(* Title: HOL/Probability/Binary_Product_Measure.thy
Author: Johannes Hölzl, TU München
*)
header {*Binary product measures*}
theory Binary_Product_Measure
imports Lebesgue_Integration
begin
lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
by auto
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
by auto
lemma Pair_vimage_times[simp]: "\<And>A B x. Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
by auto
lemma rev_Pair_vimage_times[simp]: "\<And>A B y. (\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
by auto
lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
by (cases x) simp
lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
by (auto simp: fun_eq_iff)
section "Binary products"
definition
"pair_measure_generator A B =
\<lparr> space = space A \<times> space B,
sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B},
measure = \<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A \<rparr>"
definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where
"A \<Otimes>\<^isub>M B = sigma (pair_measure_generator A B)"
locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2
for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
abbreviation (in pair_sigma_algebra)
"E \<equiv> pair_measure_generator M1 M2"
abbreviation (in pair_sigma_algebra)
"P \<equiv> M1 \<Otimes>\<^isub>M M2"
lemma sigma_algebra_pair_measure:
"sets M1 \<subseteq> Pow (space M1) \<Longrightarrow> sets M2 \<subseteq> Pow (space M2) \<Longrightarrow> sigma_algebra (pair_measure M1 M2)"
by (force simp: pair_measure_def pair_measure_generator_def intro!: sigma_algebra_sigma)
sublocale pair_sigma_algebra \<subseteq> sigma_algebra P
using M1.space_closed M2.space_closed
by (rule sigma_algebra_pair_measure)
lemma pair_measure_generatorI[intro, simp]:
"x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_measure_generator A B)"
by (auto simp add: pair_measure_generator_def)
lemma pair_measureI[intro, simp]:
"x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)"
by (auto simp add: pair_measure_def)
lemma space_pair_measure:
"space (A \<Otimes>\<^isub>M B) = space A \<times> space B"
by (simp add: pair_measure_def pair_measure_generator_def)
lemma sets_pair_measure_generator:
"sets (pair_measure_generator N M) = (\<lambda>(x, y). x \<times> y) ` (sets N \<times> sets M)"
unfolding pair_measure_generator_def by auto
lemma pair_measure_generator_sets_into_space:
assumes "sets M \<subseteq> Pow (space M)" "sets N \<subseteq> Pow (space N)"
shows "sets (pair_measure_generator M N) \<subseteq> Pow (space (pair_measure_generator M N))"
using assms by (auto simp: pair_measure_generator_def)
lemma pair_measure_generator_Int_snd:
assumes "sets S1 \<subseteq> Pow (space S1)"
shows "sets (pair_measure_generator S1 (algebra.restricted_space S2 A)) =
sets (algebra.restricted_space (pair_measure_generator S1 S2) (space S1 \<times> A))"
(is "?L = ?R")
apply (auto simp: pair_measure_generator_def image_iff)
using assms
apply (rule_tac x="a \<times> xa" in exI)
apply force
using assms
apply (rule_tac x="a" in exI)
apply (rule_tac x="b \<inter> A" in exI)
apply auto
done
lemma (in pair_sigma_algebra)
shows measurable_fst[intro!, simp]:
"fst \<in> measurable P M1" (is ?fst)
and measurable_snd[intro!, simp]:
"snd \<in> measurable P M2" (is ?snd)
proof -
{ fix X assume "X \<in> sets M1"
then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"])
using M1.sets_into_space by force+ }
moreover
{ fix X assume "X \<in> sets M2"
then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X])
using M2.sets_into_space by force+ }
ultimately have "?fst \<and> ?snd"
by (fastsimp simp: measurable_def sets_sigma space_pair_measure
intro!: sigma_sets.Basic)
then show ?fst ?snd by auto
qed
lemma (in pair_sigma_algebra) measurable_pair_iff:
assumes "sigma_algebra M"
shows "f \<in> measurable M P \<longleftrightarrow>
(fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
proof -
interpret M: sigma_algebra M by fact
from assms show ?thesis
proof (safe intro!: measurable_comp[where b=P])
assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"
show "f \<in> measurable M P" unfolding pair_measure_def
proof (rule M.measurable_sigma)
show "sets (pair_measure_generator M1 M2) \<subseteq> Pow (space E)"
unfolding pair_measure_generator_def using M1.sets_into_space M2.sets_into_space by auto
show "f \<in> space M \<rightarrow> space E"
using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma pair_measure_generator_def)
fix A assume "A \<in> sets E"
then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C"
unfolding pair_measure_generator_def by auto
moreover have "(fst \<circ> f) -` B \<inter> space M \<in> sets M"
using f `B \<in> sets M1` unfolding measurable_def by auto
moreover have "(snd \<circ> f) -` C \<inter> space M \<in> sets M"
using s `C \<in> sets M2` unfolding measurable_def by auto
