_pair_measure sets_Pair1)
qed

proposition (in sigma_finite_measure) emeasure_pair_measure_Times:
assumes A: "A ∈ sets N" and B: "B ∈ sets M"
shows "emeasure (N ⨂⇩M M) (A × B) = emeasure N A * emeasure M B"
proof -
have "emeasure (N ⨂⇩M M) (A × B) = (∫⇧+x. emeasure M B * indicator A x ∂N)"
using A B by (auto intro!: nn_integral_cong simp: emeasure_pair_measure_alt)
also have "… = emeasure M B * emeasure N A"
using A by (simp add: nn_integral_cmult_indicator)
finally show ?thesis
by (simp add: ac_simps)
qed

lemma (in sigma_finite_measure) times_in_null_sets1 [intro]:
assumes "A ∈ null_sets N" "B ∈ sets M"
shows   "A × B ∈ null_sets (N ⨂⇩M M)"
using assms by (simp add: null_sets_def emeasure_pair_measure_Times)

lemma (in sigma_finite_measure) times_in_null_sets2 [intro]:
assumes "A ∈ sets N" "B ∈ null_sets M"
shows   "A × B ∈ null_sets (N ⨂⇩M M)"
using assms by (simp add: null_sets_def emeasure_pair_measure_Times)

subsection ‹Binary products of ‹σ›-finite emeasure spaces›

locale✐‹tag unimportant› pair_sigma_finite = M1?: sigma_finite_measure M1 + M2?: sigma_finite_measure M2
for M1 :: "'a measure" and M2 :: "'b measure"

lemma (in pair_sigma_finite) measurable_emeasure_Pair1:
"Q ∈ sets (M1 ⨂⇩M M2) ⟹ (λx. emeasure M2 (Pair x -` Q)) ∈ borel_measurable M1"
using M2.measurable_emeasure_Pair .

lemma (in pair_sigma_finite) measurable_emeasure_Pair2:
assumes Q: "Q ∈ sets (M1 ⨂⇩M M2)" shows "(λy. emeasure M1 ((λx. (x, y)) -` Q)) ∈ borel_measurable M2"
proof -
have "(λ(x, y). (y, x)) -` Q ∩ space (M2 ⨂⇩M M1) ∈ sets (M2 ⨂⇩M M1)"
using Q measurable_pair_swap' by (auto intro: measurable_sets)
note M1.measurable_emeasure_Pair[OF this]
moreover have "⋀y. Pair y -` ((λ(x, y). (y, x)) -` Q ∩ space (M2 ⨂⇩M M1)) = (λx. (x, y)) -` Q"
using Q[THEN sets.sets_into_space] by (auto simp: space_pair_measure)
ultimately show ?thesis by simp
qed

proposition (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
defines "E ≡ {A × B | A B. A ∈ sets M1 ∧ B ∈ sets M2}"
shows "∃F::nat ⇒ ('a × 'b) set. range F ⊆ E ∧ incseq F ∧ (⋃i. F i) = space M1 × space M2 ∧
(∀i. emeasure (M1 ⨂⇩M M2) (F i) ≠ ∞)"
proof -
obtain F1 where F1: "range F1 ⊆ sets M1"
"⋃ (range F1) = space M1"
"⋀i. emeasure M1 (F1 i) ≠ ∞"
"incseq F1"
by (rule M1.sigma_finite_incseq) blast
obtain F2 where F2: "range F2 ⊆ sets M2"
"⋃ (range F2) = space M2"
"⋀i. emeasure M2 (F2 i) ≠ ∞"
"incseq F2"
by (rule M2.sigma_finite_incseq) blast
from F1 F2 have space: "space M1 = (⋃i. F1 i)" "space M2 = (⋃i. F2 i)" by auto
let ?F = "λi. F1 i × F2 i"
show ?thesis
proof (intro exI[of _ ?F] conjI allI)
show "range ?F ⊆ E" using F1 F2 by (auto simp: E_def) (metis range_subsetD)
next
have "space M1 × space M2 ⊆ (⋃i. ?F i)"
proof (intro subsetI)
fix x assume "x ∈ space M1 × space M2"
then obtain i j where "fst x ∈ F1 i" "snd x ∈ F2 j"
by (auto simp: space)
then have "fst x ∈ F1 (max i j)" "snd x ∈ F2 (max j i)"
using ‹incseq F1› ‹incseq F2› unfolding incseq_def
by (force split: split_max)+
then have "(fst x, snd x) ∈ F1 (max i j) × F2 (max i j)"
by (intro SigmaI) (auto simp add: max.commute)
then show "x ∈ (⋃i. ?F i)" by auto
qed
then show "(⋃i. ?F i) = space M1 × space M2"
using space by (auto simp: space)
next
fix i show "incseq (λi. F1 i × F2 i)"
using ‹incseq F1› ‹incseq F2› unfolding incseq_Suc_iff by auto
next
fix i
from F1 F2 have "F1 i ∈ sets M1" "F2 i ∈ sets M2" by auto
with F1 F2 show "emeasure (M1 ⨂⇩M M2) (F1 i × F2 i) ≠ ∞"
by (auto simp add: emeasure_pair_measure_Times ennreal_mult_eq_top_iff)
qed
qed

sublocale✐‹tag unimportant› pair_sigma_finite ⊆ P?: sigma_finite_measure "M1 ⨂⇩M M2"
proof
obtain F1 :: "'a set set" and F2 :: "'b set set" where
"countable F1 ∧ F1 ⊆ sets M1 ∧ ⋃ F1 = space M1 ∧ (∀a∈F1. emeasure M1 a ≠ ∞)"
"countable F2 ∧ F2 ⊆ sets M2 ∧ ⋃ F2 = space M2 ∧ (∀a∈F2. emeasure M2 a ≠ ∞)"
using M1.sigma_finite_countable M2.sigma_finite_countable by auto
then show
"∃A. countable A ∧ A ⊆ sets (M1 ⨂⇩M M2) ∧ ⋃A = space (M1 ⨂⇩M M2) ∧ (∀a∈A. emeasure (M1 ⨂⇩M M2) a ≠ ∞)"
by (intro exI[of _ "(λ(a, b). a × b) ` (F1 × F2)"] conjI)
(auto simp: M2.emeasure_pair_measure_Times space_pair_measure set_eq_iff subset_eq ennreal_mult_eq_top_iff)
qed

