(* Title: HOL/Probability/Finite_Product_Measure.thy
Author: Johannes Hölzl, TU München
*)
header {*Finite product measures*}
theory Finite_Product_Measure
imports Binary_Product_Measure
begin
lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
unfolding Pi_def by auto
abbreviation
"Pi\<^isub>E A B \<equiv> Pi A B \<inter> extensional A"
syntax
"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3PIE _:_./ _)" 10)
syntax (xsymbols)
"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>\<^isub>E _\<in>_./ _)" 10)
syntax (HTML output)
"_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\<Pi>\<^isub>E _\<in>_./ _)" 10)
translations
"PIE x:A. B" == "CONST Pi\<^isub>E A (%x. B)"
abbreviation
funcset_extensional :: "['a set, 'b set] => ('a => 'b) set"
(infixr "->\<^isub>E" 60) where
"A ->\<^isub>E B \<equiv> Pi\<^isub>E A (%_. B)"
notation (xsymbols)
funcset_extensional (infixr "\<rightarrow>\<^isub>E" 60)
lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
by safe (auto simp add: extensional_def fun_eq_iff)
lemma extensional_insert[intro, simp]:
assumes "a \<in> extensional (insert i I)"
shows "a(i := b) \<in> extensional (insert i I)"
using assms unfolding extensional_def by auto
lemma extensional_Int[simp]:
"extensional I \<inter> extensional I' = extensional (I \<inter> I')"
unfolding extensional_def by auto
definition
"merge I x J y = (\<lambda>i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
lemma merge_apply[simp]:
"I \<inter> J = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
"I \<inter> J = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
"J \<inter> I = {} \<Longrightarrow> i \<in> I \<Longrightarrow> merge I x J y i = x i"
"J \<inter> I = {} \<Longrightarrow> i \<in> J \<Longrightarrow> merge I x J y i = y i"
"i \<notin> I \<Longrightarrow> i \<notin> J \<Longrightarrow> merge I x J y i = undefined"
unfolding merge_def by auto
lemma merge_commute:
"I \<inter> J = {} \<Longrightarrow> merge I x J y = merge J y I x"
by (auto simp: merge_def intro!: ext)
lemma Pi_cancel_merge_range[simp]:
"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
"I \<inter> J = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge I A J B) \<longleftrightarrow> x \<in> Pi I A"
"J \<inter> I = {} \<Longrightarrow> x \<in> Pi I (merge J B I A) \<longleftrightarrow> x \<in> Pi I A"
by (auto simp: Pi_def)
lemma Pi_cancel_merge[simp]:
"I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
"J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
"I \<inter> J = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
"J \<inter> I = {} \<Longrightarrow> merge I x J y \<in> Pi J B \<longleftrightarrow> y \<in> Pi J B"
by (auto simp: Pi_def)
lemma extensional_merge[simp]: "merge I x J y \<in> extensional (I \<union> J)"
by (auto simp: extensional_def)
lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
by (auto simp: restrict_def Pi_def)
lemma restrict_merge[simp]:
"I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
"I \<inter> J = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
"J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) I = restrict x I"
"J \<inter> I = {} \<Longrightarrow> restrict (merge I x J y) J = restrict y J"
by (auto simp: restrict_def intro!: ext)
lemma extensional_insert_undefined[intro, simp]:
assumes "a \<in> extensional (insert i I)"
shows "a(i := undefined) \<in> extensional I"
using assms unfolding extensional_def by auto
lemma extensional_insert_cancel[intro, simp]:
assumes "a \<in> extensional I"
shows "a \<in> extensional (insert i I)"
using assms unfolding extensional_def by auto
lemma merge_singleton[simp]: "i \<notin> I \<Longrightarrow> merge I x {i} y = restrict (x(i := y i)) (insert i I)"
unfolding merge_def by (auto simp: fun_eq_iff)
lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
by auto
lemma PiE_Int: "(Pi\<^isub>E I A) \<inter> (Pi\<^isub>E I B) = Pi\<^isub>E I (\<lambda>x. A x \<inter> B x)"
by auto
lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
by (auto simp: Pi_def)
lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i"
by (auto simp: Pi_def)
lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
by (auto simp: Pi_def)
lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
by (auto simp: Pi_def)
lemma restrict_vimage:
assumes "I \<inter> J = {}"
shows "(\<lambda>x. (restrict x I, restrict x J)) -` (Pi\<^isub>E I E \<times> Pi\<^isub>E J F) = Pi (I \<union> J) (merge I E J F)"
using assms by (auto simp: restrict_Pi_cancel)
lemma merge_vimage:
assumes "I \<inter> J = {}"
shows "(\<lambda>(x,y). merge I x J y) -` Pi\<^isub>E (I \<union> J) E = Pi I E \<times> Pi J E"
using assms by (auto simp: restrict_Pi_cancel)
lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
by (auto simp: restrict_def intro!: ext)
lemma merge_restrict[simp]:
"merge I (restrict x I) J y = merge I x J y"
"merge I x J (restrict y J) = merge I x J y"
unfolding merge_def by (auto intro!: ext)
lemma merge_x_x_eq_restrict[simp]:
"merge I x J x = restrict x (I \<union> J)"
unfolding merge_def by (auto intro!: ext)
lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A"
apply auto
apply (drule_tac x=x in Pi_mem)
apply (simp_all split: split_if_asm)
apply (drule_tac x=i in Pi_mem)
apply (auto dest!: Pi_mem)
done
lemma Pi_UN:
fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
proof (intro set_eqI iffI)
fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto
from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
have "f \<in> Pi I (A k)"
proof (intro Pi_I)
fix i assume "i \<in> I"
from mono[OF this, of "n i" k] k[OF this] n[OF this]
show "f i \<in> A k i" by auto
qed
then show "f \<in> (\<Union>n. Pi I (A n))" by auto
qed auto
lemma PiE_cong:
assumes "\<And>i. i\<in>I \<Longrightarrow> A i = B i"
shows "Pi\<^isub>E I A = Pi\<^isub>E I B"
using assms by (auto intro!: Pi_cong)
lemma restrict_upd[simp]:
"i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
by (auto simp: fun_eq_iff)
lemma Pi_eq_subset:
assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
assumes eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and "i \<in> I"
shows "F i \<subseteq> F' i"
proof
fix x assume "x \<in> F i"
with ne have "\<forall>j. \<exists>y. ((j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined))" by auto
from choice[OF this] guess f .. note f = this
then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
then have "f \<in> Pi\<^isub>E I F'" using assms by simp
then show "x \<in> F' i" using f `i \<in> I` by auto
qed
lemma Pi_eq_iff_not_empty:
assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
shows "Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
proof (intro iffI ballI)
fix i assume eq: "Pi\<^isub>E I F = Pi\<^isub>E I F'" and i: "i \<in> I"
show "F i = F' i"
using Pi_eq_subset[of I F F', OF ne eq i]
using Pi_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
by auto
qed auto
lemma Pi_eq_empty_iff:
"Pi\<^isub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
proof
assume "Pi\<^isub>E I F = {}"
show "\<exists>i\<in>I. F i = {}"
proof (rule ccontr)
assume "\<not> ?thesis"
then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto
from choice[OF this] guess f ..