moreover have "f -` A \<inter> space M = ((fst \<circ> f) -` B \<inter> space M) \<inter> ((snd \<circ> f) -` C \<inter> space M)"
unfolding `A = B \<times> C` by (auto simp: vimage_Times)
ultimately show "f -` A \<inter> space M \<in> sets M" by auto
qed
qed
qed
lemma (in pair_sigma_algebra) measurable_pair:
assumes "sigma_algebra M"
assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
shows "f \<in> measurable M P"
unfolding measurable_pair_iff[OF assms(1)] using assms(2,3) by simp
lemma pair_measure_generatorE:
assumes "X \<in> sets (pair_measure_generator M1 M2)"
obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2"
using assms unfolding pair_measure_generator_def by auto
lemma (in pair_sigma_algebra) pair_measure_generator_swap:
"(\<lambda>X. (\<lambda>(x,y). (y,x)) -` X \<inter> space M2 \<times> space M1) ` sets E = sets (pair_measure_generator M2 M1)"
proof (safe elim!: pair_measure_generatorE)
fix A B assume "A \<in> sets M1" "B \<in> sets M2"
moreover then have "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A"
using M1.sets_into_space M2.sets_into_space by auto
ultimately show "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_measure_generator M2 M1)"
by (auto intro: pair_measure_generatorI)
next
fix A B assume "A \<in> sets M1" "B \<in> sets M2"
then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E"
using M1.sets_into_space M2.sets_into_space
by (auto intro!: image_eqI[where x="A \<times> B"] pair_measure_generatorI)
qed
lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:
assumes Q: "Q \<in> sets P"
shows "(\<lambda>(x,y). (y, x)) -` Q \<in> sets (M2 \<Otimes>\<^isub>M M1)" (is "_ \<in> sets ?Q")
proof -
let "?f Q" = "(\<lambda>(x,y). (y, x)) -` Q \<inter> space M2 \<times> space M1"
have *: "(\<lambda>(x,y). (y, x)) -` Q = ?f Q"
using sets_into_space[OF Q] by (auto simp: space_pair_measure)
have "sets (M2 \<Otimes>\<^isub>M M1) = sets (sigma (pair_measure_generator M2 M1))"
unfolding pair_measure_def ..
also have "\<dots> = sigma_sets (space M2 \<times> space M1) (?f ` sets E)"
unfolding sigma_def pair_measure_generator_swap[symmetric]
by (simp add: pair_measure_generator_def)
also have "\<dots> = ?f ` sigma_sets (space M1 \<times> space M2) (sets E)"
using M1.sets_into_space M2.sets_into_space
by (intro sigma_sets_vimage) (auto simp: pair_measure_generator_def)
also have "\<dots> = ?f ` sets P"
unfolding pair_measure_def pair_measure_generator_def sigma_def by simp
finally show ?thesis
using Q by (subst *) auto
qed
lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:
shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (M2 \<Otimes>\<^isub>M M1)"
(is "?f \<in> measurable ?P ?Q")
unfolding measurable_def
proof (intro CollectI conjI Pi_I ballI)
fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q"
unfolding pair_measure_generator_def pair_measure_def by auto
next
fix A assume "A \<in> sets (M2 \<Otimes>\<^isub>M M1)"
interpret Q: pair_sigma_algebra M2 M1 by default
with Q.sets_pair_sigma_algebra_swap[OF `A \<in> sets (M2 \<Otimes>\<^isub>M M1)`]
show "?f -` A \<inter> space ?P \<in> sets ?P" by simp
qed
lemma (in pair_sigma_algebra) measurable_cut_fst[simp,intro]:
assumes "Q \<in> sets P" shows "Pair x -` Q \<in> sets M2"
proof -
let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x -` Q \<in> sets M2}"
let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>"
interpret Q: sigma_algebra ?Q
proof qed (auto simp: vimage_UN vimage_Diff space_pair_measure)
have "sets E \<subseteq> sets ?Q"
using M1.sets_into_space M2.sets_into_space
by (auto simp: pair_measure_generator_def space_pair_measure)
then have "sets P \<subseteq> sets ?Q"
apply (subst pair_measure_def, intro Q.sets_sigma_subset)
by (simp add: pair_measure_def)
with assms show ?thesis by auto
qed
lemma (in pair_sigma_algebra) measurable_cut_snd:
assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) -` Q \<in> sets M1" (is "?cut Q \<in> sets M1")
proof -
interpret Q: pair_sigma_algebra M2 M1 by default
with Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]
show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
qed
lemma (in pair_sigma_algebra) measurable_pair_image_snd:
assumes m: "f \<in> measurable P M" and "x \<in> space M1"
shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
unfolding measurable_def
proof (intro CollectI conjI Pi_I ballI)
fix y assume "y \<in> space M2" with `f \<in> measurable P M` `x \<in> space M1`
show "f (x, y) \<in> space M"
unfolding measurable_def pair_measure_generator_def pair_measure_def by auto
next
fix A assume "A \<in> sets M"
then have "Pair x -` (f -` A \<inter> space P) \<in> sets M2" (is "?C \<in> _")
using `f \<in> measurable P M`
by (intro measurable_cut_fst) (auto simp: measurable_def)
also have "?C = (\<lambda>y. f (x, y)) -` A \<inter> space M2"
using `x \<in> space M1` by (auto simp: pair_measure_generator_def pair_measure_def)
finally show "(\<lambda>y. f (x, y)) -` A \<inter> space M2 \<in> sets M2" .