lemma sigma_finite_pair_measure:
assumes A: "sigma_finite_measure A" and B: "sigma_finite_measure B"
shows "sigma_finite_measure (A ⨂⇩M B)"
proof -
interpret A: sigma_finite_measure A by fact
interpret B: sigma_finite_measure B by fact
interpret AB: pair_sigma_finite A  B ..
show ?thesis ..
qed

lemma sets_pair_swap:
assumes "A ∈ sets (M1 ⨂⇩M M2)"
shows "(λ(x, y). (y, x)) -` A ∩ space (M2 ⨂⇩M M1) ∈ sets (M2 ⨂⇩M M1)"
using measurable_pair_swap' assms by (rule measurable_sets)

lemma (in pair_sigma_finite) distr_pair_swap:
"M1 ⨂⇩M M2 = distr (M2 ⨂⇩M M1) (M1 ⨂⇩M M2) (λ(x, y). (y, x))" (is "?P = ?D")
proof -
let ?E = "{a × b |a b. a ∈ sets M1 ∧ b ∈ sets M2}"
obtain F :: "nat ⇒ ('a × 'b) set" where F: "range F ⊆ ?E"
"incseq F" "⋃ (range F) = space M1 × space M2" "∀i. emeasure (M1 ⨂⇩M M2) (F i) ≠ ∞"
using sigma_finite_up_in_pair_measure_generator by auto
show ?thesis
proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
show "?E ⊆ Pow (space ?P)"
using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)
show "sets ?P = sigma_sets (space ?P) ?E"
by (simp add: sets_pair_measure space_pair_measure)
then show "sets ?D = sigma_sets (space ?P) ?E"
by simp
from F show "range F ⊆ ?E" "(⋃i. F i) = space ?P" "⋀i. emeasure ?P (F i) ≠ ∞"
by (auto simp: space_pair_measure)
next
fix X assume "X ∈ ?E"
then obtain A B where X[simp]: "X = A × B" and A: "A ∈ sets M1" and B: "B ∈ sets M2" by auto
have "(λ(y, x). (x, y)) -` X ∩ space (M2 ⨂⇩M M1) = B × A"
using sets.sets_into_space[OF A] sets.sets_into_space[OF B] by (auto simp: space_pair_measure)
with A B show "emeasure (M1 ⨂⇩M M2) X = emeasure ?D X"
by (simp add: M2.emeasure_pair_measure_Times M1.emeasure_pair_measure_Times emeasure_distr
measurable_pair_swap' ac_simps)
qed
qed

lemma (in pair_sigma_finite) emeasure_pair_measure_alt2:
assumes A: "A ∈ sets (M1 ⨂⇩M M2)"
shows "emeasure (M1 ⨂⇩M M2) A = (∫⇧+y. emeasure M1 ((λx. (x, y)) -` A) ∂M2)"
(is "_ = ?ν A")
proof -
have [simp]: "⋀y. (Pair y -` ((λ(x, y). (y, x)) -` A ∩ space (M2 ⨂⇩M M1))) = (λx. (x, y)) -` A"
using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)
show ?thesis using A
by (subst distr_pair_swap)
(simp_all del: vimage_Int add: measurable_sets[OF measurable_pair_swap']
M1.emeasure_pair_measure_alt emeasure_distr[OF measurable_pair_swap' A])
qed

lemma (in pair_sigma_finite) AE_pair:
assumes "AE x in (M1 ⨂⇩M M2). Q x"
shows "AE x in M1. (AE y in M2. Q (x, y))"
proof -
obtain N where N: "N ∈ sets (M1 ⨂⇩M M2)" "emeasure (M1 ⨂⇩M M2) N = 0" "{x∈space (M1 ⨂⇩M M2). ¬ Q x} ⊆ N"
using assms unfolding eventually_ae_filter by auto
show ?thesis
proof (rule AE_I)
from N measurable_emeasure_Pair1[OF ‹N ∈ sets (M1 ⨂⇩M M2)›]
show "emeasure M1 {x∈space M1. emeasure M2 (Pair x -` N) ≠ 0} = 0"
by (auto simp: M2.emeasure_pair_measure_alt nn_integral_0_iff)
show "{x ∈ space M1. emeasure M2 (Pair x -` N) ≠ 0} ∈ sets M1"
by (intro borel_measurable_eq measurable_emeasure_Pair1 N sets.sets_Collect_neg N) simp
{ fix x assume "x ∈ space M1" "emeasure M2 (Pair x -` N) = 0"
have "AE y in M2. Q (x, y)"
proof (rule AE_I)
show "emeasure M2 (Pair x -` N) = 0" by fact
show "Pair x -` N ∈ sets M2" using N(1) by (rule sets_Pair1)
show "{y ∈ space M2. ¬ Q (x, y)} ⊆ Pair x -` N"
using N ‹x ∈ space M1› unfolding space_pair_measure by auto
qed }
then show "{x ∈ space M1. ¬ (AE y in M2. Q (x, y))} ⊆ {x ∈ space M1. emeasure M2 (Pair x -` N) ≠ 0}"
by auto
qed
qed

lemma (in pair_sigma_finite) AE_pair_measure:
assumes "{x∈space (M1 ⨂⇩M M2). P x} ∈ sets (M1 ⨂⇩M M2)"
assumes ae: "AE x in M1. AE y in M2. P (x, y)"
shows "AE x in M1 ⨂⇩M M2. P x"
proof (subst AE_iff_measurable[OF _ refl])
show "{x∈space (M1 ⨂⇩M M2). ¬ P x} ∈ sets (M1 ⨂⇩M M2)"
by (rule sets.sets_Collect) fact
then have "emeasure (M1 ⨂⇩M M2) {x ∈ space (M1 ⨂⇩M M2). ¬ P x} =
(∫⇧+ x. ∫⇧+ y. indicator {x ∈ space (M1 ⨂⇩M M2). ¬ P x} (x, y) ∂M2 ∂M1)"
by (simp add: M2.emeasure_pair_measure)
also have "… = (∫⇧+ x. ∫⇧+ y. 0 ∂M2 ∂M1)"
using ae
apply (safe intro!: nn_integral_cong_AE)
apply (intro AE_I2)
apply (safe intro!: nn_integral_cong_AE)
apply auto
done
finally show "emeasure (M1 ⨂⇩M M2) {x ∈ space (M1 ⨂⇩M M2). ¬ P x} = 0" by simp
qed

lemma (in pair_sigma_finite) AE_pair_iff:
"{x∈space (M1 ⨂⇩M M2). P (fst x) (snd x)} ∈ sets (M1 ⨂⇩M M2) ⟹
(AE x in M1. AE y in M2. P x y) ⟷ (AE x in (M1 ⨂⇩M M2). P (fst x) (snd x))"
using AE_pair[of "λx. P (fst x) (snd x)"] AE_pair_measure[of "λx. P (fst x) (snd x)"] by auto

