(* Title: HOL/Probability/Infinite_Product_Measure.thy
Author: Johannes Hölzl, TU München
*)
header {*Infinite Product Measure*}
theory Infinite_Product_Measure
imports Probability_Measure
begin
lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
unfolding restrict_def extensional_def by auto
lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)"
unfolding restrict_def by (simp add: fun_eq_iff)
lemma split_merge: "P (merge I x J y i) \<longleftrightarrow> (i \<in> I \<longrightarrow> P (x i)) \<and> (i \<in> J - I \<longrightarrow> P (y i)) \<and> (i \<notin> I \<union> J \<longrightarrow> P undefined)"
unfolding merge_def by auto
lemma extensional_merge_sub: "I \<union> J \<subseteq> K \<Longrightarrow> merge I x J y \<in> extensional K"
unfolding merge_def extensional_def by auto
lemma injective_vimage_restrict:
assumes J: "J \<subseteq> I"
and sets: "A \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^isub>E i\<in>J. S i)" and ne: "(\<Pi>\<^isub>E i\<in>I. S i) \<noteq> {}"
and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^isub>E i\<in>I. S i)"
shows "A = B"
proof (intro set_eqI)
fix x
from ne obtain y where y: "\<And>i. i \<in> I \<Longrightarrow> y i \<in> S i" by auto
have "J \<inter> (I - J) = {}" by auto
show "x \<in> A \<longleftrightarrow> x \<in> B"
proof cases
assume x: "x \<in> (\<Pi>\<^isub>E i\<in>J. S i)"
have "x \<in> A \<longleftrightarrow> merge J x (I - J) y \<in> (\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^isub>E i\<in>I. S i)"
using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub split: split_merge)
then show "x \<in> A \<longleftrightarrow> x \<in> B"
using y x `J \<subseteq> I` by (auto simp add: Pi_iff extensional_restrict extensional_merge_sub eq split: split_merge)
next
assume "x \<notin> (\<Pi>\<^isub>E i\<in>J. S i)" with sets show "x \<in> A \<longleftrightarrow> x \<in> B" by auto
qed
qed
lemma (in product_prob_space) measure_preserving_restrict:
assumes "J \<noteq> {}" "J \<subseteq> K" "finite K"
shows "(\<lambda>f. restrict f J) \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)" (is "?R \<in> _")
proof -
interpret K: finite_product_prob_space M K by default fact
have J: "J \<noteq> {}" "finite J" using assms by (auto simp add: finite_subset)
interpret J: finite_product_prob_space M J
by default (insert J, auto)
from J.sigma_finite_pairs guess F .. note F = this
then have [simp,intro]: "\<And>k i. k \<in> J \<Longrightarrow> F k i \<in> sets (M k)"
by auto
let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. F k i"
let ?J = "product_algebra_generator J M \<lparr> measure := measure (Pi\<^isub>M J M) \<rparr>"
have "?R \<in> measure_preserving (\<Pi>\<^isub>M i\<in>K. M i) (sigma ?J)"
proof (rule K.measure_preserving_Int_stable)
show "Int_stable ?J"
by (auto simp: Int_stable_def product_algebra_generator_def PiE_Int)
show "range ?F \<subseteq> sets ?J" "incseq ?F" "(\<Union>i. ?F i) = space ?J"
using F by auto
show "\<And>i. measure ?J (?F i) \<noteq> \<infinity>"
using F by (simp add: J.measure_times setprod_PInf)
have "measure_space (Pi\<^isub>M J M)" by default
then show "measure_space (sigma ?J)"
by (simp add: product_algebra_def sigma_def)
show "?R \<in> measure_preserving (Pi\<^isub>M K M) ?J"
proof (simp add: measure_preserving_def measurable_def product_algebra_generator_def del: vimage_Int,
safe intro!: restrict_extensional)
fix x k assume "k \<in> J" "x \<in> (\<Pi> i\<in>K. space (M i))"
then show "x k \<in> space (M k)" using `J \<subseteq> K` by auto
next
fix E assume "E \<in> (\<Pi> i\<in>J. sets (M i))"
then have E: "\<And>j. j \<in> J \<Longrightarrow> E j \<in> sets (M j)" by auto
then have *: "?R -` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i)) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))"
(is "?X = Pi\<^isub>E K ?M")
using `J \<subseteq> K` sets_into_space by (auto simp: Pi_iff split: split_if_asm) blast+
with E show "?X \<in> sets (Pi\<^isub>M K M)"
by (auto intro!: product_algebra_generatorI)
have "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = (\<Prod>i\<in>J. measure (M i) (?M i))"
using E by (simp add: J.measure_times)
also have "\<dots> = measure (Pi\<^isub>M K M) ?X"
unfolding * using E `finite K` `J \<subseteq> K`
by (auto simp: K.measure_times M.measure_space_1
cong del: setprod_cong
intro!: setprod_mono_one_left)
finally show "measure (Pi\<^isub>M J M) (Pi\<^isub>E J E) = measure (Pi\<^isub>M K M) ?X" .
