theory Product_Measure
imports Lebesgue_Integration
begin
definition dynkin
where "dynkin M =
((\<forall> A \<in> sets M. A \<subseteq> space M) \<and>
space M \<in> sets M \<and> (\<forall> a \<in> sets M. \<forall> b \<in> sets M. b - a \<in> sets M) \<and>
(\<forall> a. (\<forall> i j :: nat. i \<noteq> j \<longrightarrow> a i \<inter> a j = {}) \<and>
(\<forall> i :: nat. a i \<in> sets M) \<longrightarrow> UNION UNIV a \<in> sets M))"
lemma dynkinI:
assumes "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M"
assumes "space M \<in> sets M" and "\<forall> a \<in> sets M. \<forall> b \<in> sets M. b - a \<in> sets M"
assumes "\<And> a. (\<And> i j :: nat. i \<noteq> j \<Longrightarrow> a i \<inter> a j = {})
\<Longrightarrow> (\<And> i :: nat. a i \<in> sets M) \<Longrightarrow> UNION UNIV a \<in> sets M"
shows "dynkin M"
using assms unfolding dynkin_def by auto
lemma dynkin_subset:
assumes "dynkin M"
shows "\<And> A. A \<in> sets M \<Longrightarrow> A \<subseteq> space M"
using assms unfolding dynkin_def by auto
lemma dynkin_space:
assumes "dynkin M"
shows "space M \<in> sets M"
using assms unfolding dynkin_def by auto
lemma dynkin_diff:
assumes "dynkin M"
shows "\<And> a b. \<lbrakk> a \<in> sets M ; b \<in> sets M \<rbrakk> \<Longrightarrow> b - a \<in> sets M"
using assms unfolding dynkin_def by auto
lemma dynkin_UN:
assumes "dynkin M"
assumes "\<And> i j :: nat. i \<noteq> j \<Longrightarrow> a i \<inter> a j = {}"
assumes "\<forall> i :: nat. a i \<in> sets M"
shows "UNION UNIV a \<in> sets M"
using assms unfolding dynkin_def by auto
definition Int_stable
where "Int_stable M = (\<forall> a \<in> sets M. (\<forall> b \<in> sets M. a \<inter> b \<in> sets M))"
lemma dynkin_trivial:
shows "dynkin \<lparr> space = A, sets = Pow A \<rparr>"
by (rule dynkinI) auto
lemma
assumes stab: "Int_stable E"
and spac: "space E = space D"
and subsED: "sets E \<subseteq> sets D"
and subsDE: "sets D \<subseteq> sigma_sets (space E) (sets E)"
and dyn: "dynkin D"
shows "sigma (space E) (sets E) = D"
proof -
def sets_\<delta>E == "\<Inter> {sets d | d :: 'a algebra. dynkin d \<and> space d = space E \<and> sets E \<subseteq> sets d}"
def \<delta>E == "\<lparr> space = space E, sets = sets_\<delta>E \<rparr>"
have "\<lparr> space = space E, sets = Pow (space E) \<rparr> \<in> {d | d. dynkin d \<and> space d = space E \<and> sets E \<subseteq> sets d}"
using dynkin_trivial spac subsED dynkin_subset[OF dyn] by fastsimp
hence not_empty: "{sets (d :: 'a algebra) | d. dynkin d \<and> space d = space E \<and> sets E \<subseteq> sets d} \<noteq> {}"
using exI[of "\<lambda> x. space x = space E \<and> dynkin x \<and> sets E \<subseteq> sets x" "\<lparr> space = space E, sets = Pow (space E) \<rparr>", simplified]
by auto
have "sets_\<delta>E \<subseteq> sets D"
unfolding sets_\<delta>E_def using assms by auto
have \<delta>ynkin: "dynkin \<delta>E"
proof (rule dynkinI, safe)
fix A x assume asm: "A \<in> sets \<delta>E" "x \<in> A"
{ fix d :: "('a, 'b) algebra_scheme" assume "A \<in> sets d" "dynkin d \<and> space d = space E"
hence "A \<subseteq> space d"
using dynkin_subset by auto }
show "x \<in> space \<delta>E" using asm
unfolding \<delta>E_def sets_\<delta>E_def using not_empty
proof auto
fix x A fix d :: "'a algebra"
assume asm: "\<forall>x. (\<exists>d :: 'a algebra. x = sets d \<and>
dynkin d \<and>