lemma (in pair_sigma_finite) AE_commute:
assumes P: "{x∈space (M1 ⨂⇩M M2). P (fst x) (snd x)} ∈ sets (M1 ⨂⇩M M2)"
shows "(AE x in M1. AE y in M2. P x y) ⟷ (AE y in M2. AE x in M1. P x y)"
proof -
interpret Q: pair_sigma_finite M2 M1 ..
have [simp]: "⋀x. (fst (case x of (x, y) ⇒ (y, x))) = snd x" "⋀x. (snd (case x of (x, y) ⇒ (y, x))) = fst x"
by auto
have "{x ∈ space (M2 ⨂⇩M M1). P (snd x) (fst x)} =
(λ(x, y). (y, x)) -` {x ∈ space (M1 ⨂⇩M M2). P (fst x) (snd x)} ∩ space (M2 ⨂⇩M M1)"
by (auto simp: space_pair_measure)
also have "… ∈ sets (M2 ⨂⇩M M1)"
by (intro sets_pair_swap P)
finally show ?thesis
apply (subst AE_pair_iff[OF P])
apply (subst distr_pair_swap)
apply (subst AE_distr_iff[OF measurable_pair_swap' P])
apply (subst Q.AE_pair_iff)
apply simp_all
done
qed

subsection "Fubinis theorem"

lemma measurable_compose_Pair1:
"x ∈ space M1 ⟹ g ∈ measurable (M1 ⨂⇩M M2) L ⟹ (λy. g (x, y)) ∈ measurable M2 L"
by simp

lemma (in sigma_finite_measure) borel_measurable_nn_integral_fst:
assumes f: "f ∈ borel_measurable (M1 ⨂⇩M M)"
shows "(λx. ∫⇧+ y. f (x, y) ∂M) ∈ borel_measurable M1"
using f proof induct
case (cong u v)
then have "⋀w x. w ∈ space M1 ⟹ x ∈ space M ⟹ u (w, x) = v (w, x)"
by (auto simp: space_pair_measure)
show ?case
apply (subst measurable_cong)
apply (rule nn_integral_cong)
apply fact+
done
next
case (set Q)
have [simp]: "⋀x y. indicator Q (x, y) = indicator (Pair x -` Q) y"
by (auto simp: indicator_def)
have "⋀x. x ∈ space M1 ⟹ emeasure M (Pair x -` Q) = ∫⇧+ y. indicator Q (x, y) ∂M"
by (simp add: sets_Pair1[OF set])
from this measurable_emeasure_Pair[OF set] show ?case
by (rule measurable_cong[THEN iffD1])
qed (simp_all add: nn_integral_add nn_integral_cmult measurable_compose_Pair1
nn_integral_monotone_convergence_SUP incseq_def le_fun_def image_comp
cong: measurable_cong)

lemma (in sigma_finite_measure) nn_integral_fst:
assumes f: "f ∈ borel_measurable (M1 ⨂⇩M M)"
shows "(∫⇧+ x. ∫⇧+ y. f (x, y) ∂M ∂M1) = integral⇧N (M1 ⨂⇩M M) f" (is "?I f = _")
using f proof induct
case (cong u v)
then have "?I u = ?I v"
by (intro nn_integral_cong) (auto simp: space_pair_measure)
with cong show ?case
by (simp cong: nn_integral_cong)
qed (simp_all add: emeasure_pair_measure nn_integral_cmult nn_integral_add
nn_integral_monotone_convergence_SUP measurable_compose_Pair1
borel_measurable_nn_integral_fst nn_integral_mono incseq_def le_fun_def image_comp
cong: nn_integral_cong)

lemma (in sigma_finite_measure) borel_measurable_nn_integral[measurable (raw)]:
"case_prod f ∈ borel_measurable (N ⨂⇩M M) ⟹ (λx. ∫⇧+ y. f x y ∂M) ∈ borel_measurable N"
using borel_measurable_nn_integral_fst[of "case_prod f" N] by simp

proposition (in pair_sigma_finite) nn_integral_snd:
assumes f[measurable]: "f ∈ borel_measurable (M1 ⨂⇩M M2)"
shows "(∫⇧+ y. (∫⇧+ x. f (x, y) ∂M1) ∂M2) = integral⇧N (M1 ⨂⇩M M2) f"
proof -
note measurable_pair_swap[OF f]
from M1.nn_integral_fst[OF this]
have "(∫⇧+ y. (∫⇧+ x. f (x, y) ∂M1) ∂M2) = (∫⇧+ (x, y). f (y, x) ∂(M2 ⨂⇩M M1))"
by simp
also have "(∫⇧+ (x, y). f (y, x) ∂(M2 ⨂⇩M M1)) = integral⇧N (M1 ⨂⇩M M2) f"
by (subst distr_pair_swap) (auto simp add: nn_integral_distr intro!: nn_integral_cong)
finally show ?thesis .
qed

theorem (in pair_sigma_finite) Fubini:
assumes f: "f ∈ borel_measurable (M1 ⨂⇩M M2)"
shows "(∫⇧+ y. (∫⇧+ x. f (x, y) ∂M1) ∂M2) = (∫⇧+ x. (∫⇧+ y. f (x, y) ∂M2) ∂M1)"
unfolding nn_integral_snd[OF assms] M2.nn_integral_fst[OF assms] ..

theorem (in pair_sigma_finite) Fubini':
assumes f: "case_prod f ∈ borel_measurable (M1 ⨂⇩M M2)"
shows "(∫⇧+ y. (∫⇧+ x. f x y ∂M1) ∂M2) = (∫⇧+ x. (∫⇧+ y. f x y ∂M2) ∂M1)"
using Fubini[OF f] by simp

