--- a/src/HOL/Probability/Infinite_Product_Measure.thy Mon Apr 23 12:23:23 2012 +0100
+++ b/src/HOL/Probability/Infinite_Product_Measure.thy Mon Apr 23 12:14:35 2012 +0200
@@ -5,9 +5,49 @@
header {*Infinite Product Measure*}
theory Infinite_Product_Measure
- imports Probability_Measure
+ imports Probability_Measure Caratheodory
begin
+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_sigma_finite)
+ assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^isub>M (I \<union> J) M)"
+ shows emeasure_fold_integral:
+ "emeasure (Pi\<^isub>M (I \<union> J) M) A = (\<integral>\<^isup>+x. emeasure (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 emeasure_fold_measurable:
+ "(\<lambda>x. emeasure (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 "Pi\<^isub>M I M" "Pi\<^isub>M J M" ..
+ have merge: "(\<lambda>(x, y). merge I x J y) -` A \<inter> space (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M) \<in> sets (Pi\<^isub>M I M \<Otimes>\<^isub>M Pi\<^isub>M J M)"
+ by (intro measurable_sets[OF _ A] measurable_merge assms)
+
+ show ?I
+ apply (subst distr_merge[symmetric, OF IJ])
+ apply (subst emeasure_distr[OF measurable_merge[OF IJ(1)] A])
+ apply (subst IJ.emeasure_pair_measure_alt[OF merge])
+ apply (auto intro!: positive_integral_cong arg_cong2[where f=emeasure] simp: space_pair_measure)
+ done
+
+ show ?B
+ using IJ.measurable_emeasure_Pair1[OF merge]
+ by (simp add: vimage_compose[symmetric] comp_def space_pair_measure cong: measurable_cong)
+qed
+
lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
unfolding restrict_def extensional_def by auto
@@ -41,189 +81,178 @@
qed
qed
-lemma (in product_prob_space) measure_preserving_restrict:
+lemma prod_algebraI_finite:
+ "finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^isub>E I E) \<in> prod_algebra I M"
+ using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets_into_space] by simp
+
+lemma Int_stable_PiE: "Int_stable {Pi\<^isub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
+proof (safe intro!: Int_stableI)
+ fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
+ then show "\<exists>G. Pi\<^isub>E J E \<inter> Pi\<^isub>E J F = Pi\<^isub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))"
+ by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"])
+qed
+
+lemma prod_emb_trans[simp]:
+ "J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> prod_emb L M K (prod_emb K M J X) = prod_emb L M J X"
+ by (auto simp add: Int_absorb1 prod_emb_def)
+
+lemma prod_emb_Pi:
+ assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
+ shows "prod_emb K M 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: prod_emb_def Pi_iff split: split_if_asm) blast+
+
+lemma prod_emb_id:
+ "B \<subseteq> (\<Pi>\<^isub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
+ by (auto simp: prod_emb_def Pi_iff subset_eq extensional_restrict)
+
+lemma measurable_prod_emb[intro, simp]:
+ "J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^isub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^isub>M L M)"
+ unfolding prod_emb_def space_PiM[symmetric]
+ by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
+
+lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^isub>M L M) (Pi\<^isub>M J M)"
+ by (intro measurable_restrict measurable_component_singleton) auto
+
+lemma (in product_prob_space) distr_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)
+ shows "(\<Pi>\<^isub>M i\<in>J. M i) = distr (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i) (\<lambda>f. restrict f J)" (is "?P = ?D")
+proof (rule measure_eqI_generator_eq)
+ have "finite J" using `J \<subseteq> K` `finite K` by (auto simp add: finite_subset)
+ interpret J: finite_product_prob_space M J proof qed fact
+ interpret K: finite_product_prob_space M K proof qed fact
+
+ let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
+ let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)"
+ let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
+ show "Int_stable ?J"
+ by (rule Int_stable_PiE)
+ show "range ?F \<subseteq> ?J" "incseq ?F" "(\<Union>i. ?F i) = ?\<Omega>"
+ using `finite J` by (auto intro!: prod_algebraI_finite)
+ { fix i show "emeasure ?P (?F i) \<noteq> \<infinity>" by simp }
+ show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space)
+ show "sets (\<Pi>\<^isub>M i\<in>J. M i) = sigma_sets ?\<Omega> ?J" "sets ?D = sigma_sets ?\<Omega> ?J"
+ using `finite J` by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)
+
+ fix X assume "X \<in> ?J"
+ then obtain E where [simp]: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto
+ with `finite J` have X: "X \<in> sets (Pi\<^isub>M J M)" by auto
+
+ have "emeasure ?P X = (\<Prod> i\<in>J. emeasure (M i) (E i))"
+ using E by (simp add: J.measure_times)
+ also have "\<dots> = (\<Prod> i\<in>J. emeasure (M i) (if i \<in> J then E i else space (M i)))"
+ by simp
+ also have "\<dots> = (\<Prod> i\<in>K. emeasure (M i) (if i \<in> J then E i else space (M i)))"
+ using `finite K` `J \<subseteq> K`
+ by (intro setprod_mono_one_left) (auto simp: M.emeasure_space_1)
+ also have "\<dots> = emeasure (Pi\<^isub>M K M) (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))"
+ using E by (simp add: K.measure_times)
+ also have "(\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i)) = (\<lambda>f. restrict f J) -` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i))"
+ using `J \<subseteq> K` sets_into_space E by (force simp: Pi_iff split: split_if_asm)
+ finally show "emeasure (Pi\<^isub>M J M) X = emeasure ?D X"
+ using X `J \<subseteq> K` apply (subst emeasure_distr)
+ by (auto intro!: measurable_restrict_subset simp: space_PiM)
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)
+abbreviation (in product_prob_space)
+ "emb L K X \<equiv> prod_emb L M K X"
+
+lemma (in product_prob_space) emeasure_prod_emb[simp]:
+ assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" and X: "X \<in> sets (Pi\<^isub>M J M)"
+ shows "emeasure (Pi\<^isub>M L M) (emb L J X) = emeasure (Pi\<^isub>M J M) X"
+ by (subst distr_restrict[OF L])
+ (simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X)
-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) prod_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 "prod_emb L M J X = prod_emb L M J Y"
+ shows "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 sets[THEN sets_into_space] by (auto simp: space_PiM)
+ 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 `prod_emb L M J X = prod_emb L M J Y` by (simp add: prod_emb_def)
+qed fact
-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)
+definition (in product_prob_space) generator :: "('i \<Rightarrow> 'a) set set" where
+ "generator = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M))"
-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) 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> generator"
+ unfolding generator_def by auto
-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
+lemma (in product_prob_space) algebra_generator:
+ assumes "I \<noteq> {}" shows "algebra (\<Pi>\<^isub>E i\<in>I. space (M i)) generator" (is "algebra ?\<Omega> ?G")
+proof
+ let ?G = generator
+ show "?G \<subseteq> Pow ?\<Omega>"
+ by (auto simp: generator_def prod_emb_def)
+ from `I \<noteq> {}` obtain i where "i \<in> I" by auto
+ then show "{} \<in> ?G"
+ by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
+ simp: sigma_sets.Empty generator_def prod_emb_def)
+ from `i \<in> I` show "?\<Omega> \<in> ?G"
+ by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
+ simp: generator_def prod_emb_def)
+ fix A assume "A \<in> ?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> ?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"
+ 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
+ show "A - B \<in> ?G" "A \<union> B \<in> ?G"
+ unfolding * using XA XB by (safe intro!: generatorI') auto
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) sets_PiM_generator:
+ assumes "I \<noteq> {}" shows "sets (PiM I M) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
+proof
+ show "sets (Pi\<^isub>M I M) \<subseteq> sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
+ unfolding sets_PiM
+ proof (safe intro!: sigma_sets_subseteq)
+ fix A assume "A \<in> prod_algebra I M" with `I \<noteq> {}` show "A \<in> generator"
+ by (auto intro!: generatorI' elim!: prod_algebraE)
+ qed
+qed (auto simp: generator_def space_PiM[symmetric] intro!: sigma_sets_subset)
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"
+ "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> 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))"
+ (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 = emeasure (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"
+ shows "\<mu>G A = emeasure (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)"
+ have "emeasure (Pi\<^isub>M K M) Y = emeasure (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 add: prod_emb_injective[of "K \<union> J" I])
+ also have "emeasure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) J X) = emeasure (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" ..