qed
lemma (in pair_sigma_algebra) measurable_pair_image_fst:
assumes m: "f \<in> measurable P M" and "y \<in> space M2"
shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
proof -
interpret Q: pair_sigma_algebra M2 M1 by default
from Q.measurable_pair_image_snd[OF measurable_comp `y \<in> space M2`,
OF Q.pair_sigma_algebra_swap_measurable m]
show ?thesis by simp
qed
lemma (in pair_sigma_algebra) Int_stable_pair_measure_generator: "Int_stable E"
unfolding Int_stable_def
proof (intro ballI)
fix A B assume "A \<in> sets E" "B \<in> sets E"
then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2"
"A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2"
unfolding pair_measure_generator_def by auto
then show "A \<inter> B \<in> sets E"
by (auto simp add: times_Int_times pair_measure_generator_def)
qed
lemma finite_measure_cut_measurable:
fixes M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
assumes "sigma_finite_measure M1" "finite_measure M2"
assumes "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)"
shows "(\<lambda>x. measure M2 (Pair x -` Q)) \<in> borel_measurable M1"
(is "?s Q \<in> _")
proof -
interpret M1: sigma_finite_measure M1 by fact
interpret M2: finite_measure M2 by fact
interpret pair_sigma_algebra M1 M2 by default
have [intro]: "sigma_algebra M1" by fact
have [intro]: "sigma_algebra M2" by fact
let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1} \<rparr>"
note space_pair_measure[simp]
interpret dynkin_system ?D
proof (intro dynkin_systemI)
fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D"
using sets_into_space by simp
next
from top show "space ?D \<in> sets ?D"
by (auto simp add: if_distrib intro!: M1.measurable_If)
next
fix A assume "A \<in> sets ?D"
with sets_into_space have "\<And>x. measure M2 (Pair x -` (space M1 \<times> space M2 - A)) =
(if x \<in> space M1 then measure M2 (space M2) - ?s A x else 0)"
by (auto intro!: M2.measure_compl simp: vimage_Diff)
with `A \<in> sets ?D` top show "space ?D - A \<in> sets ?D"
by (auto intro!: Diff M1.measurable_If M1.borel_measurable_extreal_diff)
next
fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"
moreover then have "\<And>x. measure M2 (\<Union>i. Pair x -` F i) = (\<Sum>i. ?s (F i) x)"
by (intro M2.measure_countably_additive[symmetric])
(auto simp: disjoint_family_on_def)
ultimately show "(\<Union>i. F i) \<in> sets ?D"
by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)
qed
have "sets P = sets ?D" apply (subst pair_measure_def)
proof (intro dynkin_lemma)
show "Int_stable E" by (rule Int_stable_pair_measure_generator)
from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A"
by auto
then show "sets E \<subseteq> sets ?D"
by (auto simp: pair_measure_generator_def sets_sigma if_distrib
intro: sigma_sets.Basic intro!: M1.measurable_If)
qed (auto simp: pair_measure_def)
with `Q \<in> sets P` have "Q \<in> sets ?D" by simp
then show "?s Q \<in> borel_measurable M1" by simp
qed
subsection {* Binary products of $\sigma$-finite measure spaces *}
locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2
by default
lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
proof
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros)
qed
lemma (in pair_sigma_finite) measure_cut_measurable_fst:
assumes "Q \<in> sets P" shows "(\<lambda>x. measure M2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
proof -
have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+
have M1: "sigma_finite_measure M1" by default
from M2.disjoint_sigma_finite guess F .. note F = this
then have F_sets: "\<And>i. F i \<in> sets M2" by auto
let "?C x i" = "F i \<inter> Pair x -` Q"
{ fix i
let ?R = "M2.restricted_space (F i)"
have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"
using F M2.sets_into_space by auto
let ?R2 = "M2.restricted_space (F i)"
have "(\<lambda>x. measure ?R2 (Pair x -` (space M1 \<times> space ?R2 \<inter> Q))) \<in> borel_measurable M1"
proof (intro finite_measure_cut_measurable[OF M1])
show "finite_measure ?R2"
using F by (intro M2.restricted_to_finite_measure) auto
have "(space M1 \<times> space ?R2) \<inter> Q \<in> (op \<inter> (space M1 \<times> F i)) ` sets P"
using `Q \<in> sets P` by (auto simp: image_iff)
also have "\<dots> = sigma_sets (space M1 \<times> F i) ((op \<inter> (space M1 \<times> F i)) ` sets E)"
unfolding pair_measure_def pair_measure_generator_def sigma_def
using `F i \<in> sets M2` M2.sets_into_space
by (auto intro!: sigma_sets_Int sigma_sets.Basic)
also have "\<dots> \<subseteq> sets (M1 \<Otimes>\<^isub>M ?R2)"
using M1.sets_into_space
apply (auto simp: times_Int_times pair_measure_def pair_measure_generator_def sigma_def
intro!: sigma_sets_subseteq)
apply (rule_tac x="a" in exI)
apply (rule_tac x="b \<inter> F i" in exI)
by auto
finally show "(space M1 \<times> space ?R2) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R2)" .