subsection ‹Products on counting spaces, densities and distributions›

proposition sigma_prod:
assumes X_cover: "∃E⊆A. countable E ∧ X = ⋃E" and A: "A ⊆ Pow X"
assumes Y_cover: "∃E⊆B. countable E ∧ Y = ⋃E" and B: "B ⊆ Pow Y"
shows "sigma X A ⨂⇩M sigma Y B = sigma (X × Y) {a × b | a b. a ∈ A ∧ b ∈ B}"
(is "?P = ?S")
proof (rule measure_eqI)
have [simp]: "snd ∈ X × Y → Y" "fst ∈ X × Y → X"
by auto
let ?XY = "{{fst -` a ∩ X × Y | a. a ∈ A}, {snd -` b ∩ X × Y | b. b ∈ B}}"
have "sets ?P = sets (SUP xy∈?XY. sigma (X × Y) xy)"
by (simp add: vimage_algebra_sigma sets_pair_eq_sets_fst_snd A B)
also have "… = sets (sigma (X × Y) (⋃?XY))"
by (intro Sup_sigma arg_cong[where f=sets]) auto
also have "… = sets ?S"
proof (intro arg_cong[where f=sets] sigma_eqI sigma_sets_eqI)
show "⋃?XY ⊆ Pow (X × Y)" "{a × b |a b. a ∈ A ∧ b ∈ B} ⊆ Pow (X × Y)"
using A B by auto
next
interpret XY: sigma_algebra "X × Y" "sigma_sets (X × Y) {a × b |a b. a ∈ A ∧ b ∈ B}"
using A B by (intro sigma_algebra_sigma_sets) auto
fix Z assume "Z ∈ ⋃?XY"
then show "Z ∈ sigma_sets (X × Y) {a × b |a b. a ∈ A ∧ b ∈ B}"
proof safe
fix a assume "a ∈ A"
from Y_cover obtain E where E: "E ⊆ B" "countable E" and "Y = ⋃E"
by auto
with ‹a ∈ A› A have eq: "fst -` a ∩ X × Y = (⋃e∈E. a × e)"
by auto
show "fst -` a ∩ X × Y ∈ sigma_sets (X × Y) {a × b |a b. a ∈ A ∧ b ∈ B}"
using ‹a ∈ A› E unfolding eq by (auto intro!: XY.countable_UN')
next
fix b assume "b ∈ B"
from X_cover obtain E where E: "E ⊆ A" "countable E" and "X = ⋃E"
by auto
with ‹b ∈ B› B have eq: "snd -` b ∩ X × Y = (⋃e∈E. e × b)"
by auto
show "snd -` b ∩ X × Y ∈ sigma_sets (X × Y) {a × b |a b. a ∈ A ∧ b ∈ B}"
using ‹b ∈ B› E unfolding eq by (auto intro!: XY.countable_UN')
qed
next
fix Z assume "Z ∈ {a × b |a b. a ∈ A ∧ b ∈ B}"
then obtain a b where "Z = a × b" and ab: "a ∈ A" "b ∈ B"
by auto
then have Z: "Z = (fst -` a ∩ X × Y) ∩ (snd -` b ∩ X × Y)"
using A B by auto
interpret XY: sigma_algebra "X × Y" "sigma_sets (X × Y) (⋃?XY)"
by (intro sigma_algebra_sigma_sets) auto
show "Z ∈ sigma_sets (X × Y) (⋃?XY)"
unfolding Z by (rule XY.Int) (blast intro: ab)+
qed
finally show "sets ?P = sets ?S" .
next
interpret finite_measure "sigma X A" for X A
proof qed (simp add: emeasure_sigma)
fix A assume "A ∈ sets ?P" then show "emeasure ?P A = emeasure ?S A"
by (simp add: emeasure_pair_measure_alt emeasure_sigma)
qed

lemma sigma_sets_pair_measure_generator_finite:
assumes "finite A" and "finite B"
shows "sigma_sets (A × B) { a × b | a b. a ⊆ A ∧ b ⊆ B} = Pow (A × B)"
(is "sigma_sets ?prod ?sets = _")
proof safe
have fin: "finite (A × B)" using assms by (rule finite_cartesian_product)
fix x assume subset: "x ⊆ A × B"
hence "finite x" using fin by (rule finite_subset)
from this subset show "x ∈ sigma_sets ?prod ?sets"
proof (induct x)
case empty show ?case by (rule sigma_sets.Empty)
next
case (insert a x)
hence "{a} ∈ sigma_sets ?prod ?sets" by auto
moreover have "x ∈ sigma_sets ?prod ?sets" using insert by auto
ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
qed
next
fix x a b
assume "x ∈ sigma_sets ?prod ?sets" and "(a, b) ∈ x"
from sigma_sets_into_sp[OF _ this(1)] this(2)
show "a ∈ A" and "b ∈ B" by auto
qed

proposition  sets_pair_eq:
assumes Ea: "Ea ⊆ Pow (space A)" "sets A = sigma_sets (space A) Ea"
and Ca: "countable Ca" "Ca ⊆ Ea" "⋃Ca = space A"
and Eb: "Eb ⊆ Pow (space B)" "sets B = sigma_sets (space B) Eb"
and Cb: "countable Cb" "Cb ⊆ Eb" "⋃Cb = space B"
shows "sets (A ⨂⇩M B) = sets (sigma (space A × space B) { a × b | a b. a ∈ Ea ∧ b ∈ Eb })"
(is "_ = sets (sigma ?Ω ?E)")
proof
show "sets (sigma ?Ω ?E) ⊆ sets (A ⨂⇩M B)"
using Ea(1) Eb(1) by (subst sigma_le_sets) (auto simp: Ea(2) Eb(2))
have "?E ⊆ Pow ?Ω"
using Ea(1) Eb(1) by auto
then have E: "a ∈ Ea ⟹ b ∈ Eb ⟹ a × b ∈ sets (sigma ?Ω ?E)" for a b
by auto
have "sets (A ⨂⇩M B) ⊆ sets (Sup {vimage_algebra ?Ω fst A, vimage_algebra ?Ω snd B})"
unfolding sets_pair_eq_sets_fst_snd ..
also have "vimage_algebra ?Ω fst A = vimage_algebra ?Ω fst (sigma (space A) Ea)"
by (intro vimage_algebra_cong[OF refl refl]) (simp add: Ea)
also have "… = sigma ?Ω {fst -` A ∩ ?Ω |A. A ∈ Ea}"
by (intro Ea vimage_algebra_sigma) auto
also have "vimage_algebra ?Ω snd B = vimage_algebra ?Ω snd (sigma (space B) Eb)"
by (intro vimage_algebra_cong[OF refl refl]) (simp add: Eb)
also have "… = sigma ?Ω {snd -` A ∩ ?Ω |A. A ∈ Eb}"
by (intro Eb vimage_algebra_sigma) auto
also have "{sigma ?Ω {fst -` Aa ∩ ?Ω |Aa. Aa ∈ Ea}, sigma ?Ω {snd -` Aa ∩ ?Ω |Aa. Aa ∈ Eb}} =
sigma ?Ω ` {{fst -` Aa ∩ ?Ω |Aa. Aa ∈ Ea}, {snd -` Aa ∩ ?Ω |Aa. Aa ∈ Eb}}"
by auto
also have "sets (SUP S∈{{fst -` Aa ∩ ?Ω |Aa. Aa ∈ Ea}, {snd -` Aa ∩ ?Ω |Aa. Aa ∈ Eb}}. sigma ?Ω S) =
sets (sigma ?Ω (⋃{{fst -` Aa ∩ ?Ω |Aa. Aa ∈ Ea}, {snd -` Aa ∩ ?Ω |Aa. Aa ∈ Eb}}))"
using Ea(1) Eb(1) by (intro sets_Sup_sigma) auto
also have "… ⊆ sets (sigma ?Ω ?E)"
proof (subst sigma_le_sets, safe intro!: space_in_measure_of)
fix a assume "a ∈ Ea"
then have "fst -` a ∩ ?Ω = (⋃b∈Cb. a × b)"
using Cb(3)[symmetric] Ea(1) by auto
then show "fst -` a ∩ ?Ω ∈ sets (sigma ?Ω ?E)"
using Cb ‹a ∈ Ea› by (auto intro!: sets.countable_UN' E)
next
fix b assume "b ∈ Eb"
then have "snd -` b ∩ ?Ω = (⋃a∈Ca. a × b)"
using Ca(3)[symmetric] Eb(1) by auto
then show "snd -` b ∩ ?Ω ∈ sets (sigma ?Ω ?E)"
using Ca ‹b ∈ Eb› by (auto intro!: sets.countable_UN' E)
qed
finally show "sets (A ⨂⇩M B) ⊆ sets (sigma ?Ω ?E)" .
qed