+ finally show "emeasure (Pi\<^isub>M J M) X = emeasure (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"
+ "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) = emeasure (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"
+ assumes *: "A \<in> 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 = emeasure (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
@@ -231,11 +260,11 @@
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"
+ assumes A: "A \<in> 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 = emeasure (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
+ "\<mu>G A = emeasure (Pi\<^isub>M J M) X" by auto
then show thesis by (intro that) auto
qed
@@ -243,11 +272,7 @@
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
+ from sets_Pair1[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])
@@ -266,75 +291,27 @@
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')
+ by (simp split: split_merge add: prod_emb_def Pi_iff extensional_merge_sub set_eq_iff space_PiM)
+ auto
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
+ interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" 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
+ by (rule prod_emb_injective[of J I]) simp_all
then show "\<mu>G {} = 0" by simp
next
- fix A assume "A \<in> sets generator"
+ fix A assume "A \<in> 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
+ show "0 \<le> \<mu>G A" by (simp add: emeasure_nonneg)
qed
qed
@@ -342,102 +319,47 @@
assumes "I \<noteq> {}"
shows "additive generator \<mu>G"
proof -
- interpret G!: algebra generator by (rule algebra_generator) fact
+ interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" 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
+ fix A assume "A \<in> generator" with generatorE guess J X . note J = this
+ fix B assume "B \<in> 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 (rule prod_emb_injective[of "J \<union> K" I])
apply (insert `A \<inter> B = {}` JK J K)
- apply (simp_all add: JK.Int emb_simps)
+ apply (simp_all add: Int prod_emb_Int)
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)
+ by simp
+ also have "\<dots> = emeasure (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 Un del: prod_emb_Un)
also have "\<dots> = \<mu>G A + \<mu>G B"
- using J K JK_disj by (simp add: JK.measure_additive[symmetric])
+ using J K JK_disj by (simp add: plus_emeasure[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>"
+lemma (in product_prob_space) emeasure_PiM_emb_not_empty:
+ assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. X i \<in> sets (M i)"
+ shows "emeasure (Pi\<^isub>M I M) (emb I J (Pi\<^isub>E J X)) = emeasure (Pi\<^isub>M J M) (Pi\<^isub>E J X)"
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
+ assume "finite I" with X show ?thesis by simp
next
+ let ?\<Omega> = "\<Pi>\<^isub>E i\<in>I. space (M i)"
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
+ interpret G!: algebra ?\<Omega> 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"
+ { fix Z J assume J: "J \<noteq> {}" "finite J" "J \<subseteq> I" and Z: "Z \<in> ?G"
from `infinite I` `finite J` obtain k where k: "k \<in> I" "k \<notin> J"
by (metis rev_finite_subset subsetI)
@@ -445,7 +367,7 @@
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"
+ "K - J \<noteq> {}" "K - J \<subseteq> I" "\<mu>G Z = emeasure (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)"
@@ -455,9 +377,9 @@
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"
+ have ***: "(merge J y (I - J) -` Z \<inter> space (Pi\<^isub>M I M)) \<in> ?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)"
+ have "\<mu>G (merge J y (I - J) -` emb I K X \<inter> space (Pi\<^isub>M I M)) = emeasure (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
@@ -467,16 +389,16 @@
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)"
+ have "\<mu>G Z = emeasure (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>+ x. emeasure (Pi\<^isub>M (K - J) M) (?M x) \<partial>Pi\<^isub>M J M)"
+ using K J by (subst emeasure_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)
+ proof (intro 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)"
+ show "emeasure (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)" .