then have "f \<in> Pi\<^isub>E I F" by (auto simp: extensional_def)
with `Pi\<^isub>E I F = {}` show False by auto
qed
qed auto
lemma Pi_eq_iff:
"Pi\<^isub>E I F = Pi\<^isub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
proof (intro iffI disjCI)
assume eq[simp]: "Pi\<^isub>E I F = Pi\<^isub>E I F'"
assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
with Pi_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto
next
assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
then show "Pi\<^isub>E I F = Pi\<^isub>E I F'"
using Pi_eq_empty_iff[of I F] Pi_eq_empty_iff[of I F'] by auto
qed
section "Finite product spaces"
section "Products"
locale product_sigma_algebra =
fixes M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme"
assumes sigma_algebras: "\<And>i. sigma_algebra (M i)"
locale finite_product_sigma_algebra = product_sigma_algebra M
for M :: "'i \<Rightarrow> ('a, 'b) measure_space_scheme" +
fixes I :: "'i set"
assumes finite_index[simp, intro]: "finite I"
definition
"product_algebra_generator I M = \<lparr> space = (\<Pi>\<^isub>E i \<in> I. space (M i)),
sets = Pi\<^isub>E I ` (\<Pi> i \<in> I. sets (M i)),
measure = \<lambda>A. (\<Prod>i\<in>I. measure (M i) ((SOME F. A = Pi\<^isub>E I F) i)) \<rparr>"
definition product_algebra_def:
"Pi\<^isub>M I M = sigma (product_algebra_generator I M)
\<lparr> measure := (SOME \<mu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<mu> \<rparr>) \<and>
(\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))))\<rparr>"
syntax
"_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
('i => 'a, 'b) measure_space_scheme" ("(3PIM _:_./ _)" 10)
syntax (xsymbols)
"_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
('i => 'a, 'b) measure_space_scheme" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10)
syntax (HTML output)
"_PiM" :: "[pttrn, 'i set, ('a, 'b) measure_space_scheme] =>
('i => 'a, 'b) measure_space_scheme" ("(3\<Pi>\<^isub>M _\<in>_./ _)" 10)
translations
"PIM x:I. M" == "CONST Pi\<^isub>M I (%x. M)"
abbreviation (in finite_product_sigma_algebra) "G \<equiv> product_algebra_generator I M"
abbreviation (in finite_product_sigma_algebra) "P \<equiv> Pi\<^isub>M I M"
sublocale product_sigma_algebra \<subseteq> M: sigma_algebra "M i" for i by (rule sigma_algebras)
lemma sigma_into_space:
assumes "sets M \<subseteq> Pow (space M)"
shows "sets (sigma M) \<subseteq> Pow (space M)"
using sigma_sets_into_sp[OF assms] unfolding sigma_def by auto
lemma (in product_sigma_algebra) product_algebra_generator_into_space:
"sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"
using M.sets_into_space unfolding product_algebra_generator_def
by auto blast
lemma (in product_sigma_algebra) product_algebra_into_space:
"sets (Pi\<^isub>M I M) \<subseteq> Pow (space (Pi\<^isub>M I M))"
using product_algebra_generator_into_space
by (auto intro!: sigma_into_space simp add: product_algebra_def)
lemma (in product_sigma_algebra) sigma_algebra_product_algebra: "sigma_algebra (Pi\<^isub>M I M)"
using product_algebra_generator_into_space unfolding product_algebra_def
by (rule sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) simp_all
sublocale finite_product_sigma_algebra \<subseteq> sigma_algebra P
using sigma_algebra_product_algebra .