proposition  borel_prod:
"(borel ⨂⇩M borel) = (borel :: ('a::second_countable_topology × 'b::second_countable_topology) measure)"
(is "?P = ?B")
proof -
have "?B = sigma UNIV {A × B | A B. open A ∧ open B}"
by (rule second_countable_borel_measurable[OF open_prod_generated])
also have "… = ?P"
unfolding borel_def
by (subst sigma_prod) (auto intro!: exI[of _ "{UNIV}"])
finally show ?thesis ..
qed

proposition pair_measure_count_space:
assumes A: "finite A" and B: "finite B"
shows "count_space A ⨂⇩M count_space B = count_space (A × B)" (is "?P = ?C")
proof (rule measure_eqI)
interpret A: finite_measure "count_space A" by (rule finite_measure_count_space) fact
interpret B: finite_measure "count_space B" by (rule finite_measure_count_space) fact
interpret P: pair_sigma_finite "count_space A" "count_space B" ..
show eq: "sets ?P = sets ?C"
by (simp add: sets_pair_measure sigma_sets_pair_measure_generator_finite A B)
fix X assume X: "X ∈ sets ?P"
with eq have X_subset: "X ⊆ A × B" by simp
with A B have fin_Pair: "⋀x. finite (Pair x -` X)"
by (intro finite_subset[OF _ B]) auto
have fin_X: "finite X" using X_subset by (rule finite_subset) (auto simp: A B)
have card: "0 < card (Pair a -` X)" if "(a, b) ∈ X" for a b
using card_gt_0_iff fin_Pair that by auto
then have "emeasure ?P X = ∫⇧+ x. emeasure (count_space B) (Pair x -` X)
∂count_space A"
by (simp add: B.emeasure_pair_measure_alt X)
also have "... = emeasure ?C X"
apply (subst emeasure_count_space)
using card X_subset A fin_Pair fin_X
apply (auto simp add: nn_integral_count_space
of_nat_sum[symmetric] card_SigmaI[symmetric]
simp del:  card_SigmaI
intro!: arg_cong[where f=card])
done
finally show "emeasure ?P X = emeasure ?C X" .
qed

lemma emeasure_prod_count_space:
assumes A: "A ∈ sets (count_space UNIV ⨂⇩M M)" (is "A ∈ sets (?A ⨂⇩M ?B)")
shows "emeasure (?A ⨂⇩M ?B) A = (∫⇧+ x. ∫⇧+ y. indicator A (x, y) ∂?B ∂?A)"
by (rule emeasure_measure_of[OF pair_measure_def])
(auto simp: countably_additive_def positive_def suminf_indicator A
nn_integral_suminf[symmetric] dest: sets.sets_into_space)

lemma emeasure_prod_count_space_single[simp]: "emeasure (count_space UNIV ⨂⇩M count_space UNIV) {x} = 1"
proof -
have [simp]: "⋀a b x y. indicator {(a, b)} (x, y) = (indicator {a} x * indicator {b} y::ennreal)"
by (auto split: split_indicator)
show ?thesis
by (cases x) (auto simp: emeasure_prod_count_space nn_integral_cmult sets_Pair)
qed

lemma emeasure_count_space_prod_eq:
fixes A :: "('a × 'b) set"
assumes A: "A ∈ sets (count_space UNIV ⨂⇩M count_space UNIV)" (is "A ∈ sets (?A ⨂⇩M ?B)")
shows "emeasure (?A ⨂⇩M ?B) A = emeasure (count_space UNIV) A"
proof -
{ fix A :: "('a × 'b) set" assume "countable A"
then have "emeasure (?A ⨂⇩M ?B) (⋃a∈A. {a}) = (∫⇧+a. emeasure (?A ⨂⇩M ?B) {a} ∂count_space A)"
by (intro emeasure_UN_countable) (auto simp: sets_Pair disjoint_family_on_def)
also have "… = (∫⇧+a. indicator A a ∂count_space UNIV)"
by (subst nn_integral_count_space_indicator) auto
finally have "emeasure (?A ⨂⇩M ?B) A = emeasure (count_space UNIV) A"
by simp }
note * = this

show ?thesis
proof cases
assume "finite A" then show ?thesis
by (intro * countable_finite)
next
assume "infinite A"
then obtain C where "countable C" and "infinite C" and "C ⊆ A"
by (auto dest: infinite_countable_subset')
with A have "emeasure (?A ⨂⇩M ?B) C ≤ emeasure (?A ⨂⇩M ?B) A"
by (intro emeasure_mono) auto
also have "emeasure (?A ⨂⇩M ?B) C = emeasure (count_space UNIV) C"
using ‹countable C› by (rule *)
finally show ?thesis
using ‹infinite C› ‹infinite A› by (simp add: top_unique)
qed
qed