qed
qed
then show ?thesis
by (simp add: product_algebra_def sigma_def)
qed
lemma (in product_prob_space) measurable_restrict:
assumes *: "J \<noteq> {}" "J \<subseteq> K" "finite K"
shows "(\<lambda>f. restrict f J) \<in> measurable (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i)"
using measure_preserving_restrict[OF *]
by (rule measure_preservingD2)
definition (in product_prob_space)
"emb J K X = (\<lambda>x. restrict x K) -` X \<inter> space (Pi\<^isub>M J M)"
lemma (in product_prob_space) emb_trans[simp]:
"J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> emb L K (emb K J X) = emb L J X"
by (auto simp add: Int_absorb1 emb_def)
lemma (in product_prob_space) emb_empty[simp]:
"emb K J {} = {}"
by (simp add: emb_def)
lemma (in product_prob_space) emb_Pi:
assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
shows "emb K J (Pi\<^isub>E J X) = (\<Pi>\<^isub>E i\<in>K. if i \<in> J then X i else space (M i))"
using assms space_closed
by (auto simp: emb_def Pi_iff split: split_if_asm) blast+
lemma (in product_prob_space) emb_injective:
assumes "J \<noteq> {}" "J \<subseteq> L" "finite J" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)"
assumes "emb L J X = emb L J Y"
shows "X = Y"
proof -
interpret J: finite_product_sigma_finite M J by default fact
show "X = Y"
proof (rule injective_vimage_restrict)
show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
using J.sets_into_space sets by auto
have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)"
using M.not_empty by auto
from bchoice[OF this]
show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by auto
show "(\<lambda>x. restrict x J) -` X \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i)) = (\<lambda>x. restrict x J) -` Y \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i))"
using `emb L J X = emb L J Y` by (simp add: emb_def)
qed fact
qed
lemma (in product_prob_space) emb_id:
"B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> emb L L B = B"
by (auto simp: emb_def Pi_iff subset_eq extensional_restrict)
lemma (in product_prob_space) emb_simps:
shows "emb L K (A \<union> B) = emb L K A \<union> emb L K B"
and "emb L K (A \<inter> B) = emb L K A \<inter> emb L K B"
and "emb L K (A - B) = emb L K A - emb L K B"
by (auto simp: emb_def)
lemma (in product_prob_space) measurable_emb[intro,simp]:
assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)"
shows "emb L J X \<in> sets (Pi\<^isub>M L M)"
using measurable_restrict[THEN measurable_sets, OF *] by (simp add: emb_def)
lemma (in product_prob_space) measure_emb[intro,simp]:
assumes *: "J \<noteq> {}" "J \<subseteq> L" "finite L" "X \<in> sets (Pi\<^isub>M J M)"
shows "measure (Pi\<^isub>M L M) (emb L J X) = measure (Pi\<^isub>M J M) X"
using measure_preserving_restrict[THEN measure_preservingD, OF *]
by (simp add: emb_def)
definition (in product_prob_space) generator :: "('i \<Rightarrow> 'a) measure_space" where
"generator = \<lparr>
space = (\<Pi>\<^isub>E i\<in>I. space (M i)),
sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)),
measure = undefined
\<rparr>"
lemma (in product_prob_space) generatorI:
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> A = emb I J X \<Longrightarrow> A \<in> sets generator"
unfolding generator_def by auto
lemma (in product_prob_space) generatorI':
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> emb I J X \<in> sets generator"
unfolding generator_def by auto
lemma (in product_sigma_finite)
assumes "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
shows measure_fold_integral:
"measure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. measure (Pi\<^isub>M J M) (merge I x J -` A \<inter> space (Pi\<^isub>M J M)) \<partial>Pi\<^isub>M I M)" (is ?I)
and measure_fold_measurable:
"(\<lambda>x. measure (Pi\<^isub>M J M) (merge I x J -` A \<inter> space (Pi\<^isub>M J M))) \<in> borel_measurable (Pi\<^isub>M I M)" (is ?B)
proof -
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
interpret IJ: pair_sigma_finite I.P J.P ..
show ?I
unfolding measure_fold[OF assms]
apply (subst IJ.pair_measure_alt)
apply (intro measurable_sets[OF _ A] measurable_merge assms)
apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure
intro!: I.positive_integral_cong)
done
have "(\<lambda>(x, y). merge I x J y) -` A \<inter> space (I.P \<Otimes>\<^isub>M J.P) \<in> sets (I.P \<Otimes>\<^isub>M J.P)"
by (intro measurable_sets[OF _ A] measurable_merge assms)
from IJ.measure_cut_measurable_fst[OF this]
show ?B
apply (auto simp: vimage_compose[symmetric] comp_def space_pair_measure)
apply (subst (asm) measurable_cong)
apply auto
done
qed
definition (in product_prob_space)
"\<mu>G A =
(THE x. \<forall>J. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> J \<subseteq> I \<longrightarrow> (\<forall>X\<in>sets (Pi\<^isub>M J M). A = emb I J X \<longrightarrow> x = measure (Pi\<^isub>M J M) X))"
lemma (in product_prob_space) \<mu>G_spec:
assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
shows "\<mu>G A = measure (Pi\<^isub>M J M) X"
unfolding \<mu>G_def
proof (intro the_equality allI impI ballI)
fix K Y assume K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "A = emb I K Y" "Y \<in> sets (Pi\<^isub>M K M)"
have "measure (Pi\<^isub>M K M) Y = measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) K Y)"
using K J by simp
also have "emb (K \<union> J) K Y = emb (K \<union> J) J X"
using K J by (simp add: emb_injective[of "K \<union> J" I])
also have "measure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) J X) = measure (Pi\<^isub>M J M) X"
using K J by simp
finally show "measure (Pi\<^isub>M J M) X = measure (Pi\<^isub>M K M) Y" ..
qed (insert J, force)
lemma (in product_prob_space) \<mu>G_eq:
"J \<noteq> {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> \<mu>G (emb I J X) = measure (Pi\<^isub>M J M) X"
by (intro \<mu>G_spec) auto
lemma (in product_prob_space) generator_Ex:
assumes *: "A \<in> sets generator"
shows "\<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A = emb I J X \<and> \<mu>G A = measure (Pi\<^isub>M J M) X"
proof -
from * obtain J X where J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
unfolding generator_def by auto
with \<mu>G_spec[OF this] show ?thesis by auto
qed
lemma (in product_prob_space) generatorE:
assumes A: "A \<in> sets generator"
obtains J X where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A" "\<mu>G A = measure (Pi\<^isub>M J M) X"
proof -
from generator_Ex[OF A] obtain X J where "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)" "emb I J X = A"
"\<mu>G A = measure (Pi\<^isub>M J M) X" by auto
then show thesis by (intro that) auto
qed
lemma (in product_prob_space) merge_sets:
assumes "finite J" "finite K" "J \<inter> K = {}" and A: "A \<in> sets (Pi\<^isub>M (J \<union> K) M)" and x: "x \<in> space (Pi\<^isub>M J M)"
shows "merge J x K -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)"
proof -
interpret J: finite_product_sigma_algebra M J by default fact
interpret K: finite_product_sigma_algebra M K by default fact
interpret JK: pair_sigma_algebra J.P K.P ..