space d = space E \<and>
sets E \<subseteq> sets d) \<longrightarrow>
A \<in> x" "x \<in> A"
and asm': "space d = space E" "dynkin d" "sets E \<subseteq> sets d"
have "A \<in> sets d"
apply (rule impE[OF spec[OF asm(1), of "sets d"]])
using exI[of _ d] asm' by auto
thus "x \<in> space E" using asm' dynkin_subset[OF asm'(2), of A] asm(2) by auto
qed
next
show "space \<delta>E \<in> sets \<delta>E"
unfolding \<delta>E_def sets_\<delta>E_def
using dynkin_space by fastsimp
next
fix a b assume "a \<in> sets \<delta>E" "b \<in> sets \<delta>E"
thus "b - a \<in> sets \<delta>E"
unfolding \<delta>E_def sets_\<delta>E_def by (auto intro:dynkin_diff)
next
fix a assume asm: "\<And>i j :: nat. i \<noteq> j \<Longrightarrow> a i \<inter> a j = {}" "\<And>i. a i \<in> sets \<delta>E"
thus "UNION UNIV a \<in> sets \<delta>E"
unfolding \<delta>E_def sets_\<delta>E_def apply (auto intro!:dynkin_UN[OF _ asm(1)])
by blast
qed
def Dy == "\<lambda> d. {A | A. A \<in> sets_\<delta>E \<and> A \<inter> d \<in> sets_\<delta>E}"
{ fix d assume dasm: "d \<in> sets_\<delta>E"
have "dynkin \<lparr> space = space E, sets = Dy d \<rparr>"
proof (rule dynkinI, auto)
fix A x assume "A \<in> Dy d" "x \<in> A"
thus "x \<in> space E" unfolding Dy_def sets_\<delta>E_def using not_empty
proof auto fix x A fix da :: "'a algebra"
assume asm: "x \<in> A"
"\<forall>x. (\<exists>d :: 'a algebra. x = sets d \<and>
dynkin d \<and> space d = space E \<and>
sets E \<subseteq> sets d) \<longrightarrow> A \<in> x"
"\<forall>x. (\<exists>d. x = sets d \<and>
dynkin d \<and> space d = space E \<and>
sets E \<subseteq> sets d) \<longrightarrow> A \<inter> d \<in> x"
"space da = space E" "dynkin da"
"sets E \<subseteq> sets da"
have "A \<in> sets da"
apply (rule impE[OF spec[OF asm(2)], of "sets da"])
apply (rule exI[of _ da])
using exI[of _ da] asm(4,5,6) by auto
thus "x \<in> space E" using dynkin_subset[OF asm(5)] asm by auto
qed
next
show "space E \<in> Dy d"
unfolding Dy_def \<delta>E_def sets_\<delta>E_def
proof auto
fix d assume asm: "dynkin d" "space d = space E" "sets E \<subseteq> sets d"
hence "space d \<in> sets d" using dynkin_space[OF asm(1)] by auto
thus "space E \<in> sets d" using asm by auto
next
fix da :: "'a algebra" assume asm: "dynkin da" "space da = space E" "sets E \<subseteq> sets da"
have d: "d = space E \<inter> d"
using dasm dynkin_subset[OF asm(1)] asm(2) dynkin_subset[OF \<delta>ynkin]
unfolding \<delta>E_def by auto
hence "space E \<inter> d \<in> sets \<delta>E" unfolding \<delta>E_def
using dasm by auto
have "sets \<delta>E \<subseteq> sets da" unfolding \<delta>E_def sets_\<delta>E_def using asm
by auto
thus "space E \<inter> d \<in> sets da" using dasm asm d dynkin_subset[OF \<delta>ynkin]
unfolding \<delta>E_def by auto
qed
next
fix a b assume absm: "a \<in> Dy d" "b \<in> Dy d"
hence "a \<in> sets \<delta>E" "b \<in> sets \<delta>E"
unfolding Dy_def \<delta>E_def by auto
hence *: "b - a \<in> sets \<delta>E"
using dynkin_diff[OF \<delta>ynkin] by auto
have "a \<inter> d \<in> sets \<delta>E" "b \<inter> d \<in> sets \<delta>E"
using absm unfolding Dy_def \<delta>E_def by auto
hence "(b \<inter> d) - (a \<inter> d) \<in> sets \<delta>E"
using dynkin_diff[OF \<delta>ynkin] by auto
hence **: "(b - a) \<inter> d \<in> sets \<delta>E" by (auto simp add:Diff_Int_distrib2)
thus "b - a \<in> Dy d"