qed
moreover have "\<And>x. Pair x -` (space M1 \<times> F i \<inter> Q) = ?C x i"
using `Q \<in> sets P` sets_into_space by (auto simp: space_pair_measure)
ultimately have "(\<lambda>x. measure M2 (?C x i)) \<in> borel_measurable M1"
by simp }
moreover
{ fix x
have "(\<Sum>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)"
proof (intro M2.measure_countably_additive)
show "range (?C x) \<subseteq> sets M2"
using F `Q \<in> sets P` by (auto intro!: M2.Int)
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 "(\<Union>i. ?C x i) = Pair x -` Q"
using F sets_into_space `Q \<in> sets P`
by (auto simp: space_pair_measure)
finally have "measure M2 (Pair x -` Q) = (\<Sum>i. measure M2 (?C x i))"
by simp }
ultimately show ?thesis using `Q \<in> sets P` F_sets
by (auto intro!: M1.borel_measurable_psuminf M2.Int)
qed
lemma (in pair_sigma_finite) measure_cut_measurable_snd:
assumes "Q \<in> sets P" shows "(\<lambda>y. M1.\<mu> ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
note sets_pair_sigma_algebra_swap[OF assms]
from Q.measure_cut_measurable_fst[OF this]
show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
qed
lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:
assumes "f \<in> measurable P M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"
proof -
interpret Q: pair_sigma_algebra M2 M1 by default
have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)
show ?thesis
using Q.pair_sigma_algebra_swap_measurable assms
unfolding * by (rule measurable_comp)
qed
lemma (in pair_sigma_finite) pair_measure_alt:
assumes "A \<in> sets P"
shows "measure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+ x. measure M2 (Pair x -` A) \<partial>M1)"
apply (simp add: pair_measure_def pair_measure_generator_def)
proof (rule M1.positive_integral_cong)
fix x assume "x \<in> space M1"
have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: extreal)"
unfolding indicator_def by auto
show "(\<integral>\<^isup>+ y. indicator A (x, y) \<partial>M2) = measure M2 (Pair x -` A)"
unfolding *
apply (subst M2.positive_integral_indicator)
apply (rule measurable_cut_fst[OF assms])
by simp
qed
lemma (in pair_sigma_finite) pair_measure_times:
assumes A: "A \<in> sets M1" and "B \<in> sets M2"
shows "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = M1.\<mu> A * measure M2 B"
proof -
have "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+ x. measure M2 B * indicator A x \<partial>M1)"
using assms by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)
with assms show ?thesis
by (simp add: M1.positive_integral_cmult_indicator ac_simps)
qed
lemma (in measure_space) measure_not_negative[simp,intro]:
assumes A: "A \<in> sets M" shows "\<mu> A \<noteq> - \<infinity>"
using positive_measure[OF A] by auto
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
"\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> incseq F \<and> (\<Union>i. F i) = space E \<and>
(\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
proof -
obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
F1: "range F1 \<subseteq> sets M1" "incseq F1" "(\<Union>i. F1 i) = space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<infinity>" and
F2: "range F2 \<subseteq> sets M2" "incseq F2" "(\<Union>i. F2 i) = space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<infinity>"
using M1.sigma_finite_up M2.sigma_finite_up by auto
then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
let ?F = "\<lambda>i. F1 i \<times> F2 i"
show ?thesis unfolding space_pair_measure
proof (intro exI[of _ ?F] conjI allI)
show "range ?F \<subseteq> sets E" using F1 F2
by (fastsimp intro!: pair_measure_generatorI)
next
have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
proof (intro subsetI)
fix x assume "x \<in> space M1 \<times> space M2"
then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
by (auto simp: space)
then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
using `incseq F1` `incseq F2` unfolding incseq_def
by (force split: split_max)+
then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
by (intro SigmaI) (auto simp add: min_max.sup_commute)
then show "x \<in> (\<Union>i. ?F i)" by auto
qed
then show "(\<Union>i. ?F i) = space E"
using space by (auto simp: space pair_measure_generator_def)
next
fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto
next
fix i
from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
with F1 F2 M1.positive_measure[OF this(1)] M2.positive_measure[OF this(2)]
show "measure P (F1 i \<times> F2 i) \<noteq> \<infinity>"
by (simp add: pair_measure_times)
qed
qed
sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P
proof
show "positive P (measure P)"
unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def
by (auto intro: M1.positive_integral_positive M2.positive_integral_positive)
show "countably_additive P (measure P)"
unfolding countably_additive_def
proof (intro allI impI)
fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
assume F: "range F \<subseteq> sets P" "disjoint_family F"
from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
moreover from F have "\<And>i. (\<lambda>x. measure M2 (Pair x -` F i)) \<in> borel_measurable M1"
by (intro measure_cut_measurable_fst) auto
moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2"
using F by auto
ultimately show "(\<Sum>n. measure P (F n)) = measure P (\<Union>i. F i)"