from JK.measurable_cut_fst[OF
measurable_merge[THEN measurable_sets, OF `J \<inter> K = {}`], OF A, of x] x
show ?thesis
by (simp add: space_pair_measure comp_def vimage_compose[symmetric])
qed
lemma (in product_prob_space) merge_emb:
assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)"
shows "(merge J y (I - J) -` emb I K X \<inter> space (Pi\<^isub>M I M)) =
emb I (K - J) (merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M))"
proof -
have [simp]: "\<And>x J K L. merge J y K (restrict x L) = merge J y (K \<inter> L) x"
by (auto simp: restrict_def merge_def)
have [simp]: "\<And>x J K L. restrict (merge J y K x) L = merge (J \<inter> L) y (K \<inter> L) x"
by (auto simp: restrict_def merge_def)
have [simp]: "(I - J) \<inter> K = K - J" using `K \<subseteq> I` `J \<subseteq> I` by auto
have [simp]: "(K - J) \<inter> (K \<union> J) = K - J" by auto
have [simp]: "(K - J) \<inter> K = K - J" by auto
from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis
by (simp split: split_merge add: emb_def Pi_iff extensional_merge_sub set_eq_iff) auto
qed
definition (in product_prob_space) infprod_algebra :: "('i \<Rightarrow> 'a) measure_space" where
"infprod_algebra = sigma generator \<lparr> measure :=
(SOME \<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
prob_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>)\<rparr>"
syntax
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3PIP _:_./ _)" 10)
syntax (xsymbols)
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3\<Pi>\<^isub>P _\<in>_./ _)" 10)
syntax (HTML output)
"_PiP" :: "[pttrn, 'i set, ('b, 'd) measure_space_scheme] => ('i => 'b, 'd) measure_space_scheme" ("(3\<Pi>\<^isub>P _\<in>_./ _)" 10)
abbreviation
"Pi\<^isub>P I M \<equiv> product_prob_space.infprod_algebra M I"
translations
"PIP x:I. M" == "CONST Pi\<^isub>P I (%x. M)"
lemma (in product_prob_space) algebra_generator:
assumes "I \<noteq> {}" shows "algebra generator"
proof
let ?G = generator
show "sets ?G \<subseteq> Pow (space ?G)"
by (auto simp: generator_def emb_def)
from `I \<noteq> {}` obtain i where "i \<in> I" by auto
then show "{} \<in> sets ?G"
by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
simp: product_algebra_def sigma_def sigma_sets.Empty generator_def emb_def)
from `i \<in> I` show "space ?G \<in> sets ?G"
by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
simp: generator_def emb_def)
fix A assume "A \<in> sets ?G"
then obtain JA XA where XA: "JA \<noteq> {}" "finite JA" "JA \<subseteq> I" "XA \<in> sets (Pi\<^isub>M JA M)" and A: "A = emb I JA XA"
by (auto simp: generator_def)
fix B assume "B \<in> sets ?G"
then obtain JB XB where XB: "JB \<noteq> {}" "finite JB" "JB \<subseteq> I" "XB \<in> sets (Pi\<^isub>M JB M)" and B: "B = emb I JB XB"
by (auto simp: generator_def)
let ?RA = "emb (JA \<union> JB) JA XA"
let ?RB = "emb (JA \<union> JB) JB XB"
interpret JAB: finite_product_sigma_algebra M "JA \<union> JB"
by default (insert XA XB, auto)
have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
using XA A XB B by (auto simp: emb_simps)
then show "A - B \<in> sets ?G" "A \<union> B \<in> sets ?G"
using XA XB by (auto intro!: generatorI')
qed
lemma (in product_prob_space) positive_\<mu>G:
assumes "I \<noteq> {}"
shows "positive generator \<mu>G"
proof -
interpret G!: algebra generator by (rule algebra_generator) fact
show ?thesis
proof (intro positive_def[THEN iffD2] conjI ballI)
from generatorE[OF G.empty_sets] guess J X . note this[simp]
interpret J: finite_product_sigma_finite M J by default fact
have "X = {}"
by (rule emb_injective[of J I]) simp_all
then show "\<mu>G {} = 0" by simp
next
fix A assume "A \<in> sets generator"
from generatorE[OF this] guess J X . note this[simp]
interpret J: finite_product_sigma_finite M J by default fact
show "0 \<le> \<mu>G A" by simp
qed
qed
lemma (in product_prob_space) additive_\<mu>G:
assumes "I \<noteq> {}"
shows "additive generator \<mu>G"
proof -
interpret G!: algebra generator by (rule algebra_generator) fact
show ?thesis
proof (intro additive_def[THEN iffD2] ballI impI)
fix A assume "A \<in> sets generator" with generatorE guess J X . note J = this
fix B assume "B \<in> sets generator" with generatorE guess K Y . note K = this
assume "A \<inter> B = {}"
have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)"
using J K by auto
interpret JK: finite_product_sigma_finite M "J \<union> K" by default fact
have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}"
apply (rule emb_injective[of "J \<union> K" I])
apply (insert `A \<inter> B = {}` JK J K)
apply (simp_all add: JK.Int emb_simps)
done
have AB: "A = emb I (J \<union> K) (emb (J \<union> K) J X)" "B = emb I (J \<union> K) (emb (J \<union> K) K Y)"
using J K by simp_all
then have "\<mu>G (A \<union> B) = \<mu>G (emb I (J \<union> K) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y))"
by (simp add: emb_simps)
also have "\<dots> = measure (Pi\<^isub>M (J \<union> K) M) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq JK.Un)
also have "\<dots> = \<mu>G A + \<mu>G B"
using J K JK_disj by (simp add: JK.measure_additive[symmetric])
finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" .
qed
qed
lemma (in product_prob_space) finite_index_eq_finite_product:
assumes "finite I"
shows "sets (sigma generator) = sets (Pi\<^isub>M I M)"
proof safe
interpret I: finite_product_sigma_algebra M I by default fact
have space_generator[simp]: "space generator = space (Pi\<^isub>M I M)"
by (simp add: generator_def product_algebra_def)
{ fix A assume "A \<in> sets (sigma generator)"
then show "A \<in> sets I.P" unfolding sets_sigma
proof induct
case (Basic A)
from generatorE[OF this] guess J X . note J = this
with `finite I` have "emb I J X \<in> sets I.P" by auto
with `emb I J X = A` show "A \<in> sets I.P" by simp
qed auto }
{ fix A assume A: "A \<in> sets I.P"
show "A \<in> sets (sigma generator)"
proof cases
assume "I = {}"
with I.P_empty[OF this] A
have "A = space generator \<or> A = {}"
unfolding space_generator by auto
then show ?thesis
by (auto simp: sets_sigma simp del: space_generator
intro: sigma_sets.Empty sigma_sets_top)
next
assume "I \<noteq> {}"
note A this
moreover with I.sets_into_space have "emb I I A = A" by (intro emb_id) auto
ultimately show "A \<in> sets (sigma generator)"
using `finite I` unfolding sets_sigma
by (intro sigma_sets.Basic generatorI[of I A]) auto
qed }
qed
lemma (in product_prob_space) extend_\<mu>G:
"\<exists>\<mu>. (\<forall>s\<in>sets generator. \<mu> s = \<mu>G s) \<and>
prob_space \<lparr>space = space generator, sets = sets (sigma generator), measure = \<mu>\<rparr>"
proof cases
assume "finite I"
interpret I: finite_product_prob_space M I by default fact
show ?thesis
proof (intro exI[of _ "measure (Pi\<^isub>M I M)"] ballI conjI)
fix A assume "A \<in> sets generator"
from generatorE[OF this] guess J X . note J = this
from J(1-4) `finite I` show "measure I.P A = \<mu>G A"