lemma product_algebraE:
assumes "A \<in> sets (product_algebra_generator I M)"
obtains E where "A = Pi\<^isub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
using assms unfolding product_algebra_generator_def by auto
lemma product_algebra_generatorI[intro]:
assumes "E \<in> (\<Pi> i\<in>I. sets (M i))"
shows "Pi\<^isub>E I E \<in> sets (product_algebra_generator I M)"
using assms unfolding product_algebra_generator_def by auto
lemma space_product_algebra_generator[simp]:
"space (product_algebra_generator I M) = Pi\<^isub>E I (\<lambda>i. space (M i))"
unfolding product_algebra_generator_def by simp
lemma space_product_algebra[simp]:
"space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E i\<in>I. space (M i))"
unfolding product_algebra_def product_algebra_generator_def by simp
lemma sets_product_algebra:
"sets (Pi\<^isub>M I M) = sets (sigma (product_algebra_generator I M))"
unfolding product_algebra_def sigma_def by simp
lemma product_algebra_generator_sets_into_space:
assumes "\<And>i. i\<in>I \<Longrightarrow> sets (M i) \<subseteq> Pow (space (M i))"
shows "sets (product_algebra_generator I M) \<subseteq> Pow (space (product_algebra_generator I M))"
using assms by (auto simp: product_algebra_generator_def) blast
lemma (in finite_product_sigma_algebra) in_P[simp, intro]:
"\<lbrakk> \<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i) \<rbrakk> \<Longrightarrow> Pi\<^isub>E I A \<in> sets P"
by (auto simp: sets_product_algebra)
lemma Int_stable_product_algebra_generator:
"(\<And>i. i \<in> I \<Longrightarrow> Int_stable (M i)) \<Longrightarrow> Int_stable (product_algebra_generator I M)"
by (auto simp add: product_algebra_generator_def Int_stable_def PiE_Int Pi_iff)
section "Generating set generates also product algebra"
lemma sigma_product_algebra_sigma_eq:
assumes "finite I"
assumes mono: "\<And>i. i \<in> I \<Longrightarrow> incseq (S i)"
assumes union: "\<And>i. i \<in> I \<Longrightarrow> (\<Union>j. S i j) = space (E i)"
assumes sets_into: "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> sets (E i)"
and E: "\<And>i. sets (E i) \<subseteq> Pow (space (E i))"
shows "sets (\<Pi>\<^isub>M i\<in>I. sigma (E i)) = sets (\<Pi>\<^isub>M i\<in>I. E i)"
(is "sets ?S = sets ?E")
proof cases
assume "I = {}" then show ?thesis
by (simp add: product_algebra_def product_algebra_generator_def)
next
assume "I \<noteq> {}"
interpret E: sigma_algebra "sigma (E i)" for i
using E by (rule sigma_algebra_sigma)
have into_space[intro]: "\<And>i x A. A \<in> sets (E i) \<Longrightarrow> x i \<in> A \<Longrightarrow> x i \<in> space (E i)"
using E by auto
interpret G: sigma_algebra ?E
unfolding product_algebra_def product_algebra_generator_def using E
by (intro sigma_algebra.sigma_algebra_cong[OF sigma_algebra_sigma]) (auto dest: Pi_mem)
{ fix A i assume "i \<in> I" and A: "A \<in> sets (E i)"
then have "(\<lambda>x. x i) -` A \<inter> space ?E = (\<Pi>\<^isub>E j\<in>I. if j = i then A else \<Union>n. S j n) \<inter> space ?E"
using mono union unfolding incseq_Suc_iff space_product_algebra
by (auto dest: Pi_mem)
also have "\<dots> = (\<Union>n. (\<Pi>\<^isub>E j\<in>I. if j = i then A else S j n))"
unfolding space_product_algebra
apply simp
apply (subst Pi_UN[OF `finite I`])
using mono[THEN incseqD] apply simp
apply (simp add: PiE_Int)
apply (intro PiE_cong)
using A sets_into by (auto intro!: into_space)
also have "\<dots> \<in> sets ?E"
using sets_into `A \<in> sets (E i)`
unfolding sets_product_algebra sets_sigma
by (intro sigma_sets.Union)
(auto simp: image_subset_iff intro!: sigma_sets.Basic)
finally have "(\<lambda>x. x i) -` A \<inter> space ?E \<in> sets ?E" . }
then have proj:
"\<And>i. i\<in>I \<Longrightarrow> (\<lambda>x. x i) \<in> measurable ?E (sigma (E i))"
using E by (subst G.measurable_iff_sigma)
(auto simp: sets_product_algebra sets_sigma)
{ fix A assume A: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (sigma (E i))"
with proj have basic: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) -` (A i) \<inter> space ?E \<in> sets ?E"
unfolding measurable_def by simp
have "Pi\<^isub>E I A = (\<Inter>i\<in>I. (\<lambda>x. x i) -` (A i) \<inter> space ?E)"
using A E.sets_into_space `I \<noteq> {}` unfolding product_algebra_def by auto blast
then have "Pi\<^isub>E I A \<in> sets ?E"
using G.finite_INT[OF `finite I` `I \<noteq> {}` basic, of "\<lambda>i. i"] by simp }
then have "sigma_sets (space ?E) (sets (product_algebra_generator I (\<lambda>i. sigma (E i)))) \<subseteq> sets ?E"
by (intro G.sigma_sets_subset) (auto simp add: product_algebra_generator_def)
then have subset: "sets ?S \<subseteq> sets ?E"
by (simp add: sets_sigma sets_product_algebra)
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 sets_product_algebra
proof induct
case (Basic A) then show ?case
by (auto simp: sets_sigma product_algebra_generator_def intro: sigma_sets.Basic)
qed (auto intro: sigma_sets.intros simp: product_algebra_generator_def)
next
fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
qed
qed
lemma product_algebraI[intro]:
"E \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> Pi\<^isub>E I E \<in> sets (Pi\<^isub>M I M)"
using assms unfolding product_algebra_def by (auto intro: product_algebra_generatorI)
lemma (in product_sigma_algebra) measurable_component_update:
assumes "x \<in> space (Pi\<^isub>M I M)" and "i \<notin> I"
shows "(\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^isub>M (insert i I) M)" (is "?f \<in> _")
unfolding product_algebra_def apply simp
proof (intro measurable_sigma)
let ?G = "product_algebra_generator (insert i I) M"
show "sets ?G \<subseteq> Pow (space ?G)" using product_algebra_generator_into_space .
show "?f \<in> space (M i) \<rightarrow> space ?G"
using M.sets_into_space assms by auto
fix A assume "A \<in> sets ?G"
from product_algebraE[OF this] guess E . note E = this
then have "?f -` A \<inter> space (M i) = E i \<or> ?f -` A \<inter> space (M i) = {}"
using M.sets_into_space assms by auto
then show "?f -` A \<inter> space (M i) \<in> sets (M i)"
using E by (auto intro!: product_algebraI)
qed
lemma (in product_sigma_algebra) measurable_add_dim:
assumes "i \<notin> I"
shows "(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) (Pi\<^isub>M (insert i I) M)"
proof -
let ?f = "(\<lambda>(f, y). f(i := y))" and ?G = "product_algebra_generator (insert i I) M"
interpret Ii: pair_sigma_algebra "Pi\<^isub>M I M" "M i"
unfolding pair_sigma_algebra_def
by (intro sigma_algebra_product_algebra sigma_algebras conjI)
have "?f \<in> measurable Ii.P (sigma ?G)"
proof (rule Ii.measurable_sigma)
show "sets ?G \<subseteq> Pow (space ?G)"
using product_algebra_generator_into_space .