using * ** unfolding Dy_def \<delta>E_def by auto
next
fix a assume aasm: "\<And>i j :: nat. i \<noteq> j \<Longrightarrow> a i \<inter> a j = {}" "\<And>i. a i \<in> Dy d"
hence "\<forall> i. a i \<in> sets \<delta>E"
unfolding Dy_def \<delta>E_def by auto
from dynkin_UN[OF \<delta>ynkin aasm(1) this]
have *: "UNION UNIV a \<in> sets \<delta>E" by auto
from aasm
have aE: "\<forall> i. a i \<inter> d \<in> sets \<delta>E"
unfolding Dy_def \<delta>E_def by auto
from aasm
have "\<And>i j :: nat. i \<noteq> j \<Longrightarrow> (a i \<inter> d) \<inter> (a j \<inter> d) = {}" by auto
from dynkin_UN[OF \<delta>ynkin this]
have "UNION UNIV (\<lambda> i. a i \<inter> d) \<in> sets \<delta>E"
using aE by auto
hence **: "UNION UNIV a \<inter> d \<in> sets \<delta>E" by auto
from * ** show "UNION UNIV a \<in> Dy d" unfolding Dy_def \<delta>E_def by auto
qed } note Dy_nkin = this
have E_\<delta>E: "sets E \<subseteq> sets \<delta>E"
unfolding \<delta>E_def sets_\<delta>E_def by auto
{ fix d assume dasm: "d \<in> sets \<delta>E"
{ fix e assume easm: "e \<in> sets E"
hence deasm: "e \<in> sets \<delta>E"
unfolding \<delta>E_def sets_\<delta>E_def by auto
have subset: "Dy e \<subseteq> sets \<delta>E"
unfolding Dy_def \<delta>E_def by auto
{ fix e' assume e'asm: "e' \<in> sets E"
have "e' \<inter> e \<in> sets E"
using easm e'asm stab unfolding Int_stable_def by auto
hence "e' \<inter> e \<in> sets \<delta>E"
unfolding \<delta>E_def sets_\<delta>E_def by auto
hence "e' \<in> Dy e" using e'asm unfolding Dy_def \<delta>E_def sets_\<delta>E_def by auto }
hence E_Dy: "sets E \<subseteq> Dy e" by auto
have "\<lparr> space = space E, sets = Dy e \<rparr> \<in> {d | d. dynkin d \<and> space d = space E \<and> sets E \<subseteq> sets d}"
using Dy_nkin[OF deasm[unfolded \<delta>E_def, simplified]] E_\<delta>E E_Dy by auto
hence "sets_\<delta>E \<subseteq> Dy e"
unfolding sets_\<delta>E_def
proof auto fix x
assume asm: "\<forall>xa. (\<exists>d :: 'a algebra. xa = sets d \<and>
dynkin d \<and>
space d = space E \<and>
sets E \<subseteq> sets d) \<longrightarrow>
x \<in> xa"
"dynkin \<lparr>space = space E, sets = Dy e\<rparr>"
"sets E \<subseteq> Dy e"
show "x \<in> Dy e"
apply (rule impE[OF spec[OF asm(1), of "Dy e"]])
apply (rule exI[of _ "\<lparr>space = space E, sets = Dy e\<rparr>"])
using asm by auto
qed
hence "sets \<delta>E = Dy e" using subset unfolding \<delta>E_def by auto
hence "d \<inter> e \<in> sets \<delta>E"
using dasm easm deasm unfolding Dy_def \<delta>E_def by auto
hence "e \<in> Dy d" using deasm
unfolding Dy_def \<delta>E_def
by (auto simp add:Int_commute) }
hence "sets E \<subseteq> Dy d" by auto
hence "sets \<delta>E \<subseteq> Dy d" using Dy_nkin[OF dasm[unfolded \<delta>E_def, simplified]]
unfolding \<delta>E_def sets_\<delta>E_def
proof auto
fix x
assume asm: "sets E \<subseteq> Dy d"
"dynkin \<lparr>space = space E, sets = Dy d\<rparr>"
"\<forall>xa. (\<exists>d :: 'a algebra. xa = sets d \<and> dynkin d \<and>
space d = space E \<and> sets E \<subseteq> sets d) \<longrightarrow> x \<in> xa"
show "x \<in> Dy d"
apply (rule impE[OF spec[OF asm(3), of "Dy d"]])
apply (rule exI[of _ "\<lparr>space = space E, sets = Dy d\<rparr>"])
using asm by auto
qed
hence *: "sets \<delta>E = Dy d"
unfolding Dy_def \<delta>E_def by auto
fix a assume aasm: "a \<in> sets \<delta>E"
hence "a \<inter> d \<in> sets \<delta>E"