unfolding J(6)
by (subst J(5)[symmetric]) (simp add: measure_emb)
next
have [simp]: "space generator = space (Pi\<^isub>M I M)"
by (simp add: generator_def product_algebra_def)
have "\<lparr>space = space generator, sets = sets (sigma generator), measure = measure I.P\<rparr>
= I.P" (is "?P = _")
by (auto intro!: measure_space.equality simp: finite_index_eq_finite_product[OF `finite I`])
show "prob_space ?P"
proof
show "measure_space ?P" using `?P = I.P` by simp default
show "measure ?P (space ?P) = 1"
using I.measure_space_1 by simp
qed
qed
next
let ?G = generator
assume "\<not> finite I"
then have I_not_empty: "I \<noteq> {}" by auto
interpret G!: algebra generator by (rule algebra_generator) fact
note \<mu>G_mono =
G.additive_increasing[OF positive_\<mu>G[OF I_not_empty] additive_\<mu>G[OF I_not_empty], THEN increasingD]
{ fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> sets ?G"
from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J"
by (metis rev_finite_subset subsetI)
moreover from Z guess K' X' by (rule generatorE)
moreover def K \<equiv> "insert k K'"
moreover def X \<equiv> "emb K K' X'"
ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X"
"K - J \<noteq> {}" "K - J \<subseteq> I" "\<mu>G Z = measure (Pi\<^isub>M K M) X"
by (auto simp: subset_insertI)
let ?M = "\<lambda>y. merge J y (K - J) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M)"
{ fix y assume y: "y \<in> space (Pi\<^isub>M J M)"
note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X]
moreover
have **: "?M y \<in> sets (Pi\<^isub>M (K - J) M)"
using J K y by (intro merge_sets) auto
ultimately
have ***: "(merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<in> sets ?G"
using J K by (intro generatorI) auto
have "\<mu>G (merge J y (I - J) -` emb I K X \<inter> space (Pi\<^isub>M I M)) = measure (Pi\<^isub>M (K - J) M) (?M y)"
unfolding * using K J by (subst \<mu>G_eq[OF _ _ _ **]) auto
note * ** *** this }
note merge_in_G = this
have "finite (K - J)" using K by auto
interpret J: finite_product_prob_space M J by default fact+
interpret KmJ: finite_product_prob_space M "K - J" by default fact+
have "\<mu>G Z = measure (Pi\<^isub>M (J \<union> (K - J)) M) (emb (J \<union> (K - J)) K X)"
using K J by simp
also have "\<dots> = (\<integral>\<^isup>+ x. measure (Pi\<^isub>M (K - J) M) (?M x) \<partial>Pi\<^isub>M J M)"
using K J by (subst measure_fold_integral) auto
also have "\<dots> = (\<integral>\<^isup>+ y. \<mu>G (merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<partial>Pi\<^isub>M J M)"
(is "_ = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)")
proof (intro J.positive_integral_cong)
fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
with K merge_in_G(2)[OF this]
show "measure (Pi\<^isub>M (K - J) M) (?M x) = \<mu>G (?MZ x)"
unfolding `Z = emb I K X` merge_in_G(1)[OF x] by (subst \<mu>G_eq) auto
qed
finally have fold: "\<mu>G Z = (\<integral>\<^isup>+x. \<mu>G (?MZ x) \<partial>Pi\<^isub>M J M)" .
{ fix x assume x: "x \<in> space (Pi\<^isub>M J M)"
then have "\<mu>G (?MZ x) \<le> 1"
unfolding merge_in_G(4)[OF x] `Z = emb I K X`
by (intro KmJ.measure_le_1 merge_in_G(2)[OF x]) }
note le_1 = this
let ?q = "\<lambda>y. \<mu>G (merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M))"
have "?q \<in> borel_measurable (Pi\<^isub>M J M)"
unfolding `Z = emb I K X` using J K merge_in_G(3)
by (simp add: merge_in_G \<mu>G_eq measure_fold_measurable
del: space_product_algebra cong: measurable_cong)
note this fold le_1 merge_in_G(3) }
note fold = this
have "\<exists>\<mu>. (\<forall>s\<in>sets ?G. \<mu> s = \<mu>G s) \<and>
measure_space \<lparr>space = space ?G, sets = sets (sigma ?G), measure = \<mu>\<rparr>"
(is "\<exists>\<mu>. _ \<and> measure_space (?ms \<mu>)")
proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G])
fix A assume "A \<in> sets ?G"
with generatorE guess J X . note JX = this
interpret JK: finite_product_prob_space M J by default fact+
with JX show "\<mu>G A \<noteq> \<infinity>" by simp
next
fix A assume A: "range A \<subseteq> sets ?G" "decseq A" "(\<Inter>i. A i) = {}"
then have "decseq (\<lambda>i. \<mu>G (A i))"
by (auto intro!: \<mu>G_mono simp: decseq_def)
moreover
have "(INF i. \<mu>G (A i)) = 0"
proof (rule ccontr)
assume "(INF i. \<mu>G (A i)) \<noteq> 0" (is "?a \<noteq> 0")
moreover have "0 \<le> ?a"
using A positive_\<mu>G[OF I_not_empty] by (auto intro!: INF_greatest simp: positive_def)
ultimately have "0 < ?a" by auto
have "\<forall>n. \<exists>J X. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I \<and> X \<in> sets (Pi\<^isub>M J M) \<and> A n = emb I J X \<and> \<mu>G (A n) = measure (Pi\<^isub>M J M) X"
using A by (intro allI generator_Ex) auto
then obtain J' X' where J': "\<And>n. J' n \<noteq> {}" "\<And>n. finite (J' n)" "\<And>n. J' n \<subseteq> I" "\<And>n. X' n \<in> sets (Pi\<^isub>M (J' n) M)"
and A': "\<And>n. A n = emb I (J' n) (X' n)"
unfolding choice_iff by blast
moreover def J \<equiv> "\<lambda>n. (\<Union>i\<le>n. J' i)"
moreover def X \<equiv> "\<lambda>n. emb (J n) (J' n) (X' n)"
ultimately have J: "\<And>n. J n \<noteq> {}" "\<And>n. finite (J n)" "\<And>n. J n \<subseteq> I" "\<And>n. X n \<in> sets (Pi\<^isub>M (J n) M)"
by auto
with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> sets ?G"
unfolding J_def X_def by (subst emb_trans) (insert A, auto)
have J_mono: "\<And>n m. n \<le> m \<Longrightarrow> J n \<subseteq> J m"
unfolding J_def by force
interpret J: finite_product_prob_space M "J i" for i by default fact+
have a_le_1: "?a \<le> 1"
using \<mu>G_spec[of "J 0" "A 0" "X 0"] J A_eq
by (auto intro!: INF_lower2[of 0] J.measure_le_1)
let ?M = "\<lambda>K Z y. merge K y (I - K) -` Z \<inter> space (Pi\<^isub>M I M)"
{ fix Z k assume Z: "range Z \<subseteq> sets ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)"
then have Z_sets: "\<And>n. Z n \<in> sets ?G" by auto
fix J' assume J': "J' \<noteq> {}" "finite J'" "J' \<subseteq> I"
interpret J': finite_product_prob_space M J' by default fact+
let ?q = "\<lambda>n y. \<mu>G (?M J' (Z n) y)"
let ?Q = "\<lambda>n. ?q n -` {?a / 2^(k+1) ..} \<inter> space (Pi\<^isub>M J' M)"
{ fix n
have "?q n \<in> borel_measurable (Pi\<^isub>M J' M)"
using Z J' by (intro fold(1)) auto
then have "?Q n \<in> sets (Pi\<^isub>M J' M)"
by (rule measurable_sets) auto }
note Q_sets = this
have "?a / 2^(k+1) \<le> (INF n. measure (Pi\<^isub>M J' M) (?Q n))"
proof (intro INF_greatest)
fix n
have "?a / 2^k \<le> \<mu>G (Z n)" using Z by auto
also have "\<dots> \<le> (\<integral>\<^isup>+ x. indicator (?Q n) x + ?a / 2^(k+1) \<partial>Pi\<^isub>M J' M)"
unfolding fold(2)[OF J' `Z n \<in> sets ?G`]
proof (intro J'.positive_integral_mono)
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
then have "?q n x \<le> 1 + 0"
using J' Z fold(3) Z_sets by auto
also have "\<dots> \<le> 1 + ?a / 2^(k+1)"
using `0 < ?a` by (intro add_mono) auto
finally have "?q n x \<le> 1 + ?a / 2^(k+1)" .