show "?f \<in> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<rightarrow> space ?G"
by (auto simp: space_pair_measure)
next
fix A assume "A \<in> sets ?G"
then obtain F where "A = Pi\<^isub>E (insert i I) F"
and F: "\<And>j. j \<in> I \<Longrightarrow> F j \<in> sets (M j)" "F i \<in> sets (M i)"
by (auto elim!: product_algebraE)
then have "?f -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) = Pi\<^isub>E I F \<times> (F i)"
using sets_into_space `i \<notin> I`
by (auto simp add: space_pair_measure) blast+
then show "?f -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M M i) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M M i)"
using F by (auto intro!: pair_measureI)
qed
then show ?thesis
by (simp add: product_algebra_def)
qed
lemma (in product_sigma_algebra) measurable_merge:
assumes [simp]: "I \<inter> J = {}"
shows "(\<lambda>(x, y). merge I x J y) \<in> measurable (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
proof -
let ?I = "Pi\<^isub>M I M" and ?J = "Pi\<^isub>M J M"
interpret P: sigma_algebra "?I \<Otimes>\<^isub>M ?J"
by (intro sigma_algebra_pair_measure product_algebra_into_space)
let ?f = "\<lambda>(x, y). merge I x J y"
let ?G = "product_algebra_generator (I \<union> J) M"
have "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (sigma ?G)"
proof (rule P.measurable_sigma)
fix A assume "A \<in> sets ?G"
from product_algebraE[OF this]
obtain E where E: "A = Pi\<^isub>E (I \<union> J) E" "E \<in> (\<Pi> i\<in>I \<union> J. sets (M i))" .
then have *: "?f -` A \<inter> space (?I \<Otimes>\<^isub>M ?J) = Pi\<^isub>E I E \<times> Pi\<^isub>E J E"
using sets_into_space `I \<inter> J = {}`
by (auto simp add: space_pair_measure) fast+
then show "?f -` A \<inter> space (?I \<Otimes>\<^isub>M ?J) \<in> sets (?I \<Otimes>\<^isub>M ?J)"
using E unfolding * by (auto intro!: pair_measureI in_sigma product_algebra_generatorI
simp: product_algebra_def)
qed (insert product_algebra_generator_into_space, auto simp: space_pair_measure)
then show "?f \<in> measurable (?I \<Otimes>\<^isub>M ?J) (Pi\<^isub>M (I \<union> J) M)"
unfolding product_algebra_def[of "I \<union> J"] by simp
qed
lemma (in product_sigma_algebra) measurable_component_singleton:
assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>M I M) (M i)"
proof (unfold measurable_def, intro CollectI conjI ballI)
fix A assume "A \<in> sets (M i)"
then have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) = (\<Pi>\<^isub>E j\<in>I. if i = j then A else space (M j))"
using M.sets_into_space `i \<in> I` by (fastforce dest: Pi_mem split: split_if_asm)
then show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>M I M) \<in> sets (Pi\<^isub>M I M)"
using `A \<in> sets (M i)` by (auto intro!: product_algebraI)
qed (insert `i \<in> I`, auto)
lemma (in sigma_algebra) measurable_restrict:
assumes I: "finite I"
assumes "\<And>i. i \<in> I \<Longrightarrow> sets (N i) \<subseteq> Pow (space (N i))"
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable M (N i)"
shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)"
unfolding product_algebra_def
proof (simp, rule measurable_sigma)
show "sets (product_algebra_generator I N) \<subseteq> Pow (space (product_algebra_generator I N))"
by (rule product_algebra_generator_sets_into_space) fact
show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> space M \<rightarrow> space (product_algebra_generator I N)"
using X by (auto simp: measurable_def)
fix E assume "E \<in> sets (product_algebra_generator I N)"
then obtain F where "E = Pi\<^isub>E I F" and F: "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets (N i)"
by (auto simp: product_algebra_generator_def)
then have "(\<lambda>x. \<lambda>i\<in>I. X i x) -` E \<inter> space M = (\<Inter>i\<in>I. X i -` F i \<inter> space M) \<inter> space M"
by (auto simp: Pi_iff)
also have "\<dots> \<in> sets M"
proof cases
assume "I = {}" then show ?thesis by simp
next
assume "I \<noteq> {}" with X F I show ?thesis
by (intro finite_INT measurable_sets Int) auto
qed
finally show "(\<lambda>x. \<lambda>i\<in>I. X i x) -` E \<inter> space M \<in> sets M" .
qed
locale product_sigma_finite = product_sigma_algebra M
for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" +
assumes sigma_finite_measures: "\<And>i. sigma_finite_measure (M i)"
sublocale product_sigma_finite \<subseteq> M: sigma_finite_measure "M i" for i
by (rule sigma_finite_measures)
locale finite_product_sigma_finite = finite_product_sigma_algebra M I + product_sigma_finite M
for M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme" and I :: "'i set"
lemma (in finite_product_sigma_finite) product_algebra_generator_measure:
assumes "Pi\<^isub>E I F \<in> sets G"
shows "measure G (Pi\<^isub>E I F) = (\<Prod>i\<in>I. M.\<mu> i (F i))"
proof cases
assume ne: "\<forall>i\<in>I. F i \<noteq> {}"
have "\<forall>i\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') i = F i"
by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
(insert ne, auto simp: Pi_eq_iff)
then show ?thesis
unfolding product_algebra_generator_def by simp
next
assume empty: "\<not> (\<forall>j\<in>I. F j \<noteq> {})"
then have "(\<Prod>j\<in>I. M.\<mu> j (F j)) = 0"
by (auto simp: setprod_ereal_0 intro!: bexI)
moreover
have "\<exists>j\<in>I. (SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j = {}"
by (rule someI2[where P="\<lambda>F'. Pi\<^isub>E I F = Pi\<^isub>E I F'"])
(insert empty, auto simp: Pi_eq_empty_iff[symmetric])
then have "(\<Prod>j\<in>I. M.\<mu> j ((SOME F'. Pi\<^isub>E I F = Pi\<^isub>E I F') j)) = 0"
by (auto simp: setprod_ereal_0 intro!: bexI)
ultimately show ?thesis
unfolding product_algebra_generator_def by simp
qed
lemma (in finite_product_sigma_finite) sigma_finite_pairs:
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
(\<forall>k. \<forall>i\<in>I. \<mu> i (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k) \<and>
(\<Union>k. \<Pi>\<^isub>E i\<in>I. F i k) = space G"
proof -
have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. \<mu> i (F k) \<noteq> \<infinity>)"
using M.sigma_finite_up by simp
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
then have F: "\<And>i. range (F i) \<subseteq> sets (M i)" "\<And>i. incseq (F i)" "\<And>i. (\<Union>j. F i j) = space (M i)" "\<And>i k. \<mu> i (F i k) \<noteq> \<infinity>"
by auto
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I. F i k"
note space_product_algebra[simp]
show ?thesis
proof (intro exI[of _ F] conjI allI incseq_SucI set_eqI iffI ballI)
fix i show "range (F i) \<subseteq> sets (M i)" by fact
next
fix i k show "\<mu> i (F i k) \<noteq> \<infinity>" by fact
next
fix A assume "A \<in> (\<Union>i. ?F i)" then show "A \<in> space G"
using `\<And>i. range (F i) \<subseteq> sets (M i)` M.sets_into_space
by (force simp: image_subset_iff)
next
fix f assume "f \<in> space G"
with Pi_UN[OF finite_index, of "\<lambda>k i. F i k"] F
show "f \<in> (\<Union>i. ?F i)" by (auto simp: incseq_def)
next
fix i show "?F i \<subseteq> ?F (Suc i)"
using `\<And>i. incseq (F i)`[THEN incseq_SucD] by auto
qed
qed
lemma sets_pair_cancel_measure[simp]:
"sets (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) = sets (M1 \<Otimes>\<^isub>M M2)"
"sets (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) = sets (M1 \<Otimes>\<^isub>M M2)"
unfolding pair_measure_def pair_measure_generator_def sets_sigma
by simp_all
lemma measurable_pair_cancel_measure[simp]:
"measurable (M1\<lparr>measure := m1\<rparr> \<Otimes>\<^isub>M M2) M = measurable (M1 \<Otimes>\<^isub>M M2) M"
"measurable (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m2\<rparr>) M = measurable (M1 \<Otimes>\<^isub>M M2) M"
"measurable M (M1\<lparr>measure := m3\<rparr> \<Otimes>\<^isub>M M2) = measurable M (M1 \<Otimes>\<^isub>M M2)"
"measurable M (M1 \<Otimes>\<^isub>M M2\<lparr>measure := m4\<rparr>) = measurable M (M1 \<Otimes>\<^isub>M M2)"
unfolding measurable_def by (auto simp add: space_pair_measure)
lemma (in product_sigma_finite) product_measure_exists:
assumes "finite I"
shows "\<exists>\<nu>. sigma_finite_measure (sigma (product_algebra_generator I M) \<lparr> measure := \<nu> \<rparr>) \<and>
(\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i)))"
using `finite I` proof induct
case empty
interpret finite_product_sigma_finite M "{}" by default simp
let ?\<nu> = "(\<lambda>A. if A = {} then 0 else 1) :: 'd set \<Rightarrow> ereal"
show ?case
proof (intro exI conjI ballI)
have "sigma_algebra (sigma G \<lparr>measure := ?\<nu>\<rparr>)"
by (rule sigma_algebra_cong) (simp_all add: product_algebra_def)
then have "measure_space (sigma G\<lparr>measure := ?\<nu>\<rparr>)"
by (rule finite_additivity_sufficient)
(simp_all add: positive_def additive_def sets_sigma
product_algebra_generator_def image_constant)
then show "sigma_finite_measure (sigma G\<lparr>measure := ?\<nu>\<rparr>)"
by (auto intro!: exI[of _ "\<lambda>i. {\<lambda>_. undefined}"]
simp: sigma_finite_measure_def sigma_finite_measure_axioms_def
product_algebra_generator_def)
qed auto
next
case (insert i I)
interpret finite_product_sigma_finite M I by default fact
have "finite (insert i I)" using `finite I` by auto
interpret I': finite_product_sigma_finite M "insert i I" by default fact
from insert obtain \<nu> where
prod: "\<And>A. A \<in> (\<Pi> i\<in>I. sets (M i)) \<Longrightarrow> \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))" and
"sigma_finite_measure (sigma G\<lparr> measure := \<nu> \<rparr>)" by auto
then interpret I: sigma_finite_measure "P\<lparr> measure := \<nu>\<rparr>" unfolding product_algebra_def by simp
interpret P: pair_sigma_finite "P\<lparr> measure := \<nu>\<rparr>" "M i" ..
let ?h = "(\<lambda>(f, y). f(i := y))"
let ?\<nu> = "\<lambda>A. P.\<mu> (?h -` A \<inter> space P.P)"
have I': "sigma_algebra (I'.P\<lparr> measure := ?\<nu> \<rparr>)"
by (rule I'.sigma_algebra_cong) simp_all
interpret I'': measure_space "I'.P\<lparr> measure := ?\<nu> \<rparr>"
using measurable_add_dim[OF `i \<notin> I`]
by (intro P.measure_space_vimage[OF I']) (auto simp add: measure_preserving_def)
show ?case
proof (intro exI[of _ ?\<nu>] conjI ballI)
let ?m = "\<lambda>A. measure (Pi\<^isub>M I M\<lparr>measure := \<nu>\<rparr> \<Otimes>\<^isub>M M i) (?h -` A \<inter> space P.P)"
{ fix A assume A: "A \<in> (\<Pi> i\<in>insert i I. sets (M i))"
then have *: "?h -` Pi\<^isub>E (insert i I) A \<inter> space P.P = Pi\<^isub>E I A \<times> A i"
using `i \<notin> I` M.sets_into_space by (auto simp: space_pair_measure space_product_algebra) blast
show "?m (Pi\<^isub>E (insert i I) A) = (\<Prod>i\<in>insert i I. M.\<mu> i (A i))"
unfolding * using A
apply (subst P.pair_measure_times)
using A apply fastforce
using A apply fastforce
using `i \<notin> I` `finite I` prod[of A] A by (auto simp: ac_simps) }
note product = this
have *: "sigma I'.G\<lparr> measure := ?\<nu> \<rparr> = I'.P\<lparr> measure := ?\<nu> \<rparr>"
by (simp add: product_algebra_def)
show "sigma_finite_measure (sigma I'.G\<lparr> measure := ?\<nu> \<rparr>)"
proof (unfold *, default, simp)
from I'.sigma_finite_pairs guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
then have F: "\<And>j. j \<in> insert i I \<Longrightarrow> range (F j) \<subseteq> sets (M j)"
"incseq (\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k)"
"(\<Union>k. \<Pi>\<^isub>E j \<in> insert i I. F j k) = space I'.G"
"\<And>k. \<And>j. j \<in> insert i I \<Longrightarrow> \<mu> j (F j k) \<noteq> \<infinity>"
by blast+
let ?F = "\<lambda>k. \<Pi>\<^isub>E j \<in> insert i I. F j k"