using * dasm unfolding Dy_def \<delta>E_def by auto } note \<delta>E_stab = this
have "sigma_algebra D"
apply unfold_locales
using dynkin_subset[OF dyn]
using dynkin_diff[OF dyn, of _ "space D", OF _ dynkin_space[OF dyn]]
using dynkin_diff[OF dyn, of "space D" "space D", OF dynkin_space[OF dyn] dynkin_space[OF dyn]]
using dynkin_space[OF dyn]
proof auto
fix A :: "nat \<Rightarrow> 'a set" assume Asm: "range A \<subseteq> sets D" "\<And>A. A \<in> sets D \<Longrightarrow> A \<subseteq> space D"
"\<And>a. a \<in> sets D \<Longrightarrow> space D - a \<in> sets D"
"{} \<in> sets D" "space D \<in> sets D"
let "?A i" = "A i - (\<Inter> j \<in> {..< i}. A j)"
{ fix i :: nat assume "i > 0"
have "(\<Inter> j \<in> {..< i}. A j) \<in> sets \<delta>E"
apply (induct i)
apply auto
}
from dynkin_UN
qed
qed
lemma
(*
definition prod_sets where
"prod_sets A B = {z. \<exists>x \<in> A. \<exists>y \<in> B. z = x \<times> y}"
definition
"prod_measure M \<mu> N \<nu> = (\<lambda>A. measure_space.positive_integral M \<mu> (\<lambda>s0. \<nu> ((\<lambda>s1. (s0, s1)) -` A)))"
definition
"prod_measure_space M1 M2 = sigma (space M1 \<times> space M2) (prod_sets (sets M1) (sets M2))"
lemma prod_setsI: "x \<in> A \<Longrightarrow> y \<in> B \<Longrightarrow> (x \<times> y) \<in> prod_sets A B"
by (auto simp add: prod_sets_def)
lemma sigma_prod_sets_finite:
assumes "finite A" and "finite B"
shows "sigma_sets (A \<times> B) (prod_sets (Pow A) (Pow B)) = Pow (A \<times> B)"
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 (A\<times>B) (prod_sets (Pow A) (Pow B))"
(is "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: prod_sets_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 (A\<times>B) (prod_sets (Pow A) (Pow B))" 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 simp: prod_sets_def)
qed
lemma (in sigma_algebra) measurable_prod_sigma:
assumes sa1: "sigma_algebra a1" and sa2: "sigma_algebra a2"
assumes 1: "(fst o f) \<in> measurable M a1" and 2: "(snd o f) \<in> measurable M a2"
shows "f \<in> measurable M (sigma ((space a1) \<times> (space a2))
(prod_sets (sets a1) (sets a2)))"
proof -
from 1 have fn1: "fst \<circ> f \<in> space M \<rightarrow> space a1"
and q1: "\<forall>y\<in>sets a1. (fst \<circ> f) -` y \<inter> space M \<in> sets M"
by (auto simp add: measurable_def)
from 2 have fn2: "snd \<circ> f \<in> space M \<rightarrow> space a2"
and q2: "\<forall>y\<in>sets a2. (snd \<circ> f) -` y \<inter> space M \<in> sets M"
by (auto simp add: measurable_def)
show ?thesis
proof (rule measurable_sigma)
show "prod_sets (sets a1) (sets a2) \<subseteq> Pow (space a1 \<times> space a2)" using sa1 sa2
by (auto simp add: prod_sets_def sigma_algebra_iff dest: algebra.space_closed)
next
show "f \<in> space M \<rightarrow> space a1 \<times> space a2"
by (rule prod_final [OF fn1 fn2])
next
fix z
assume z: "z \<in> prod_sets (sets a1) (sets a2)"
thus "f -` z \<inter> space M \<in> sets M"
proof (auto simp add: prod_sets_def vimage_Times)
fix x y
assume x: "x \<in> sets a1" and y: "y \<in> sets a2"
have "(fst \<circ> f) -` x \<inter> (snd \<circ> f) -` y \<inter> space M =
((fst \<circ> f) -` x \<inter> space M) \<inter> ((snd \<circ> f) -` y \<inter> space M)"
by blast
also have "... \<in> sets M" using x y q1 q2
by blast
finally show "(fst \<circ> f) -` x \<inter> (snd \<circ> f) -` y \<inter> space M \<in> sets M" .