with x show "?q n x \<le> indicator (?Q n) x + ?a / 2^(k+1)"
by (auto split: split_indicator simp del: power_Suc)
qed
also have "\<dots> = measure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)"
using `0 \<le> ?a` Q_sets J'.measure_space_1
by (subst J'.positive_integral_add) auto
finally show "?a / 2^(k+1) \<le> measure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1`
by (cases rule: ereal2_cases[of ?a "measure (Pi\<^isub>M J' M) (?Q n)"])
(auto simp: field_simps)
qed
also have "\<dots> = measure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)"
proof (intro J'.continuity_from_above)
show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto
show "decseq ?Q"
unfolding decseq_def
proof (safe intro!: vimageI[OF refl])
fix m n :: nat assume "m \<le> n"
fix x assume x: "x \<in> space (Pi\<^isub>M J' M)"
assume "?a / 2^(k+1) \<le> ?q n x"
also have "?q n x \<le> ?q m x"
proof (rule \<mu>G_mono)
from fold(4)[OF J', OF Z_sets x]
show "?M J' (Z n) x \<in> sets ?G" "?M J' (Z m) x \<in> sets ?G" by auto
show "?M J' (Z n) x \<subseteq> ?M J' (Z m) x"
using `decseq Z`[THEN decseqD, OF `m \<le> n`] by auto
qed
finally show "?a / 2^(k+1) \<le> ?q m x" .
qed
qed (intro J'.finite_measure Q_sets)
finally have "(\<Inter>n. ?Q n) \<noteq> {}"
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
then have "\<exists>w\<in>space (Pi\<^isub>M J' M). \<forall>n. ?a / 2 ^ (k + 1) \<le> ?q n w" by auto }
note Ex_w = this
let ?q = "\<lambda>k n y. \<mu>G (?M (J k) (A n) y)"
have "\<forall>n. ?a / 2 ^ 0 \<le> \<mu>G (A n)" by (auto intro: INF_lower)
from Ex_w[OF A(1,2) this J(1-3), of 0] guess w0 .. note w0 = this
let ?P =
"\<lambda>k wk w. w \<in> space (Pi\<^isub>M (J (Suc k)) M) \<and> restrict w (J k) = wk \<and>
(\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n w)"
def w \<equiv> "nat_rec w0 (\<lambda>k wk. Eps (?P k wk))"
{ fix k have w: "w k \<in> space (Pi\<^isub>M (J k) M) \<and>
(\<forall>n. ?a / 2 ^ (k + 1) \<le> ?q k n (w k)) \<and> (k \<noteq> 0 \<longrightarrow> restrict (w k) (J (k - 1)) = w (k - 1))"
proof (induct k)
case 0 with w0 show ?case
unfolding w_def nat_rec_0 by auto
next
case (Suc k)
then have wk: "w k \<in> space (Pi\<^isub>M (J k) M)" by auto
have "\<exists>w'. ?P k (w k) w'"
proof cases
assume [simp]: "J k = J (Suc k)"
show ?thesis
proof (intro exI[of _ "w k"] conjI allI)
fix n
have "?a / 2 ^ (Suc k + 1) \<le> ?a / 2 ^ (k + 1)"
using `0 < ?a` `?a \<le> 1` by (cases ?a) (auto simp: field_simps)
also have "\<dots> \<le> ?q k n (w k)" using Suc by auto
finally show "?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n (w k)" by simp
next
show "w k \<in> space (Pi\<^isub>M (J (Suc k)) M)"
using Suc by simp
then show "restrict (w k) (J k) = w k"
by (simp add: extensional_restrict)
qed
next
assume "J k \<noteq> J (Suc k)"
with J_mono[of k "Suc k"] have "J (Suc k) - J k \<noteq> {}" (is "?D \<noteq> {}") by auto
have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> sets ?G"
"decseq (\<lambda>n. ?M (J k) (A n) (w k))"
"\<forall>n. ?a / 2 ^ (k + 1) \<le> \<mu>G (?M (J k) (A n) (w k))"
using `decseq A` fold(4)[OF J(1-3) A_eq(2), of "w k" k] Suc
by (auto simp: decseq_def)
from Ex_w[OF this `?D \<noteq> {}`] J[of "Suc k"]
obtain w' where w': "w' \<in> space (Pi\<^isub>M ?D M)"
"\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> \<mu>G (?M ?D (?M (J k) (A n) (w k)) w')" by auto
let ?w = "merge (J k) (w k) ?D w'"
have [simp]: "\<And>x. merge (J k) (w k) (I - J k) (merge ?D w' (I - ?D) x) =
merge (J (Suc k)) ?w (I - (J (Suc k))) x"
using J(3)[of "Suc k"] J(3)[of k] J_mono[of k "Suc k"]
by (auto intro!: ext split: split_merge)
have *: "\<And>n. ?M ?D (?M (J k) (A n) (w k)) w' = ?M (J (Suc k)) (A n) ?w"
using w'(1) J(3)[of "Suc k"]
by (auto split: split_merge intro!: extensional_merge_sub) force+
show ?thesis
apply (rule exI[of _ ?w])
using w' J_mono[of k "Suc k"] wk unfolding *
apply (auto split: split_merge intro!: extensional_merge_sub ext)
apply (force simp: extensional_def)
done
qed
then have "?P k (w k) (w (Suc k))"
unfolding w_def nat_rec_Suc unfolding w_def[symmetric]
by (rule someI_ex)
then show ?case by auto
qed
moreover
then have "w k \<in> space (Pi\<^isub>M (J k) M)" by auto
moreover
from w have "?a / 2 ^ (k + 1) \<le> ?q k k (w k)" by auto
then have "?M (J k) (A k) (w k) \<noteq> {}"
using positive_\<mu>G[OF I_not_empty, unfolded positive_def] `0 < ?a` `?a \<le> 1`
by (cases ?a) (auto simp: divide_le_0_iff power_le_zero_eq)
then obtain x where "x \<in> ?M (J k) (A k) (w k)" by auto
then have "merge (J k) (w k) (I - J k) x \<in> A k" by auto
then have "\<exists>x\<in>A k. restrict x (J k) = w k"
using `w k \<in> space (Pi\<^isub>M (J k) M)`
by (intro rev_bexI) (auto intro!: ext simp: extensional_def)
ultimately have "w k \<in> space (Pi\<^isub>M (J k) M)"
"\<exists>x\<in>A k. restrict x (J k) = w k"
"k \<noteq> 0 \<Longrightarrow> restrict (w k) (J (k - 1)) = w (k - 1)"
by auto }
note w = this
{ fix k l i assume "k \<le> l" "i \<in> J k"
{ fix l have "w k i = w (k + l) i"
proof (induct l)
case (Suc l)
from `i \<in> J k` J_mono[of k "k + l"] have "i \<in> J (k + l)" by auto
with w(3)[of "k + Suc l"]
have "w (k + l) i = w (k + Suc l) i"