show "\<exists>F::nat \<Rightarrow> ('i \<Rightarrow> 'a) set. range F \<subseteq> sets I'.P \<and>
(\<Union>i. F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i)) \<and> (\<forall>i. ?m (F i) \<noteq> \<infinity>)"
proof (intro exI[of _ ?F] conjI allI)
show "range ?F \<subseteq> sets I'.P" using F(1) by auto
next
from F(3) show "(\<Union>i. ?F i) = (\<Pi>\<^isub>E i\<in>insert i I. space (M i))" by simp
next
fix j
have "\<And>k. k \<in> insert i I \<Longrightarrow> 0 \<le> measure (M k) (F k j)"
using F(1) by auto
with F `finite I` setprod_PInf[of "insert i I", OF this] show "?\<nu> (?F j) \<noteq> \<infinity>"
by (subst product) auto
qed
qed
qed
qed
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure P
unfolding product_algebra_def
using product_measure_exists[OF finite_index]
by (rule someI2_ex) auto
lemma (in finite_product_sigma_finite) measure_times:
assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)"
shows "\<mu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
unfolding product_algebra_def
using product_measure_exists[OF finite_index]
proof (rule someI2_ex, elim conjE)
fix \<nu> assume *: "\<forall>A\<in>\<Pi> i\<in>I. sets (M i). \<nu> (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
have "Pi\<^isub>E I A = Pi\<^isub>E I (\<lambda>i\<in>I. A i)" by (auto dest: Pi_mem)
then have "\<nu> (Pi\<^isub>E I A) = \<nu> (Pi\<^isub>E I (\<lambda>i\<in>I. A i))" by simp
also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i ((\<lambda>i\<in>I. A i) i))" using assms * by auto
finally show "measure (sigma G\<lparr>measure := \<nu>\<rparr>) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. M.\<mu> i (A i))"
by (simp add: sigma_def)
qed
lemma (in product_sigma_finite) product_measure_empty[simp]:
"measure (Pi\<^isub>M {} M) {\<lambda>x. undefined} = 1"
proof -
interpret finite_product_sigma_finite M "{}" by default auto
from measure_times[of "\<lambda>x. {}"] show ?thesis by simp
qed
lemma (in finite_product_sigma_algebra) P_empty:
assumes "I = {}"
shows "space P = {\<lambda>k. undefined}" "sets P = { {}, {\<lambda>k. undefined} }"
unfolding product_algebra_def product_algebra_generator_def `I = {}`
by (simp_all add: sigma_def image_constant)
lemma (in product_sigma_finite) positive_integral_empty:
assumes pos: "0 \<le> f (\<lambda>k. undefined)"
shows "integral\<^isup>P (Pi\<^isub>M {} M) f = f (\<lambda>k. undefined)"
proof -
interpret finite_product_sigma_finite M "{}" by default (fact finite.emptyI)
have "\<And>A. measure (Pi\<^isub>M {} M) (Pi\<^isub>E {} A) = 1"
using assms by (subst measure_times) auto
then show ?thesis
unfolding positive_integral_def simple_function_def simple_integral_def_raw
proof (simp add: P_empty, intro antisym)
show "f (\<lambda>k. undefined) \<le> (SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined))"
by (intro SUP_upper) (auto simp: le_fun_def split: split_max)
show "(SUP f:{g. g \<le> max 0 \<circ> f}. f (\<lambda>k. undefined)) \<le> f (\<lambda>k. undefined)" using pos
by (intro SUP_least) (auto simp: le_fun_def simp: max_def split: split_if_asm)
qed
qed
lemma (in product_sigma_finite) measure_fold:
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
assumes A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
shows "measure (Pi\<^isub>M (I \<union> J) M) A =
measure (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) ((\<lambda>(x,y). merge I x J y) -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M))"
proof -
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
have "finite (I \<union> J)" using fin by auto
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
interpret P: pair_sigma_finite I.P J.P by default
let ?g = "\<lambda>(x,y). merge I x J y"
let ?X = "\<lambda>S. ?g -` S \<inter> space P.P"
from IJ.sigma_finite_pairs obtain F where
F: "\<And>i. i\<in> I \<union> J \<Longrightarrow> range (F i) \<subseteq> sets (M i)"
"incseq (\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k)"
"(\<Union>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k) = space IJ.G"
"\<And>k. \<forall>i\<in>I\<union>J. \<mu> i (F i k) \<noteq> \<infinity>"
by auto
let ?F = "\<lambda>k. \<Pi>\<^isub>E i\<in>I \<union> J. F i k"
show "IJ.\<mu> A = P.\<mu> (?X A)"
proof (rule measure_unique_Int_stable_vimage)
show "measure_space IJ.P" "measure_space P.P" by default
show "sets (sigma IJ.G) = sets IJ.P" "space IJ.G = space IJ.P" "A \<in> sets (sigma IJ.G)"
using A unfolding product_algebra_def by auto
next
show "Int_stable IJ.G"
by (rule Int_stable_product_algebra_generator)
(auto simp: Int_stable_def)
show "range ?F \<subseteq> sets IJ.G" using F
by (simp add: image_subset_iff product_algebra_def
product_algebra_generator_def)
show "incseq ?F" "(\<Union>i. ?F i) = space IJ.G " by fact+
next
fix k
have "\<And>j. j \<in> I \<union> J \<Longrightarrow> 0 \<le> measure (M j) (F j k)"
using F(1) by auto
with F `finite I` setprod_PInf[of "I \<union> J", OF this]
show "IJ.\<mu> (?F k) \<noteq> \<infinity>" by (subst IJ.measure_times) auto
next
fix A assume "A \<in> sets IJ.G"
then obtain F where A: "A = Pi\<^isub>E (I \<union> J) F"
and F: "\<And>i. i \<in> I \<union> J \<Longrightarrow> F i \<in> sets (M i)"
by (auto simp: product_algebra_generator_def)
then have X: "?X A = (Pi\<^isub>E I F \<times> Pi\<^isub>E J F)"
using sets_into_space by (auto simp: space_pair_measure) blast+
then have "P.\<mu> (?X A) = (\<Prod>i\<in>I. \<mu> i (F i)) * (\<Prod>i\<in>J. \<mu> i (F i))"
using `finite J` `finite I` F
by (simp add: P.pair_measure_times I.measure_times J.measure_times)
also have "\<dots> = (\<Prod>i\<in>I \<union> J. \<mu> i (F i))"
using `finite J` `finite I` `I \<inter> J = {}` by (simp add: setprod_Un_one)
also have "\<dots> = IJ.\<mu> A"
using `finite J` `finite I` F unfolding A