by (simp add: pair_measure_alt vimage_UN M1.positive_integral_suminf[symmetric]
M2.measure_countably_additive
cong: M1.positive_integral_cong)
qed
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. measure P (F i) \<noteq> \<infinity>)"
proof (rule exI[of _ F], intro conjI)
show "range F \<subseteq> sets P" using F by (auto simp: pair_measure_def)
show "(\<Union>i. F i) = space P"
using F by (auto simp: pair_measure_def pair_measure_generator_def)
show "\<forall>i. measure P (F i) \<noteq> \<infinity>" using F by auto
qed
qed
lemma (in pair_sigma_algebra) sets_swap:
assumes "A \<in> sets P"
shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
(is "_ -` A \<inter> space ?Q \<in> sets ?Q")
proof -
have *: "(\<lambda>(x, y). (y, x)) -` A \<inter> space ?Q = (\<lambda>(x, y). (y, x)) -` A"
using `A \<in> sets P` sets_into_space by (auto simp: space_pair_measure)
show ?thesis
unfolding * using assms by (rule sets_pair_sigma_algebra_swap)
qed
lemma (in pair_sigma_finite) pair_measure_alt2:
assumes A: "A \<in> sets P"
shows "\<mu> A = (\<integral>\<^isup>+y. M1.\<mu> ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
(is "_ = ?\<nu> A")
proof -
interpret Q: pair_sigma_finite M2 M1 by default
from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
have [simp]: "\<And>m. \<lparr> space = space E, sets = sets (sigma E), measure = m \<rparr> = P\<lparr> measure := m \<rparr>"
unfolding pair_measure_def by simp
have "\<mu> A = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` A \<inter> space Q.P)"
proof (rule measure_unique_Int_stable_vimage[OF Int_stable_pair_measure_generator])
show "measure_space P" "measure_space Q.P" by default
show "(\<lambda>(y, x). (x, y)) \<in> measurable Q.P P" by (rule Q.pair_sigma_algebra_swap_measurable)
show "sets (sigma E) = sets P" "space E = space P" "A \<in> sets (sigma E)"
using assms unfolding pair_measure_def by auto
show "range F \<subseteq> sets E" "incseq F" "(\<Union>i. F i) = space E" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
using F `A \<in> sets P` by (auto simp: pair_measure_def)
fix X assume "X \<in> sets E"
then obtain A B where X[simp]: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
unfolding pair_measure_def pair_measure_generator_def by auto
then have "(\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P = B \<times> A"
using M1.sets_into_space M2.sets_into_space by (auto simp: space_pair_measure)
then show "\<mu> X = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P)"
using AB by (simp add: pair_measure_times Q.pair_measure_times ac_simps)
qed
then show ?thesis
using sets_into_space[OF A] Q.pair_measure_alt[OF sets_swap[OF A]]
by (auto simp add: Q.pair_measure_alt space_pair_measure
intro!: M2.positive_integral_cong arg_cong[where f="M1.\<mu>"])
qed
lemma pair_sigma_algebra_sigma:
assumes 1: "incseq S1" "(\<Union>i. S1 i) = space E1" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
assumes 2: "decseq S2" "(\<Union>i. S2 i) = space E2" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
shows "sets (sigma (pair_measure_generator (sigma E1) (sigma E2))) = sets (sigma (pair_measure_generator E1 E2))"
(is "sets ?S = sets ?E")
proof -
interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)
interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)
have P: "sets (pair_measure_generator E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"
using E1 E2 by (auto simp add: pair_measure_generator_def)
interpret E: sigma_algebra ?E unfolding pair_measure_generator_def
using E1 E2 by (intro sigma_algebra_sigma) auto
{ fix A assume "A \<in> sets E1"
then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
using E1 2 unfolding pair_measure_generator_def by auto
also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
also have "\<dots> \<in> sets ?E" unfolding pair_measure_generator_def sets_sigma
using 2 `A \<in> sets E1`
by (intro sigma_sets.Union)
(force simp: image_subset_iff intro!: sigma_sets.Basic)
finally have "fst -` A \<inter> space ?E \<in> sets ?E" . }
moreover
{ fix B assume "B \<in> sets E2"
then have "snd -` B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
using E2 1 unfolding pair_measure_generator_def by auto
also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
also have "\<dots> \<in> sets ?E"
using 1 `B \<in> sets E2` unfolding pair_measure_generator_def sets_sigma
by (intro sigma_sets.Union)
(force simp: image_subset_iff intro!: sigma_sets.Basic)
finally have "snd -` B \<inter> space ?E \<in> sets ?E" . }
ultimately have proj:
"fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
(auto simp: pair_measure_generator_def sets_sigma)
{ fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
with proj have "fst -` A \<inter> space ?E \<in> sets ?E" "snd -` B \<inter> space ?E \<in> sets ?E"
unfolding measurable_def by simp_all
moreover have "A \<times> B = (fst -` A \<inter> space ?E) \<inter> (snd -` B \<inter> space ?E)"
using A B M1.sets_into_space M2.sets_into_space
by (auto simp: pair_measure_generator_def)
ultimately have "A \<times> B \<in> sets ?E" by auto }
then have "sigma_sets (space ?E) (sets (pair_measure_generator (sigma E1) (sigma E2))) \<subseteq> sets ?E"
by (intro E.sigma_sets_subset) (auto simp add: pair_measure_generator_def sets_sigma)
then have subset: "sets ?S \<subseteq> sets ?E"
by (simp add: sets_sigma pair_measure_generator_def)