by (auto simp: restrict_def fun_eq_iff split: split_if_asm)
with Suc show ?case by simp
qed simp }
from this[of "l - k"] `k \<le> l` have "w l i = w k i" by simp }
note w_mono = this
def w' \<equiv> "\<lambda>i. if i \<in> (\<Union>k. J k) then w (LEAST k. i \<in> J k) i else if i \<in> I then (SOME x. x \<in> space (M i)) else undefined"
{ fix i k assume k: "i \<in> J k"
have "w k i = w (LEAST k. i \<in> J k) i"
by (intro w_mono Least_le k LeastI[of _ k])
then have "w' i = w k i"
unfolding w'_def using k by auto }
note w'_eq = this
have w'_simps1: "\<And>i. i \<notin> I \<Longrightarrow> w' i = undefined"
using J by (auto simp: w'_def)
have w'_simps2: "\<And>i. i \<notin> (\<Union>k. J k) \<Longrightarrow> i \<in> I \<Longrightarrow> w' i \<in> space (M i)"
using J by (auto simp: w'_def intro!: someI_ex[OF M.not_empty[unfolded ex_in_conv[symmetric]]])
{ fix i assume "i \<in> I" then have "w' i \<in> space (M i)"
using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq)+ }
note w'_simps[simp] = w'_eq w'_simps1 w'_simps2 this
have w': "w' \<in> space (Pi\<^isub>M I M)"
using w(1) by (auto simp add: Pi_iff extensional_def)
{ fix n
have "restrict w' (J n) = w n" using w(1)
by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def)
with w[of n] obtain x where "x \<in> A n" "restrict x (J n) = restrict w' (J n)" by auto
then have "w' \<in> A n" unfolding A_eq using w' by (auto simp: emb_def) }
then have "w' \<in> (\<Inter>i. A i)" by auto
with `(\<Inter>i. A i) = {}` show False by auto
qed
ultimately show "(\<lambda>i. \<mu>G (A i)) ----> 0"
using LIMSEQ_ereal_INFI[of "\<lambda>i. \<mu>G (A i)"] by simp
qed fact+
then guess \<mu> .. note \<mu> = this
show ?thesis
proof (intro exI[of _ \<mu>] conjI)
show "\<forall>S\<in>sets ?G. \<mu> S = \<mu>G S" using \<mu> by simp
show "prob_space (?ms \<mu>)"
proof
show "measure_space (?ms \<mu>)" using \<mu> by simp
obtain i where "i \<in> I" using I_not_empty by auto
interpret i: finite_product_sigma_finite M "{i}" by default auto
let ?X = "\<Pi>\<^isub>E i\<in>{i}. space (M i)"
have X: "?X \<in> sets (Pi\<^isub>M {i} M)"
by auto
with `i \<in> I` have "emb I {i} ?X \<in> sets generator"
by (intro generatorI') auto
with \<mu> have "\<mu> (emb I {i} ?X) = \<mu>G (emb I {i} ?X)" by auto
with \<mu>G_eq[OF _ _ _ X] `i \<in> I`
have "\<mu> (emb I {i} ?X) = measure (M i) (space (M i))"
by (simp add: i.measure_times)
also have "emb I {i} ?X = space (Pi\<^isub>P I M)"
using `i \<in> I` by (auto simp: emb_def infprod_algebra_def generator_def)
finally show "measure (?ms \<mu>) (space (?ms \<mu>)) = 1"
using M.measure_space_1 by (simp add: infprod_algebra_def)
qed
qed
qed
lemma (in product_prob_space) infprod_spec:
"(\<forall>s\<in>sets generator. measure (Pi\<^isub>P I M) s = \<mu>G s) \<and> prob_space (Pi\<^isub>P I M)"
(is "?Q infprod_algebra")
unfolding infprod_algebra_def
by (rule someI2_ex[OF extend_\<mu>G])
(auto simp: sigma_def generator_def)
sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>P I M"
using infprod_spec by simp
lemma (in product_prob_space) measure_infprod_emb:
assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
shows "\<mu> (emb I J X) = measure (Pi\<^isub>M J M) X"
proof -
have "emb I J X \<in> sets generator"
using assms by (rule generatorI')
with \<mu>G_eq[OF assms] infprod_spec show ?thesis by auto
qed
lemma (in product_prob_space) measurable_component:
assumes "i \<in> I"
shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)"
proof (unfold measurable_def, safe)
fix x assume "x \<in> space (Pi\<^isub>P I M)"
then show "x i \<in> space (M i)"
using `i \<in> I` by (auto simp: infprod_algebra_def generator_def)
next
fix A assume "A \<in> sets (M i)"
with `i \<in> I` have
"(\<Pi>\<^isub>E x \<in> {i}. A) \<in> sets (Pi\<^isub>M {i} M)"
"(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E x \<in> {i}. A)"
by (auto simp: infprod_algebra_def generator_def emb_def)
from generatorI[OF _ _ _ this] `i \<in> I`
show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)"
unfolding infprod_algebra_def by auto
qed
lemma (in product_prob_space) emb_in_infprod_algebra[intro]:
fixes J assumes J: "finite J" "J \<subseteq> I" and X: "X \<in> sets (Pi\<^isub>M J M)"
shows "emb I J X \<in> sets (\<Pi>\<^isub>P i\<in>I. M i)"
proof cases
assume "J = {}"
with X have "emb I J X = space (\<Pi>\<^isub>P i\<in>I. M i) \<or> emb I J X = {}"
by (auto simp: emb_def infprod_algebra_def generator_def
product_algebra_def product_algebra_generator_def image_constant sigma_def)
then show ?thesis by auto
next
assume "J \<noteq> {}"
show ?thesis unfolding infprod_algebra_def
by simp (intro in_sigma generatorI' `J \<noteq> {}` J X)
qed
lemma (in product_prob_space) finite_measure_infprod_emb:
assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" "X \<in> sets (Pi\<^isub>M J M)"
shows "\<mu>' (emb I J X) = finite_measure.\<mu>' (Pi\<^isub>M J M) X"
proof -
interpret J: finite_product_prob_space M J by default fact+
from assms have "emb I J X \<in> sets (Pi\<^isub>P I M)" by auto
with assms show "\<mu>' (emb I J X) = J.\<mu>' X"
unfolding \<mu>'_def J.\<mu>'_def
unfolding measure_infprod_emb[OF assms]
by auto