by (intro IJ.measure_times[symmetric]) auto
finally show "IJ.\<mu> A = P.\<mu> (?X A)" by (rule sym)
qed (rule measurable_merge[OF IJ])
qed
lemma (in product_sigma_finite) measure_preserving_merge:
assumes IJ: "I \<inter> J = {}" and "finite I" "finite J"
shows "(\<lambda>(x,y). merge I x J y) \<in> measure_preserving (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) (Pi\<^isub>M (I \<union> J) M)"
by (intro measure_preservingI measurable_merge[OF IJ] measure_fold[symmetric] assms)
lemma (in product_sigma_finite) product_positive_integral_fold:
assumes IJ[simp]: "I \<inter> J = {}" "finite I" "finite J"
and f: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
shows "integral\<^isup>P (Pi\<^isub>M (I \<union> J) M) f =
(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (merge I x J y) \<partial>(Pi\<^isub>M J M)) \<partial>(Pi\<^isub>M I M))"
proof -
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
interpret P: pair_sigma_finite "Pi\<^isub>M I M" "Pi\<^isub>M J M" by default
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default simp
have P_borel: "(\<lambda>x. f (case x of (x, y) \<Rightarrow> merge I x J y)) \<in> borel_measurable P.P"
using measurable_comp[OF measurable_merge[OF IJ(1)] f] by (simp add: comp_def)
show ?thesis
unfolding P.positive_integral_fst_measurable[OF P_borel, simplified]
proof (rule P.positive_integral_vimage)
show "sigma_algebra IJ.P" by default
show "(\<lambda>(x, y). merge I x J y) \<in> measure_preserving P.P IJ.P"
using IJ by (rule measure_preserving_merge)
show "f \<in> borel_measurable IJ.P" using f by simp
qed
qed
lemma (in product_sigma_finite) measure_preserving_component_singelton:
"(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"
proof (intro measure_preservingI measurable_component_singleton)
interpret I: finite_product_sigma_finite M "{i}" by default simp
fix A let ?P = "(\<lambda>x. x i) -` A \<inter> space I.P"
assume A: "A \<in> sets (M i)"
then have *: "?P = {i} \<rightarrow>\<^isub>E A" using sets_into_space by auto
show "I.\<mu> ?P = M.\<mu> i A" unfolding *
using A I.measure_times[of "\<lambda>_. A"] by auto
qed auto
lemma (in product_sigma_finite) product_positive_integral_singleton:
assumes f: "f \<in> borel_measurable (M i)"
shows "integral\<^isup>P (Pi\<^isub>M {i} M) (\<lambda>x. f (x i)) = integral\<^isup>P (M i) f"
proof -
interpret I: finite_product_sigma_finite M "{i}" by default simp
show ?thesis
proof (rule I.positive_integral_vimage[symmetric])
show "sigma_algebra (M i)" by (rule sigma_algebras)
show "(\<lambda>x. x i) \<in> measure_preserving (Pi\<^isub>M {i} M) (M i)"
by (rule measure_preserving_component_singelton)
show "f \<in> borel_measurable (M i)" by fact
qed
qed
lemma (in product_sigma_finite) product_positive_integral_insert:
assumes [simp]: "finite I" "i \<notin> I"
and f: "f \<in> borel_measurable (Pi\<^isub>M (insert i I) M)"
shows "integral\<^isup>P (Pi\<^isub>M (insert i I) M) f = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^isub>M I M))"
proof -
interpret I: finite_product_sigma_finite M I by default auto
interpret i: finite_product_sigma_finite M "{i}" by default auto
interpret P: pair_sigma_algebra I.P i.P ..
have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
using f by auto
show ?thesis
unfolding product_positive_integral_fold[OF IJ, unfolded insert, simplified, OF f]
proof (rule I.positive_integral_cong, subst product_positive_integral_singleton)
fix x assume x: "x \<in> space I.P"
let ?f = "\<lambda>y. f (restrict (x(i := y)) (insert i I))"
have f'_eq: "\<And>y. ?f y = f (x(i := y))"
using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
show "?f \<in> borel_measurable (M i)" unfolding f'_eq
using measurable_comp[OF measurable_component_update f] x `i \<notin> I`
by (simp add: comp_def)
show "integral\<^isup>P (M i) ?f = \<integral>\<^isup>+ y. f (x(i:=y)) \<partial>M i"
unfolding f'_eq by simp
qed
qed
lemma (in product_sigma_finite) product_positive_integral_setprod:
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "finite I" and borel: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
and pos: "\<And>i x. i \<in> I \<Longrightarrow> 0 \<le> f i x"
shows "(\<integral>\<^isup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>P (M i) (f i))"
using assms proof induct
case empty
interpret finite_product_sigma_finite M "{}" by default auto
show ?case by simp
next
case (insert i I)
note `finite I`[intro, simp]
interpret I: finite_product_sigma_finite M I by default auto
have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
using insert by (auto intro!: setprod_cong)
have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow> (\<lambda>x. (\<Prod>i\<in>J. f i (x i))) \<in> borel_measurable (Pi\<^isub>M J M)"
using sets_into_space insert
by (intro sigma_algebra.borel_measurable_ereal_setprod sigma_algebra_product_algebra
measurable_comp[OF measurable_component_singleton, unfolded comp_def])
auto
then show ?case
apply (simp add: product_positive_integral_insert[OF insert(1,2) prod])
apply (simp add: insert * pos borel setprod_ereal_pos M.positive_integral_multc)
apply (subst I.positive_integral_cmult)
apply (auto simp add: pos borel insert setprod_ereal_pos positive_integral_positive)
done
qed
lemma (in product_sigma_finite) product_integral_singleton:
assumes f: "f \<in> borel_measurable (M i)"
shows "(\<integral>x. f (x i) \<partial>Pi\<^isub>M {i} M) = integral\<^isup>L (M i) f"
proof -
interpret I: finite_product_sigma_finite M "{i}" by default simp
have *: "(\<lambda>x. ereal (f x)) \<in> borel_measurable (M i)"
"(\<lambda>x. ereal (- f x)) \<in> borel_measurable (M i)"
using assms by auto
show ?thesis
unfolding lebesgue_integral_def *[THEN product_positive_integral_singleton] ..