show "sets ?S = sets ?E"
proof (intro set_eqI iffI)
fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
unfolding sets_sigma
proof induct
case (Basic A) then show ?case
by (auto simp: pair_measure_generator_def sets_sigma intro: sigma_sets.Basic)
qed (auto intro: sigma_sets.intros simp: pair_measure_generator_def)
next
fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
qed
qed
section "Fubinis theorem"
lemma (in pair_sigma_finite) simple_function_cut:
assumes f: "simple_function P f" "\<And>x. 0 \<le> f x"
shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
proof -
have f_borel: "f \<in> borel_measurable P"
using f(1) by (rule borel_measurable_simple_function)
let "?F z" = "f -` {z} \<inter> space P"
let "?F' x z" = "Pair x -` ?F z"
{ fix x assume "x \<in> space M1"
have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
by (auto simp: indicator_def)
have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using `x \<in> space M1`
by (simp add: space_pair_measure)
moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
by (intro borel_measurable_vimage measurable_cut_fst)
ultimately have "simple_function M2 (\<lambda> y. f (x, y))"
apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
apply (rule simple_function_indicator_representation[OF f(1)])
using `x \<in> space M1` by (auto simp del: space_sigma) }
note M2_sf = this
{ fix x assume x: "x \<in> space M1"
then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f ` space P. z * M2.\<mu> (?F' x z))"
unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]
unfolding simple_integral_def
proof (safe intro!: setsum_mono_zero_cong_left)
from f(1) show "finite (f ` space P)" by (rule simple_functionD)
next
fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space P"
using `x \<in> space M1` by (auto simp: space_pair_measure)
next
fix x' y assume "(x', y) \<in> space P"
"f (x', y) \<notin> (\<lambda>y. f (x, y)) ` space M2"
then have *: "?F' x (f (x', y)) = {}"
by (force simp: space_pair_measure)
show "f (x', y) * M2.\<mu> (?F' x (f (x', y))) = 0"
unfolding * by simp
qed (simp add: vimage_compose[symmetric] comp_def
space_pair_measure) }
note eq = this
moreover have "\<And>z. ?F z \<in> sets P"
by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
moreover then have "\<And>z. (\<lambda>x. M2.\<mu> (?F' x z)) \<in> borel_measurable M1"
by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
moreover have *: "\<And>i x. 0 \<le> M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"
using f(1)[THEN simple_functionD(2)] f(2) by (intro M2.positive_measure measurable_cut_fst)
moreover { fix i assume "i \<in> f`space P"
with * have "\<And>x. 0 \<le> i * M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"
using f(2) by auto }
ultimately
show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" using f(2)
by (auto simp del: vimage_Int cong: measurable_cong
intro!: M1.borel_measurable_extreal_setsum setsum_cong
simp add: M1.positive_integral_setsum simple_integral_def
M1.positive_integral_cmult
M1.positive_integral_cong[OF eq]
positive_integral_eq_simple_integral[OF f]
pair_measure_alt[symmetric])
qed
lemma (in pair_sigma_finite) positive_integral_fst_measurable:
assumes f: "f \<in> borel_measurable P"
shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
(is "?C f \<in> borel_measurable M1")
and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
proof -
from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this
then have F_borel: "\<And>i. F i \<in> borel_measurable P"
by (auto intro: borel_measurable_simple_function)
note sf = simple_function_cut[OF F(1,5)]
then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"
using F(1) by auto
moreover
{ fix x assume "x \<in> space M1"
from F measurable_pair_image_snd[OF F_borel`x \<in> space M1`]
have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)"
by (intro M2.positive_integral_monotone_convergence_SUP)
(auto simp: incseq_Suc_iff le_fun_def)
then have "(SUP i. ?C (F i) x) = ?C f x"
unfolding F(4) positive_integral_max_0 by simp }
note SUPR_C = this
ultimately show "?C f \<in> borel_measurable M1"
by (simp cong: measurable_cong)
have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>P) = (SUP i. integral\<^isup>P P (F i))"
using F_borel F
by (intro positive_integral_monotone_convergence_SUP) auto
also have "(SUP i. integral\<^isup>P P (F i)) = (SUP i. \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1)"
unfolding sf(2) by simp
also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1)
by (intro M1.positive_integral_monotone_convergence_SUP[symmetric])
(auto intro!: M2.positive_integral_mono M2.positive_integral_positive
simp: incseq_Suc_iff le_fun_def)
also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1"
using F_borel F(2,5)
by (auto intro!: M1.positive_integral_cong M2.positive_integral_monotone_convergence_SUP[symmetric]
simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd)
finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
using F by (simp add: positive_integral_max_0)
qed
lemma (in pair_sigma_finite) measure_preserving_swap:
"(\<lambda>(x,y). (y, x)) \<in> measure_preserving (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
proof
interpret Q: pair_sigma_finite M2 M1 by default
show *: "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
using pair_sigma_algebra_swap_measurable .