qed
lemma (in finite_product_prob_space) 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))"
using assms
unfolding \<mu>'_def M.\<mu>'_def
by (subst measure_times[OF assms])
(auto simp: finite_measure_eq M.finite_measure_eq setprod_ereal)
lemma (in product_prob_space) finite_measure_infprod_emb_Pi:
assumes J: "finite J" "J \<subseteq> I" "\<And>j. j \<in> J \<Longrightarrow> X j \<in> sets (M j)"
shows "\<mu>' (emb I J (Pi\<^isub>E J X)) = (\<Prod>j\<in>J. M.\<mu>' j (X j))"
proof cases
assume "J = {}"
then have "emb I J (Pi\<^isub>E J X) = space infprod_algebra"
by (auto simp: infprod_algebra_def generator_def sigma_def emb_def)
then show ?thesis using `J = {}` P.prob_space
by simp
next
assume "J \<noteq> {}"
interpret J: finite_product_prob_space M J by default fact+
have "(\<Prod>i\<in>J. M.\<mu>' i (X i)) = J.\<mu>' (Pi\<^isub>E J X)"
using J `J \<noteq> {}` by (subst J.finite_measure_times) auto
also have "\<dots> = \<mu>' (emb I J (Pi\<^isub>E J X))"
using J `J \<noteq> {}` by (intro finite_measure_infprod_emb[symmetric]) auto
finally show ?thesis by simp
qed
lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X 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> sigma_sets X B`, auto intro: sigma_sets.intros)
qed
lemma sigma_sets_mono': 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 sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A"
by (auto intro: sigma_sets.Basic)
lemma (in product_prob_space) infprod_algebra_alt:
"Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M),
sets = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i))),
measure = measure (Pi\<^isub>P I M) \<rparr>"
(is "_ = sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>")
proof (rule measure_space.equality)
let ?G = "\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M)"
have "sigma_sets ?O ?M = sigma_sets ?O ?G"
proof (intro equalityI sigma_sets_mono UN_least)
fix J assume J: "J \<in> {J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}"
have "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> emb I J ` sets (Pi\<^isub>M J M)" by auto
also have "\<dots> \<subseteq> ?G" using J by (rule UN_upper)
also have "\<dots> \<subseteq> sigma_sets ?O ?G" by (rule sigma_sets_superset_generator)
finally show "emb I J ` Pi\<^isub>E J ` (\<Pi> i\<in>J. sets (M i)) \<subseteq> sigma_sets ?O ?G" .
have "emb I J ` sets (Pi\<^isub>M J M) = emb I J ` sigma_sets (space (Pi\<^isub>M J M)) (Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))"
by (simp add: sets_sigma product_algebra_generator_def product_algebra_def)
also have "\<dots> = sigma_sets (space (Pi\<^isub>M I M)) (emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))"
using J M.sets_into_space
by (auto simp: emb_def_raw intro!: sigma_sets_vimage[symmetric]) blast
also have "\<dots> \<subseteq> sigma_sets (space (Pi\<^isub>M I M)) ?M"
using J by (intro sigma_sets_mono') auto
finally show "emb I J ` sets (Pi\<^isub>M J M) \<subseteq> sigma_sets ?O ?M"
by (simp add: infprod_algebra_def generator_def)
qed
then show "sets (Pi\<^isub>P I M) = sets (sigma \<lparr> space = ?O, sets = ?M, measure = ?m \<rparr>)"
by (simp_all add: infprod_algebra_def generator_def sets_sigma)
qed simp_all
lemma (in product_prob_space) infprod_algebra_alt2:
"Pi\<^isub>P I M = sigma \<lparr> space = space (Pi\<^isub>P I M),
sets = (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i)),
measure = measure (Pi\<^isub>P I M) \<rparr>"
(is "_ = ?S")
proof (rule measure_space.equality)
let "sigma \<lparr> space = ?O, sets = ?A, \<dots> = _ \<rparr>" = ?S
let ?G = "(\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` Pi\<^isub>E J ` (\<Pi> i \<in> J. sets (M i)))"
have "sets (Pi\<^isub>P I M) = sigma_sets ?O ?G"
by (subst infprod_algebra_alt) (simp add: sets_sigma)
also have "\<dots> = sigma_sets ?O ?A"
proof (intro equalityI sigma_sets_mono subsetI)
interpret A: sigma_algebra ?S
by (rule sigma_algebra_sigma) auto
fix A assume "A \<in> ?G"
then obtain J B where "finite J" "J \<noteq> {}" "J \<subseteq> I" "A = emb I J (Pi\<^isub>E J B)"
and B: "\<And>i. i \<in> J \<Longrightarrow> B i \<in> sets (M i)"
by auto
then have A: "A = (\<Inter>j\<in>J. (\<lambda>x. x j) -` (B j) \<inter> space (Pi\<^isub>P I M))"
by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff)
{ fix j assume "j\<in>J"
with `J \<subseteq> I` have "j \<in> I" by auto
with `j \<in> J` B have "(\<lambda>x. x j) -` (B j) \<inter> space (Pi\<^isub>P I M) \<in> sets ?S"
by (auto simp: sets_sigma intro: sigma_sets.Basic) }
with `finite J` `J \<noteq> {}` have "A \<in> sets ?S"
unfolding A by (intro A.finite_INT) auto
then show "A \<in> sigma_sets ?O ?A" by (simp add: sets_sigma)
next
fix A assume "A \<in> ?A"
then obtain i B where i: "i \<in> I" "B \<in> sets (M i)"
and "A = (\<lambda>x. x i) -` B \<inter> space (Pi\<^isub>P I M)"
by auto
then have "A = emb I {i} (Pi\<^isub>E {i} (\<lambda>_. B))"
by (auto simp: emb_def infprod_algebra_def generator_def Pi_iff)
with i show "A \<in> sigma_sets ?O ?G"
by (intro sigma_sets.Basic UN_I[where a="{i}"]) auto
qed
also have "\<dots> = sets ?S"
by (simp add: sets_sigma)
finally show "sets (Pi\<^isub>P I M) = sets ?S" .