@@ -490,21 +412,18 @@
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)
+ by (simp add: merge_in_G \<mu>G_eq emeasure_fold_measurable 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>)")
+ have "\<exists>\<mu>. (\<forall>s\<in>?G. \<mu> s = \<mu>G s) \<and> measure_space ?\<Omega> (sigma_sets ?\<Omega> ?G) \<mu>"
proof (rule G.caratheodory_empty_continuous[OF positive_\<mu>G additive_\<mu>G])
- fix A assume "A \<in> sets ?G"
+ fix A assume "A \<in> ?G"
with generatorE guess J X . note JX = this
interpret JK: finite_product_prob_space M J by default fact+
from 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) = {}"
+ fix A assume A: "range A \<subseteq> ?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
@@ -515,7 +434,7 @@
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"
+ 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) = emeasure (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)"
@@ -524,8 +443,8 @@
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)
+ with A' have A_eq: "\<And>n. A n = emb I (J n) (X n)" "\<And>n. A n \<in> ?G"
+ unfolding J_def X_def by (subst prod_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
@@ -538,8 +457,8 @@
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 Z k assume Z: "range Z \<subseteq> ?G" "decseq Z" "\<forall>n. ?a / 2^k \<le> \<mu>G (Z n)"
+ then have Z_sets: "\<And>n. Z n \<in> ?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+
@@ -552,13 +471,13 @@
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))"
+ have "?a / 2^(k+1) \<le> (INF n. emeasure (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)
+ unfolding fold(2)[OF J' `Z n \<in> ?G`]
+ proof (intro 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
@@ -568,15 +487,15 @@
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)"])
+ also have "\<dots> = emeasure (Pi\<^isub>M J' M) (?Q n) + ?a / 2^(k+1)"
+ using `0 \<le> ?a` Q_sets J'.emeasure_space_1
+ by (subst positive_integral_add) auto
+ finally show "?a / 2^(k+1) \<le> emeasure (Pi\<^isub>M J' M) (?Q n)" using `?a \<le> 1`
+ by (cases rule: ereal2_cases[of ?a "emeasure (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)
+ also have "\<dots> = emeasure (Pi\<^isub>M J' M) (\<Inter>n. ?Q n)"
+ proof (intro INF_emeasure_decseq)
show "range ?Q \<subseteq> sets (Pi\<^isub>M J' M)" using Q_sets by auto
show "decseq ?Q"
unfolding decseq_def
@@ -587,13 +506,13 @@
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 \<in> ?G" "?M J' (Z m) x \<in> ?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)
+ qed simp
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 }
@@ -631,12 +550,12 @@
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)
+ by (simp add: extensional_restrict space_PiM)
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"
+ have "range (\<lambda>n. ?M (J k) (A n) (w k)) \<subseteq> ?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
@@ -651,11 +570,11 @@
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+
+ by (auto simp: space_PiM 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 (auto split: split_merge intro!: extensional_merge_sub ext simp: space_PiM)
apply (force simp: extensional_def)
done
qed
@@ -675,7 +594,7 @@
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)
+ by (intro rev_bexI) (auto intro!: ext simp: extensional_def space_PiM)
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)"
@@ -707,17 +626,17 @@
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)+ }
+ using w(1) by (cases "i \<in> (\<Union>k. J k)") (force simp: w'_simps2 w'_eq space_PiM)+ }
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)
+ using w(1) by (auto simp add: Pi_iff extensional_def space_PiM)
{ 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)
+ by (auto simp add: fun_eq_iff restrict_def Pi_iff extensional_def space_PiM)
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> A n" unfolding A_eq using w' by (auto simp: prod_emb_def space_PiM) }
then have "w' \<in> (\<Inter>i. A i)" by auto
with `(\<Inter>i. A i) = {}` show False by auto
qed
@@ -726,276 +645,76 @@
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
+ proof (subst emeasure_extend_measure_Pair[OF PiM_def, of I M \<mu> J X])
+ from assms show "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))"
+ by (simp add: Pi_iff)
+ next
+ fix J X assume J: "(J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))"
+ then show "emb I J (Pi\<^isub>E J X) \<in> Pow (\<Pi>\<^isub>E i\<in>I. space (M i))"
+ by (auto simp: Pi_iff prod_emb_def dest: sets_into_space)
+ have "emb I J (Pi\<^isub>E J X) \<in> generator"
+ using J `I \<noteq> {}` by (intro generatorI') auto
+ then have "\<mu> (emb I J (Pi\<^isub>E J X)) = \<mu>G (emb I J (Pi\<^isub>E J X))"
+ using \<mu> by simp
+ also have "\<dots> = (\<Prod> j\<in>J. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
+ using J `I \<noteq> {}` by (subst \<mu>G_spec[OF _ _ _ refl]) (auto simp: emeasure_PiM Pi_iff)
+ also have "\<dots> = (\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}.
+ if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
+ using J `I \<noteq> {}` by (intro setprod_mono_one_right) (auto simp: M.emeasure_space_1)
+ finally show "\<mu> (emb I J (Pi\<^isub>E J X)) = \<dots>" .