fix X assume "X \<in> sets (M2 \<Otimes>\<^isub>M M1)"
from measurable_sets[OF * this] this Q.sets_into_space[OF this]
show "measure (M1 \<Otimes>\<^isub>M M2) ((\<lambda>(x, y). (y, x)) -` X \<inter> space P) = measure (M2 \<Otimes>\<^isub>M M1) X"
by (auto intro!: M1.positive_integral_cong arg_cong[where f="M2.\<mu>"]
simp: pair_measure_alt Q.pair_measure_alt2 space_pair_measure)
qed
lemma (in pair_sigma_finite) positive_integral_product_swap:
assumes f: "f \<in> borel_measurable P"
shows "(\<integral>\<^isup>+x. f (case x of (x,y)\<Rightarrow>(y,x)) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P P f"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
have "sigma_algebra P" by default
with f show ?thesis
by (subst Q.positive_integral_vimage[OF _ Q.measure_preserving_swap]) auto
qed
lemma (in pair_sigma_finite) positive_integral_snd_measurable:
assumes f: "f \<in> borel_measurable P"
shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P P f"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
note pair_sigma_algebra_measurable[OF f]
from Q.positive_integral_fst_measurable[OF this]
have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P)"
by simp
also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P) = integral\<^isup>P P f"
unfolding positive_integral_product_swap[OF f, symmetric]
by (auto intro!: Q.positive_integral_cong)
finally show ?thesis .
qed
lemma (in pair_sigma_finite) Fubini:
assumes f: "f \<in> borel_measurable P"
shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1)"
unfolding positive_integral_snd_measurable[OF assms]
unfolding positive_integral_fst_measurable[OF assms] ..
lemma (in pair_sigma_finite) AE_pair:
assumes "AE x in P. Q x"
shows "AE x in M1. (AE y in M2. Q (x, y))"
proof -
obtain N where N: "N \<in> sets P" "\<mu> N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
using assms unfolding almost_everywhere_def by auto
show ?thesis
proof (rule M1.AE_I)
from N measure_cut_measurable_fst[OF `N \<in> sets P`]
show "M1.\<mu> {x\<in>space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} = 0"
by (auto simp: pair_measure_alt M1.positive_integral_0_iff)
show "{x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} \<in> sets M1"
by (intro M1.borel_measurable_extreal_neq_const measure_cut_measurable_fst N)
{ fix x assume "x \<in> space M1" "M2.\<mu> (Pair x -` N) = 0"
have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
proof (rule M2.AE_I)
show "M2.\<mu> (Pair x -` N) = 0" by fact
show "Pair x -` N \<in> sets M2" by (intro measurable_cut_fst N)
show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
using N `x \<in> space M1` unfolding space_sigma space_pair_measure by auto
qed }
then show "{x \<in> space M1. \<not> M2.almost_everywhere (\<lambda>y. Q (x, y))} \<subseteq> {x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0}"
by auto
qed
qed
lemma (in pair_sigma_algebra) measurable_product_swap:
"f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"
proof -
interpret Q: pair_sigma_algebra M2 M1 by default
show ?thesis
using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]
by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)
qed
lemma (in pair_sigma_finite) integrable_product_swap:
assumes "integrable P f"
shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
show ?thesis unfolding *
using assms unfolding integrable_def
apply (subst (1 2) positive_integral_product_swap)
using `integrable P f` unfolding integrable_def
by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
qed
lemma (in pair_sigma_finite) integrable_product_swap_iff:
"integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable P f"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
show ?thesis by auto
qed
lemma (in pair_sigma_finite) integral_product_swap:
assumes "integrable P f"
shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L P f"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
show ?thesis
unfolding lebesgue_integral_def *
apply (subst (1 2) positive_integral_product_swap)
using `integrable P f` unfolding integrable_def
by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
qed
lemma (in pair_sigma_finite) integrable_fst_measurable:
assumes f: "integrable P f"
shows "M1.almost_everywhere (\<lambda>x. integrable M2 (\<lambda> y. f (x, y)))" (is "?AE")
and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L P f" (is "?INT")
proof -
let "?pf x" = "extreal (f x)" and "?nf x" = "extreal (- f x)"
have
borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and
int: "integral\<^isup>P P ?nf \<noteq> \<infinity>" "integral\<^isup>P P ?pf \<noteq> \<infinity>"
using assms by auto
have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
"(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
using borel[THEN positive_integral_fst_measurable(1)] int
unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
with borel[THEN positive_integral_fst_measurable(1)]
have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
"AE x in M1. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
by (auto intro!: M1.positive_integral_PInf_AE )