qed
lemma (in product_sigma_finite) product_integral_fold:
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
and f: "integrable (Pi\<^isub>M (I \<union> J) M) f"
shows "integral\<^isup>L (Pi\<^isub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I x J y) \<partial>Pi\<^isub>M J M) \<partial>Pi\<^isub>M I M)"
proof -
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
have "finite (I \<union> J)" using fin by auto
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
interpret P: pair_sigma_finite I.P J.P by default
let ?M = "\<lambda>(x, y). merge I x J y"
let ?f = "\<lambda>x. f (?M x)"
show ?thesis
proof (subst P.integrable_fst_measurable(2)[of ?f, simplified])
have 1: "sigma_algebra IJ.P" by default
have 2: "?M \<in> measure_preserving P.P IJ.P" using measure_preserving_merge[OF assms(1,2,3)] .
have 3: "integrable (Pi\<^isub>M (I \<union> J) M) f" by fact
then have 4: "f \<in> borel_measurable (Pi\<^isub>M (I \<union> J) M)"
by (simp add: integrable_def)
show "integrable P.P ?f"
by (rule P.integrable_vimage[where f=f, OF 1 2 3])
show "integral\<^isup>L IJ.P f = integral\<^isup>L P.P ?f"
by (rule P.integral_vimage[where f=f, OF 1 2 4])
qed
qed
lemma (in product_sigma_finite) product_integral_insert:
assumes [simp]: "finite I" "i \<notin> I"
and f: "integrable (Pi\<^isub>M (insert i I) M) f"
shows "integral\<^isup>L (Pi\<^isub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^isub>M I M)"
proof -
interpret I: finite_product_sigma_finite M I by default auto
interpret I': finite_product_sigma_finite M "insert i I" by default auto
interpret i: finite_product_sigma_finite M "{i}" by default auto
interpret P: pair_sigma_finite I.P i.P ..
have IJ: "I \<inter> {i} = {}" by auto
show ?thesis
unfolding product_integral_fold[OF IJ, simplified, OF f]
proof (rule I.integral_cong, subst product_integral_singleton)
fix x assume x: "x \<in> space I.P"
let ?f = "\<lambda>y. f (restrict (x(i := y)) (insert i I))"
have f'_eq: "\<And>y. ?f y = f (x(i := y))"
using x by (auto intro!: arg_cong[where f=f] simp: fun_eq_iff extensional_def)
have f: "f \<in> borel_measurable I'.P" using f unfolding integrable_def by auto
show "?f \<in> borel_measurable (M i)"
unfolding measurable_cong[OF f'_eq]
using measurable_comp[OF measurable_component_update f] x `i \<notin> I`
by (simp add: comp_def)
show "integral\<^isup>L (M i) ?f = integral\<^isup>L (M i) (\<lambda>y. f (x(i := y)))"
unfolding f'_eq by simp
qed
qed
lemma (in product_sigma_finite) product_integrable_setprod:
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
shows "integrable (Pi\<^isub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
proof -
interpret finite_product_sigma_finite M I by default fact
have f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
using integrable unfolding integrable_def by auto
then have borel: "?f \<in> borel_measurable P"
using measurable_comp[OF measurable_component_singleton f]
by (auto intro!: borel_measurable_setprod simp: comp_def)
moreover have "integrable (Pi\<^isub>M I M) (\<lambda>x. \<bar>\<Prod>i\<in>I. f i (x i)\<bar>)"
proof (unfold integrable_def, intro conjI)
show "(\<lambda>x. abs (?f x)) \<in> borel_measurable P"
using borel by auto
have "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. (\<Prod>i\<in>I. ereal (abs (f i (x i)))) \<partial>P)"
by (simp add: setprod_ereal abs_setprod)
also have "\<dots> = (\<Prod>i\<in>I. (\<integral>\<^isup>+x. ereal (abs (f i x)) \<partial>M i))"
using f by (subst product_positive_integral_setprod) auto
also have "\<dots> < \<infinity>"
using integrable[THEN M.integrable_abs]
by (simp add: setprod_PInf integrable_def M.positive_integral_positive)
finally show "(\<integral>\<^isup>+x. ereal (abs (?f x)) \<partial>P) \<noteq> \<infinity>" by auto
have "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>P) = (\<integral>\<^isup>+x. 0 \<partial>P)"
by (intro positive_integral_cong_pos) auto
then show "(\<integral>\<^isup>+x. ereal (- abs (?f x)) \<partial>P) \<noteq> \<infinity>" by simp
qed
ultimately show ?thesis
by (rule integrable_abs_iff[THEN iffD1])
qed
lemma (in product_sigma_finite) product_integral_setprod:
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> real"
assumes "finite I" "I \<noteq> {}" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^isub>M I M) = (\<Prod>i\<in>I. integral\<^isup>L (M i) (f i))"
using assms proof (induct rule: finite_ne_induct)
case (singleton i)
then show ?case by (simp add: product_integral_singleton integrable_def)
next
case (insert i I)
then have iI: "finite (insert i I)" by auto
then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
integrable (Pi\<^isub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
by (intro product_integrable_setprod insert(5)) (auto intro: finite_subset)
interpret I: finite_product_sigma_finite M I by default fact
have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
using `i \<notin> I` by (auto intro!: setprod_cong)
show ?case
unfolding product_integral_insert[OF insert(1,3) prod[OF subset_refl]]
by (simp add: * insert integral_multc I.integral_cmult[OF prod] subset_insertI)
qed
end