qed simp_all
lemma (in product_prob_space) measurable_into_infprod_algebra:
assumes "sigma_algebra N"
assumes f: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> measurable N (M i)"
assumes ext: "\<And>x. x \<in> space N \<Longrightarrow> f x \<in> extensional I"
shows "f \<in> measurable N (Pi\<^isub>P I M)"
proof -
interpret N: sigma_algebra N by fact
have f_in: "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. f x i) \<in> space N \<rightarrow> space (M i)"
using f by (auto simp: measurable_def)
{ fix i A assume i: "i \<in> I" "A \<in> sets (M i)"
then have "f -` (\<lambda>x. x i) -` A \<inter> f -` space infprod_algebra \<inter> space N = (\<lambda>x. f x i) -` A \<inter> space N"
using f_in ext by (auto simp: infprod_algebra_def generator_def)
also have "\<dots> \<in> sets N"
by (rule measurable_sets f i)+
finally have "f -` (\<lambda>x. x i) -` A \<inter> f -` space infprod_algebra \<inter> space N \<in> sets N" . }
with f_in ext show ?thesis
by (subst infprod_algebra_alt2)
(auto intro!: N.measurable_sigma simp: Pi_iff infprod_algebra_def generator_def)
qed
lemma (in product_prob_space) measurable_singleton_infprod:
assumes "i \<in> I"
shows "(\<lambda>x. x i) \<in> measurable (Pi\<^isub>P I M) (M i)"
proof (unfold measurable_def, intro CollectI conjI ballI)
show "(\<lambda>x. x i) \<in> space (Pi\<^isub>P I M) \<rightarrow> space (M i)"
using M.sets_into_space `i \<in> I`
by (auto simp: infprod_algebra_def generator_def)
fix A assume "A \<in> sets (M i)"
have "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) = emb I {i} (\<Pi>\<^isub>E _\<in>{i}. A)"
by (auto simp: infprod_algebra_def generator_def emb_def)
also have "\<dots> \<in> sets (Pi\<^isub>P I M)"
using `i \<in> I` `A \<in> sets (M i)`
by (intro emb_in_infprod_algebra product_algebraI) auto
finally show "(\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M) \<in> sets (Pi\<^isub>P I M)" .
qed
lemma (in product_prob_space) sigma_product_algebra_sigma_eq:
assumes M: "\<And>i. i \<in> I \<Longrightarrow> M i = sigma (E i)"
shows "sets (Pi\<^isub>P I M) = sigma_sets (space (Pi\<^isub>P I M)) (\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (E i))"
proof -
let ?E = "(\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (E i))"
let ?M = "(\<Union>i\<in>I. (\<lambda>A. (\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) ` sets (M i))"
{ fix i A assume "i\<in>I" "A \<in> sets (E i)"
then have "A \<in> sets (M i)" using M by auto
then have "A \<in> Pow (space (M i))" using M.sets_into_space by auto
then have "A \<in> Pow (space (E i))" using M[OF `i \<in> I`] by auto }
moreover
have "\<And>i. i \<in> I \<Longrightarrow> (\<lambda>x. x i) \<in> space infprod_algebra \<rightarrow> space (E i)"
by (auto simp: M infprod_algebra_def generator_def Pi_iff)
ultimately have "sigma_sets (space (Pi\<^isub>P I M)) ?M \<subseteq> sigma_sets (space (Pi\<^isub>P I M)) ?E"
apply (intro sigma_sets_mono UN_least)
apply (simp add: sets_sigma M)
apply (subst sigma_sets_vimage[symmetric])
apply (auto intro!: sigma_sets_mono')
done
moreover have "sigma_sets (space (Pi\<^isub>P I M)) ?E \<subseteq> sigma_sets (space (Pi\<^isub>P I M)) ?M"
by (intro sigma_sets_mono') (auto simp: M)
ultimately show ?thesis
by (subst infprod_algebra_alt2) (auto simp: sets_sigma)
qed
lemma (in product_prob_space) Int_proj_eq_emb:
assumes "J \<noteq> {}" "J \<subseteq> I"
shows "(\<Inter>i\<in>J. (\<lambda>x. x i) -` A i \<inter> space (Pi\<^isub>P I M)) = emb I J (Pi\<^isub>E J A)"
using assms by (auto simp: infprod_algebra_def generator_def emb_def Pi_iff)
lemma (in product_prob_space) emb_insert:
"i \<notin> J \<Longrightarrow> emb I J (Pi\<^isub>E J f) \<inter> ((\<lambda>x. x i) -` A \<inter> space (Pi\<^isub>P I M)) =
emb I (insert i J) (Pi\<^isub>E (insert i J) (f(i := A)))"
by (auto simp: emb_def Pi_iff infprod_algebra_def generator_def split: split_if_asm)
subsection {* Sequence space *}
locale sequence_space = product_prob_space M "UNIV :: nat set" for M
lemma (in sequence_space) infprod_in_sets[intro]:
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
shows "Pi UNIV E \<in> sets (Pi\<^isub>P UNIV M)"
proof -
have "Pi UNIV E = (\<Inter>i. emb UNIV {..i} (\<Pi>\<^isub>E j\<in>{..i}. E j))"
using E E[THEN M.sets_into_space]
by (auto simp: emb_def Pi_iff extensional_def) blast
with E show ?thesis
by (auto intro: emb_in_infprod_algebra)
qed
lemma (in sequence_space) measure_infprod:
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
shows "(\<lambda>n. \<Prod>i\<le>n. M.\<mu>' i (E i)) ----> \<mu>' (Pi UNIV E)"
proof -
let ?E = "\<lambda>n. emb UNIV {..n} (Pi\<^isub>E {.. n} E)"
{ fix n :: nat
interpret n: finite_product_prob_space M "{..n}" by default auto
have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = n.\<mu>' (Pi\<^isub>E {.. n} E)"
using E by (subst n.finite_measure_times) auto
also have "\<dots> = \<mu>' (?E n)"
using E by (intro finite_measure_infprod_emb[symmetric]) auto
finally have "(\<Prod>i\<le>n. M.\<mu>' i (E i)) = \<mu>' (?E n)" . }
moreover have "Pi UNIV E = (\<Inter>n. ?E n)"
using E E[THEN M.sets_into_space]
by (auto simp: emb_def extensional_def Pi_iff) blast
moreover have "range ?E \<subseteq> sets (Pi\<^isub>P UNIV M)"
using E by auto
moreover have "decseq ?E"
by (auto simp: emb_def Pi_iff decseq_def)
ultimately show ?thesis
by (simp add: finite_continuity_from_above)
qed
end