then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
"AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
by (auto simp: M2.positive_integral_positive)
from AE_pos show ?AE using assms
by (simp add: measurable_pair_image_snd integrable_def)
{ fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
using M2.positive_integral_positive
by (intro M1.positive_integral_cong_pos) (auto simp: extreal_uminus_le_reorder)
then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
note this[simp]
{ fix f assume borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable P"
and int: "integral\<^isup>P P (\<lambda>x. extreal (f x)) \<noteq> \<infinity>"
and AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>)"
have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
proof (intro integrable_def[THEN iffD2] conjI)
show "?f \<in> borel_measurable M1"
using borel by (auto intro!: M1.borel_measurable_real_of_extreal positive_integral_fst_measurable)
have "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<partial>M1)"
using AE M2.positive_integral_positive
by (auto intro!: M1.positive_integral_cong_AE simp: extreal_real)
then show "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) \<noteq> \<infinity>"
using positive_integral_fst_measurable[OF borel] int by simp
have "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
by (intro M1.positive_integral_cong_pos)
(simp add: M2.positive_integral_positive real_of_extreal_pos)
then show "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
qed }
with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
show ?INT
unfolding lebesgue_integral_def[of P] lebesgue_integral_def[of M2]
borel[THEN positive_integral_fst_measurable(2), symmetric]
using AE[THEN M1.integral_real]
by simp
qed
lemma (in pair_sigma_finite) integrable_snd_measurable:
assumes f: "integrable P f"
shows "M2.almost_everywhere (\<lambda>y. integrable M1 (\<lambda>x. f (x, y)))" (is "?AE")
and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L P f" (is "?INT")
proof -
interpret Q: pair_sigma_finite M2 M1 by default
have Q_int: "integrable Q.P (\<lambda>(x, y). f (y, x))"
using f unfolding integrable_product_swap_iff .
show ?INT
using Q.integrable_fst_measurable(2)[OF Q_int]
using integral_product_swap[OF f] by simp
show ?AE
using Q.integrable_fst_measurable(1)[OF Q_int]
by simp
qed
lemma (in pair_sigma_finite) Fubini_integral:
assumes f: "integrable P f"
shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"
unfolding integrable_snd_measurable[OF assms]
unfolding integrable_fst_measurable[OF assms] ..
section "Products on finite spaces"
lemma sigma_sets_pair_measure_generator_finite:
assumes "finite A" and "finite B"
shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<in> Pow A \<and> b \<in> Pow B} = Pow (A \<times> B)"
(is "sigma_sets ?prod ?sets = _")
proof safe
have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
fix x assume subset: "x \<subseteq> A \<times> B"
hence "finite x" using fin by (rule finite_subset)
from this subset show "x \<in> sigma_sets ?prod ?sets"
proof (induct x)
case empty show ?case by (rule sigma_sets.Empty)
next
case (insert a x)
hence "{a} \<in> sigma_sets ?prod ?sets"
by (auto simp: pair_measure_generator_def intro!: sigma_sets.Basic)
moreover have "x \<in> 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 \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
from sigma_sets_into_sp[OF _ this(1)] this(2)
show "a \<in> A" and "b \<in> B" by auto
qed
locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2
for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default
lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra:
shows "P = \<lparr> space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2), \<dots> = algebra.more P \<rparr>"
proof -
show ?thesis
using sigma_sets_pair_measure_generator_finite[OF M1.finite_space M2.finite_space]
by (intro algebra.equality) (simp_all add: pair_measure_def pair_measure_generator_def sigma_def)
qed
sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P
apply default
using M1.finite_space M2.finite_space
apply (subst finite_pair_sigma_algebra) apply simp
apply (subst (1 2) finite_pair_sigma_algebra) apply simp
done
locale pair_finite_space = M1: finite_measure_space M1 + M2: finite_measure_space M2
for M1 M2
sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra
by default
sublocale pair_finite_space \<subseteq> pair_sigma_finite
by default
lemma (in pair_finite_space) pair_measure_Pair[simp]:
assumes "a \<in> space M1" "b \<in> space M2"
shows "\<mu> {(a, b)} = M1.\<mu> {a} * M2.\<mu> {b}"
proof -
have "\<mu> ({a}\<times>{b}) = M1.\<mu> {a} * M2.\<mu> {b}"
using M1.sets_eq_Pow M2.sets_eq_Pow assms
by (subst pair_measure_times) auto
then show ?thesis by simp
qed
lemma (in pair_finite_space) pair_measure_singleton[simp]:
assumes "x \<in> space M1 \<times> space M2"
shows "\<mu> {x} = M1.\<mu> {fst x} * M2.\<mu> {snd x}"
using pair_measure_Pair assms by (cases x) auto
sublocale pair_finite_space \<subseteq> finite_measure_space P
by default (auto simp: space_pair_measure)
end