lemma nn_integral_count_space_prod_eq:
"nn_integral (count_space UNIV ⨂⇩M count_space UNIV) f = nn_integral (count_space UNIV) f"
(is "nn_integral ?P f = _")
proof cases
assume cntbl: "countable {x. f x ≠ 0}"
have [simp]: "⋀x. card ({x} ∩ {x. f x ≠ 0}) = (indicator {x. f x ≠ 0} x::ennreal)"
by (auto split: split_indicator)
have [measurable]: "⋀y. (λx. indicator {y} x) ∈ borel_measurable ?P"
by (rule measurable_discrete_difference[of "λx. 0" _ borel "{y}" "λx. indicator {y} x" for y])
(auto intro: sets_Pair)

have "(∫⇧+x. f x ∂?P) = (∫⇧+x. ∫⇧+x'. f x * indicator {x} x' ∂count_space {x. f x ≠ 0} ∂?P)"
by (auto simp add: nn_integral_cmult nn_integral_indicator' intro!: nn_integral_cong split: split_indicator)
also have "… = (∫⇧+x. ∫⇧+x'. f x' * indicator {x'} x ∂count_space {x. f x ≠ 0} ∂?P)"
by (auto intro!: nn_integral_cong split: split_indicator)
also have "… = (∫⇧+x'. ∫⇧+x. f x' * indicator {x'} x ∂?P ∂count_space {x. f x ≠ 0})"
by (intro nn_integral_count_space_nn_integral cntbl) auto
also have "… = (∫⇧+x'. f x' ∂count_space {x. f x ≠ 0})"
by (intro nn_integral_cong) (auto simp: nn_integral_cmult sets_Pair)
finally show ?thesis
by (auto simp add: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator)
next
{ fix x assume "f x ≠ 0"
then have "(∃r≥0. 0 < r ∧ f x = ennreal r) ∨ f x = ∞"
by (cases "f x" rule: ennreal_cases) (auto simp: less_le)
then have "∃n. ennreal (1 / real (Suc n)) ≤ f x"
by (auto elim!: nat_approx_posE intro!: less_imp_le) }
note * = this

assume cntbl: "uncountable {x. f x ≠ 0}"
also have "{x. f x ≠ 0} = (⋃n. {x. 1/Suc n ≤ f x})"
using * by auto
finally obtain n where "infinite {x. 1/Suc n ≤ f x}"
by (meson countableI_type countable_UN uncountable_infinite)
then obtain C where C: "C ⊆ {x. 1/Suc n ≤ f x}" and "countable C" "infinite C"
by (metis infinite_countable_subset')

have [measurable]: "C ∈ sets ?P"
using sets.countable[OF _ ‹countable C›, of ?P] by (auto simp: sets_Pair)

have "(∫⇧+x. ennreal (1/Suc n) * indicator C x ∂?P) ≤ nn_integral ?P f"
using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
moreover have "(∫⇧+x. ennreal (1/Suc n) * indicator C x ∂?P) = ∞"
using ‹infinite C› by (simp add: nn_integral_cmult emeasure_count_space_prod_eq ennreal_mult_top)
moreover have "(∫⇧+x. ennreal (1/Suc n) * indicator C x ∂count_space UNIV) ≤ nn_integral (count_space UNIV) f"
using C by (intro nn_integral_mono) (auto split: split_indicator simp: zero_ereal_def[symmetric])
moreover have "(∫⇧+x. ennreal (1/Suc n) * indicator C x ∂count_space UNIV) = ∞"
using ‹infinite C› by (simp add: nn_integral_cmult ennreal_mult_top)
ultimately show ?thesis
by (simp add: top_unique)
qed

theorem pair_measure_density:
assumes f: "f ∈ borel_measurable M1"
assumes g: "g ∈ borel_measurable M2"
assumes "sigma_finite_measure M2" "sigma_finite_measure (density M2 g)"
shows "density M1 f ⨂⇩M density M2 g = density (M1 ⨂⇩M M2) (λ(x,y). f x * g y)" (is "?L = ?R")
proof (rule measure_eqI)
interpret M2: sigma_finite_measure M2 by fact
interpret D2: sigma_finite_measure "density M2 g" by fact

fix A assume A: "A ∈ sets ?L"
with f g have "(∫⇧+ x. f x * ∫⇧+ y. g y * indicator A (x, y) ∂M2 ∂M1) =
(∫⇧+ x. ∫⇧+ y. f x * g y * indicator A (x, y) ∂M2 ∂M1)"
by (intro nn_integral_cong_AE)
(auto simp add: nn_integral_cmult[symmetric] ac_simps)
with A f g show "emeasure ?L A = emeasure ?R A"
by (simp add: D2.emeasure_pair_measure emeasure_density nn_integral_density
M2.nn_integral_fst[symmetric]
cong: nn_integral_cong)
qed simp

lemma sigma_finite_measure_distr:
assumes "sigma_finite_measure (distr M N f)" and f: "f ∈ measurable M N"
shows "sigma_finite_measure M"
proof -
interpret sigma_finite_measure "distr M N f" by fact
obtain A where A: "countable A" "A ⊆ sets (distr M N f)"
"⋃ A = space (distr M N f)" "∀a∈A. emeasure (distr M N f) a ≠ ∞"
using sigma_finite_countable by auto
show ?thesis
proof
show "∃A. countable A ∧ A ⊆ sets M ∧ ⋃A = space M ∧ (∀a∈A. emeasure M a ≠ ∞)"
using A f
by (intro exI[of _ "(λa. f -` a ∩ space M) ` A"])
(auto simp: emeasure_distr set_eq_iff subset_eq intro: measurable_space)
qed
qed

lemma pair_measure_distr:
assumes f: "f ∈ measurable M S" and g: "g ∈ measurable N T"
assumes "sigma_finite_measure (distr N T g)"
shows "distr M S f ⨂⇩M distr N T g = distr (M ⨂⇩M N) (S ⨂⇩M T) (λ(x, y). (f x, g y))" (is "?P = ?D")
proof (rule measure_eqI)
interpret T: sigma_finite_measure "distr N T g" by fact
interpret N: sigma_finite_measure N by (rule sigma_finite_measure_distr) fact+

fix A assume A: "A ∈ sets ?P"
with f g show "emeasure ?P A = emeasure ?D A"
by (auto simp add: N.emeasure_pair_measure_alt space_pair_measure emeasure_distr
T.emeasure_pair_measure_alt nn_integral_distr
intro!: nn_integral_cong arg_cong[where f="emeasure N"])
qed simp