+ next
+ let ?F = "\<lambda>j. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j))"
+ have "(\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. ?F j) = (\<Prod>j\<in>J. ?F j)"
+ using X `I \<noteq> {}` by (intro setprod_mono_one_right) (auto simp: M.emeasure_space_1)
+ then show "(\<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. ?F j) =
+ emeasure (Pi\<^isub>M J M) (Pi\<^isub>E J X)"
+ using X by (auto simp add: emeasure_PiM)
+ next
+ show "positive (sets (Pi\<^isub>M I M)) \<mu>" "countably_additive (sets (Pi\<^isub>M I M)) \<mu>"
+ using \<mu> unfolding sets_PiM_generator[OF `I \<noteq> {}`] by (auto simp: measure_space_def)
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"
+sublocale product_prob_space \<subseteq> P: prob_space "Pi\<^isub>M I M"
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)
+ show "emeasure (Pi\<^isub>M I M) (space (Pi\<^isub>M I M)) = 1"
+ proof cases
+ assume "I = {}" then show ?thesis by (simp add: space_PiM_empty)
+ next
+ assume "I \<noteq> {}"
+ then obtain i where "i \<in> I" by auto
+ moreover then have "emb I {i} (\<Pi>\<^isub>E i\<in>{i}. space (M i)) = (space (Pi\<^isub>M I M))"
+ by (auto simp: prod_emb_def space_PiM)
+ ultimately show ?thesis
+ using emeasure_PiM_emb_not_empty[of "{i}" "\<lambda>i. space (M i)"]
+ by (simp add: emeasure_PiM emeasure_space_1)
+ qed
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 [abs_def] 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)
+lemma (in product_prob_space) emeasure_PiM_emb:
+ assumes X: "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)"
+ shows "emeasure (Pi\<^isub>M I M) (emb I J (Pi\<^isub>E J X)) = (\<Prod> i\<in>J. emeasure (M i) (X i))"
+proof cases
+ assume "J = {}"
+ moreover have "emb I {} {\<lambda>x. undefined} = space (Pi\<^isub>M I M)"
+ by (auto simp: space_PiM prod_emb_def)
+ ultimately show ?thesis
+ by (simp add: space_PiM_empty P.emeasure_space_1)
+next
+ assume "J \<noteq> {}" with X show ?thesis
+ by (subst emeasure_PiM_emb_not_empty) (auto simp: emeasure_PiM)
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) measure_PiM_emb:
+ assumes "J \<subseteq> I" "finite J" "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)"
+ shows "measure (PiM I M) (emb I J (Pi\<^isub>E J X)) = (\<Prod> i\<in>J. measure (M i) (X i))"
+ using emeasure_PiM_emb[OF assms]
+ unfolding emeasure_eq_measure M.emeasure_eq_measure by (simp add: setprod_ereal)
-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)
+lemma (in finite_product_prob_space) finite_measure_PiM_emb:
+ "(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> measure (PiM I M) (Pi\<^isub>E I A) = (\<Prod>i\<in>I. measure (M i) (A i))"
+ using measure_PiM_emb[of I A] finite_index prod_emb_PiE_same_index[OF sets_into_space, of I A M]
+ by auto
subsection {* Sequence space *}
@@ -1003,36 +722,30 @@
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)"
+ shows "Pi UNIV E \<in> sets (Pi\<^isub>M 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)
+ using E E[THEN sets_into_space]
+ by (auto simp: prod_emb_def Pi_iff extensional_def) blast
+ with E show ?thesis by auto
qed
-lemma (in sequence_space) measure_infprod:
+lemma (in sequence_space) measure_PiM_countable:
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)"
+ shows "(\<lambda>n. \<Prod>i\<le>n. measure (M i) (E i)) ----> measure (Pi\<^isub>M UNIV M) (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)" . }
+ have "\<And>n. (\<Prod>i\<le>n. measure (M i) (E i)) = measure (Pi\<^isub>M UNIV M) (?E n)"
+ using E by (simp add: measure_PiM_emb)
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 E[THEN sets_into_space]
+ by (auto simp: prod_emb_def extensional_def Pi_iff) blast
+ moreover have "range ?E \<subseteq> sets (Pi\<^isub>M UNIV M)"
using E by auto
moreover have "decseq ?E"
- by (auto simp: emb_def Pi_iff decseq_def)
+ by (auto simp: prod_emb_def Pi_iff decseq_def)
ultimately show ?thesis
- by (simp add: finite_continuity_from_above)
+ by (simp add: finite_Lim_measure_decseq)
qed
end
\ No newline at end of file