qed
qed
qed
lemma (in sigma_finite_measure) prod_measure_times:
assumes "sigma_finite_measure N \<nu>"
and "A1 \<in> sets M" "A2 \<in> sets N"
shows "prod_measure M \<mu> N \<nu> (A1 \<times> A2) = \<mu> A1 * \<nu> A2"
oops
lemma (in sigma_finite_measure) sigma_finite_prod_measure_space:
assumes "sigma_finite_measure N \<nu>"
shows "sigma_finite_measure (prod_measure_space M N) (prod_measure M \<mu> N \<nu>)"
oops
lemma (in finite_measure_space) finite_prod_measure_times:
assumes "finite_measure_space N \<nu>"
and "A1 \<in> sets M" "A2 \<in> sets N"
shows "prod_measure M \<mu> N \<nu> (A1 \<times> A2) = \<mu> A1 * \<nu> A2"
proof -
interpret N: finite_measure_space N \<nu> by fact
have *: "\<And>x. \<nu> (Pair x -` (A1 \<times> A2)) * \<mu> {x} = (if x \<in> A1 then \<nu> A2 * \<mu> {x} else 0)"
by (auto simp: vimage_Times comp_def)
have "finite A1"
using `A1 \<in> sets M` finite_space by (auto simp: sets_eq_Pow intro: finite_subset)
then have "\<mu> A1 = (\<Sum>x\<in>A1. \<mu> {x})" using `A1 \<in> sets M`
by (auto intro!: measure_finite_singleton simp: sets_eq_Pow)
then show ?thesis using `A1 \<in> sets M`
unfolding prod_measure_def positive_integral_finite_eq_setsum *
by (auto simp add: sets_eq_Pow setsum_right_distrib[symmetric] mult_commute setsum_cases[OF finite_space])
qed
lemma (in finite_measure_space) finite_prod_measure_space:
assumes "finite_measure_space N \<nu>"
shows "prod_measure_space M N = \<lparr> space = space M \<times> space N, sets = Pow (space M \<times> space N) \<rparr>"
proof -
interpret N: finite_measure_space N \<nu> by fact
show ?thesis using finite_space N.finite_space
by (simp add: sigma_def prod_measure_space_def sigma_prod_sets_finite sets_eq_Pow N.sets_eq_Pow)
qed
lemma (in finite_measure_space) finite_measure_space_finite_prod_measure:
assumes "finite_measure_space N \<nu>"
shows "finite_measure_space (prod_measure_space M N) (prod_measure M \<mu> N \<nu>)"
unfolding finite_prod_measure_space[OF assms]
proof (rule finite_measure_spaceI)
interpret N: finite_measure_space N \<nu> by fact
let ?P = "\<lparr>space = space M \<times> space N, sets = Pow (space M \<times> space N)\<rparr>"
show "measure_space ?P (prod_measure M \<mu> N \<nu>)"
proof (rule sigma_algebra.finite_additivity_sufficient)
show "sigma_algebra ?P" by (rule sigma_algebra_Pow)
show "finite (space ?P)" using finite_space N.finite_space by auto
from finite_prod_measure_times[OF assms, of "{}" "{}"]
show "positive (prod_measure M \<mu> N \<nu>)"
unfolding positive_def by simp
show "additive ?P (prod_measure M \<mu> N \<nu>)"
unfolding additive_def
apply (auto simp add: sets_eq_Pow prod_measure_def positive_integral_add[symmetric]
intro!: positive_integral_cong)
apply (subst N.measure_additive[symmetric])
by (auto simp: N.sets_eq_Pow sets_eq_Pow)
qed
show "finite (space ?P)" using finite_space N.finite_space by auto
show "sets ?P = Pow (space ?P)" by simp
fix x assume "x \<in> space ?P"
with finite_prod_measure_times[OF assms, of "{fst x}" "{snd x}"]
finite_measure[of "{fst x}"] N.finite_measure[of "{snd x}"]
show "prod_measure M \<mu> N \<nu> {x} \<noteq> \<omega>"
by (auto simp add: sets_eq_Pow N.sets_eq_Pow elim!: SigmaE)
qed
lemma (in finite_measure_space) finite_measure_space_finite_prod_measure_alterantive:
assumes N: "finite_measure_space N \<nu>"
shows "finite_measure_space \<lparr> space = space M \<times> space N, sets = Pow (space M \<times> space N) \<rparr> (prod_measure M \<mu> N \<nu>)"
(is "finite_measure_space ?M ?m")
unfolding finite_prod_measure_space[OF N, symmetric]
using finite_measure_space_finite_prod_measure[OF N] .
*)
end