lemma pair_measure_eqI:
assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
assumes sets: "sets (M1 ⨂⇩M M2) = sets M"
assumes emeasure: "⋀A B. A ∈ sets M1 ⟹ B ∈ sets M2 ⟹ emeasure M1 A * emeasure M2 B = emeasure M (A × B)"
shows "M1 ⨂⇩M M2 = M"
proof -
interpret M1: sigma_finite_measure M1 by fact
interpret M2: sigma_finite_measure M2 by fact
interpret pair_sigma_finite M1 M2 ..
let ?E = "{a × b |a b. a ∈ sets M1 ∧ b ∈ sets M2}"
let ?P = "M1 ⨂⇩M M2"
obtain F :: "nat ⇒ ('a × 'b) set" where F:
"range F ⊆ ?E" "incseq F" "⋃ (range F) = space M1 × space M2" "∀i. emeasure ?P (F i) ≠ ∞"
using sigma_finite_up_in_pair_measure_generator
by blast
show ?thesis
proof (rule measure_eqI_generator_eq[OF Int_stable_pair_measure_generator[of M1 M2]])
show "?E ⊆ Pow (space ?P)"
using sets.space_closed[of M1] sets.space_closed[of M2] by (auto simp: space_pair_measure)
show "sets ?P = sigma_sets (space ?P) ?E"
by (simp add: sets_pair_measure space_pair_measure)
then show "sets M = sigma_sets (space ?P) ?E"
using sets[symmetric] by simp
next
show "range F ⊆ ?E" "(⋃i. F i) = space ?P" "⋀i. emeasure ?P (F i) ≠ ∞"
using F by (auto simp: space_pair_measure)
next
fix X assume "X ∈ ?E"
then obtain A B where X[simp]: "X = A × B" and A: "A ∈ sets M1" and B: "B ∈ sets M2" by auto
then have "emeasure ?P X = emeasure M1 A * emeasure M2 B"
by (simp add: M2.emeasure_pair_measure_Times)
also have "… = emeasure M (A × B)"
using A B emeasure by auto
finally show "emeasure ?P X = emeasure M X"
by simp
qed
qed

lemma sets_pair_countable:
assumes "countable S1" "countable S2"
assumes M: "sets M = Pow S1" and N: "sets N = Pow S2"
shows "sets (M ⨂⇩M N) = Pow (S1 × S2)"
proof auto
fix x a b assume x: "x ∈ sets (M ⨂⇩M N)" "(a, b) ∈ x"
from sets.sets_into_space[OF x(1)] x(2)
sets_eq_imp_space_eq[of N "count_space S2"] sets_eq_imp_space_eq[of M "count_space S1"] M N
show "a ∈ S1" "b ∈ S2"
by (auto simp: space_pair_measure)
next
fix X assume X: "X ⊆ S1 × S2"
then have "countable X"
by (metis countable_subset ‹countable S1› ‹countable S2› countable_SIGMA)
have "X = (⋃(a, b)∈X. {a} × {b})" by auto
also have "… ∈ sets (M ⨂⇩M N)"
using X
by (safe intro!: sets.countable_UN' ‹countable X› subsetI pair_measureI) (auto simp: M N)
finally show "X ∈ sets (M ⨂⇩M N)" .
qed

lemma pair_measure_countable:
assumes "countable S1" "countable S2"
shows "count_space S1 ⨂⇩M count_space S2 = count_space (S1 × S2)"
proof (rule pair_measure_eqI)
show "sigma_finite_measure (count_space S1)" "sigma_finite_measure (count_space S2)"
using assms by (auto intro!: sigma_finite_measure_count_space_countable)
show "sets (count_space S1 ⨂⇩M count_space S2) = sets (count_space (S1 × S2))"
by (subst sets_pair_countable[OF assms]) auto
next
fix A B assume "A ∈ sets (count_space S1)" "B ∈ sets (count_space S2)"
then show "emeasure (count_space S1) A * emeasure (count_space S2) B =
emeasure (count_space (S1 × S2)) (A × B)"
by (subst (1 2 3) emeasure_count_space) (auto simp: finite_cartesian_product_iff ennreal_mult_top ennreal_top_mult)
qed

proposition nn_integral_fst_count_space:
"(∫⇧+ x. ∫⇧+ y. f (x, y) ∂count_space UNIV ∂count_space UNIV) = integral⇧N (count_space UNIV) f"
(is "?lhs = ?rhs")
proof(cases)
assume *: "countable {xy. f xy ≠ 0}"
let ?A = "fst ` {xy. f xy ≠ 0}"
let ?B = "snd ` {xy. f xy ≠ 0}"
from * have [simp]: "countable ?A" "countable ?B" by(rule countable_image)+
have "?lhs = (∫⇧+ x. ∫⇧+ y. f (x, y) ∂count_space UNIV ∂count_space ?A)"
by(rule nn_integral_count_space_eq)
(auto simp add: nn_integral_0_iff_AE AE_count_space not_le intro: rev_image_eqI)
also have "… = (∫⇧+ x. ∫⇧+ y. f (x, y) ∂count_space ?B ∂count_space ?A)"
by(intro nn_integral_count_space_eq nn_integral_cong)(auto intro: rev_image_eqI)
also have "… = (∫⇧+ xy. f xy ∂count_space (?A × ?B))"
by(subst sigma_finite_measure.nn_integral_fst)
(simp_all add: sigma_finite_measure_count_space_countable pair_measure_countable)
also have "… = ?rhs"
by(rule nn_integral_count_space_eq)(auto intro: rev_image_eqI)
finally show ?thesis .
next
{ fix xy assume "f xy ≠ 0"
then have "(∃r≥0. 0 < r ∧ f xy = ennreal r) ∨ f xy = ∞"
by (cases "f xy" rule: ennreal_cases) (auto simp: less_le)
then have "∃n. ennreal (1 / real (Suc n)) ≤ f xy"
by (auto elim!: nat_approx_posE intro!: less_imp_le) }
note * = this

assume cntbl: "uncountable {xy. f xy ≠ 0}"
also have "{xy. f xy ≠ 0} = (⋃n. {xy. 1/Suc n ≤ f xy})"
using * by auto
finally obtain n where "infinite {xy. 1/Suc n ≤ f xy}"
by (meson countableI_type countable_UN uncountable_infinite)
then obtain C where C: "C ⊆ {xy. 1/Suc n ≤ f xy}" and "countable C" "infinite C"
by (metis infinite_countable_subset')

have "∞ = (∫⇧+ xy. ennreal (1 / Suc n) * indicator C xy ∂count_space UNIV)"
using ‹infinite C› by(simp add: nn_integral_cmult ennreal_mult_top)
also have "… ≤ ?rhs" using C
by(intro nn_integral_mono)(auto split: split_indicator)
finally have "?rhs = ∞" by (simp add: top_unique)
moreover have "?lhs = ∞"
proof(cases "finite (fst ` C)")
case True
then obtain x C' where x: "x ∈ fst ` C"
and C': "C' = fst -` {x} ∩ C"
and "infinite C'"
using ‹infinite C› by(auto elim!: inf_img_fin_domE')
from x C C' have **: "C' ⊆ {xy. 1 / Suc n ≤ f xy}" by auto

from C' ‹infinite C'› have "infinite (snd ` C')"
by(auto dest!: finite_imageD simp add: inj_on_def)
then have "∞ = (∫⇧+ y. ennreal (1 / Suc n) * indicator (snd ` C') y ∂count_space UNIV)"
by(simp add: nn_integral_cmult ennreal_mult_top)
also have "… = (∫⇧+ y. ennreal (1 / Suc n) * indicator C' (x, y) ∂count_space UNIV)"
by(rule nn_integral_cong)(force split: split_indicator intro: rev_image_eqI simp add: C')
also have "… = (∫⇧+ x'. (∫⇧+ y. ennreal (1 / Suc n) * indicator C' (x, y) ∂count_space UNIV) * indicator {x} x' ∂count_space UNIV)"
by(simp add: one_ereal_def[symmetric])
also have "… ≤ (∫⇧+ x. ∫⇧+ y. ennreal (1 / Suc n) * indicator C' (x, y) ∂count_space UNIV ∂count_space UNIV)"
by(rule nn_integral_mono)(simp split: split_indicator)
also have "… ≤ ?lhs" using **
by(intro nn_integral_mono)(auto split: split_indicator)
finally show ?thesis by (simp add: top_unique)
next
case False
define C' where "C' = fst ` C"
have "∞ = ∫⇧+ x. ennreal (1 / Suc n) * indicator C' x ∂count_space UNIV"
using C'_def False by(simp add: nn_integral_cmult ennreal_mult_top)
also have "… = ∫⇧+ x. ∫⇧+ y. ennreal (1 / Suc n) * indicator C' x * indicator {SOME y. (x, y) ∈ C} y ∂count_space UNIV ∂count_space UNIV"
by(auto simp add: one_ereal_def[symmetric] max_def intro: nn_integral_cong)
also have "… ≤ ∫⇧+ x. ∫⇧+ y. ennreal (1 / Suc n) * indicator C (x, y) ∂count_space UNIV ∂count_space UNIV"
by(intro nn_integral_mono)(auto simp add: C'_def split: split_indicator intro: someI)
also have "… ≤ ?lhs" using C
by(intro nn_integral_mono)(auto split: split_indicator)
finally show ?thesis by (simp add: top_unique)
qed
ultimately show ?thesis by simp
qed

proposition nn_integral_snd_count_space:
"(∫⇧+ y. ∫⇧+ x. f (x, y) ∂count_space UNIV ∂count_space UNIV) = integral⇧N (count_space UNIV) f"
(is "?lhs = ?rhs")
proof -
have "?lhs = (∫⇧+ y. ∫⇧+ x. (λ(y, x). f (x, y)) (y, x) ∂count_space UNIV ∂count_space UNIV)"
by(simp)
also have "… = ∫⇧+ yx. (λ(y, x). f (x, y)) yx ∂count_space UNIV"
by(rule nn_integral_fst_count_space)
also have "… = ∫⇧+ xy. f xy ∂count_space ((λ(x, y). (y, x)) ` UNIV)"
by(subst nn_integral_bij_count_space[OF inj_on_imp_bij_betw, symmetric])
(simp_all add: inj_on_def split_def)
also have "… = ?rhs" by(rule nn_integral_count_space_eq) auto
finally show ?thesis .
qed

lemma measurable_pair_measure_countable1:
assumes "countable A"
and [measurable]: "⋀x. x ∈ A ⟹ (λy. f (x, y)) ∈ measurable N K"
shows "f ∈ measurable (count_space A ⨂⇩M N) K"
using _ _ assms(1)
by(rule measurable_compose_countable'[where f="λa b. f (a, snd b)" and g=fst and I=A, simplified])simp_all

subsection ‹Product of Borel spaces›

theorem borel_Times:
fixes A :: "'a::topological_space set" and B :: "'b::topological_space set"
assumes A: "A ∈ sets borel" and B: "B ∈ sets borel"
shows "A × B ∈ sets borel"
proof -
have "A × B = (A×UNIV) ∩ (UNIV × B)"
by auto
moreover
{ have "A ∈ sigma_sets UNIV {S. open S}" using A by (simp add: sets_borel)
then have "A×UNIV ∈ sets borel"
proof (induct A)
case (Basic S) then show ?case
by (auto intro!: borel_open open_Times)
next
case (Compl A)
moreover have *: "(UNIV - A) × UNIV = UNIV - (A × UNIV)"
by auto
ultimately show ?case
unfolding * by auto
next
case (Union A)
moreover have *: "(⋃(A ` UNIV)) × UNIV = ⋃((λi. A i × UNIV) ` UNIV)"
by auto
ultimately show ?case
unfolding * by auto
qed simp }
moreover
{ have "B ∈ sigma_sets UNIV {S. open S}" using B by (simp add: sets_borel)
then have "UNIV×B ∈ sets borel"
proof (induct B)
case (Basic S) then show ?case
by (auto intro!: borel_open open_Times)
next
case (Compl B)
moreover have *: "UNIV × (UNIV - B) = UNIV - (UNIV × B)"
by auto
ultimately show ?case
unfolding * by auto
next
case (Union B)
moreover have *: "UNIV × (⋃(B ` UNIV)) = ⋃((λi. UNIV × B i) ` UNIV)"
by auto
ultimately show ?case
unfolding * by auto
qed simp }
ultimately show ?thesis
by auto
qed

lemma finite_measure_pair_measure:
assumes "finite_measure M" "finite_measure N"
shows "finite_measure (N  ⨂⇩M M)"
proof (rule finite_measureI)
interpret M: finite_measure M by fact
interpret N: finite_measure N by fact
show "emeasure (N  ⨂⇩M M) (space (N  ⨂⇩M M)) ≠ ∞"
by (auto simp: space_pair_measure M.emeasure_pair_measure_Times ennreal_mult_eq_top_iff)
qed

end
```