added projective_family; generalized generator in product_prob_space to projective_family
authorimmler@in.tum.de
Wed Nov 07 11:33:27 2012 +0100 (2012-11-07)
changeset 50039bfd5198cbe40
parent 50038 8e32c9254535
child 50040 5da32dc55cd8
added projective_family; generalized generator in product_prob_space to projective_family
src/HOL/Probability/Infinite_Product_Measure.thy
src/HOL/Probability/Projective_Family.thy
     1.1 --- a/src/HOL/Probability/Infinite_Product_Measure.thy	Tue Nov 06 11:03:28 2012 +0100
     1.2 +++ b/src/HOL/Probability/Infinite_Product_Measure.thy	Wed Nov 07 11:33:27 2012 +0100
     1.3 @@ -5,7 +5,7 @@
     1.4  header {*Infinite Product Measure*}
     1.5  
     1.6  theory Infinite_Product_Measure
     1.7 -  imports Probability_Measure Caratheodory
     1.8 +  imports Probability_Measure Caratheodory Projective_Family
     1.9  begin
    1.10  
    1.11  lemma split_merge: "P (merge I J (x,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)"
    1.12 @@ -76,16 +76,20 @@
    1.13      by (auto intro!: measurable_restrict_subset simp: space_PiM)
    1.14  qed
    1.15  
    1.16 -abbreviation (in product_prob_space)
    1.17 -  "emb L K X \<equiv> prod_emb L M K X"
    1.18 -
    1.19  lemma (in product_prob_space) emeasure_prod_emb[simp]:
    1.20    assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" and X: "X \<in> sets (Pi\<^isub>M J M)"
    1.21 -  shows "emeasure (Pi\<^isub>M L M) (emb L J X) = emeasure (Pi\<^isub>M J M) X"
    1.22 +  shows "emeasure (Pi\<^isub>M L M) (prod_emb L M J X) = emeasure (Pi\<^isub>M J M) X"
    1.23    by (subst distr_restrict[OF L])
    1.24       (simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X)
    1.25  
    1.26 -lemma (in product_prob_space) prod_emb_injective:
    1.27 +sublocale product_prob_space \<subseteq> projective_family I "\<lambda>J. PiM J M" M
    1.28 +proof
    1.29 +  fix J::"'i set" assume "finite J"
    1.30 +  interpret f: finite_product_prob_space M J proof qed fact
    1.31 +  show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) \<noteq> \<infinity>" by simp
    1.32 +qed simp_all
    1.33 +
    1.34 +lemma (in projective_family) prod_emb_injective:
    1.35    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)"
    1.36    assumes "prod_emb L M J X = prod_emb L M J Y"
    1.37    shows "X = Y"
    1.38 @@ -100,14 +104,17 @@
    1.39      using `prod_emb L M J X = prod_emb L M J Y` by (simp add: prod_emb_def)
    1.40  qed fact
    1.41  
    1.42 -definition (in product_prob_space) generator :: "('i \<Rightarrow> 'a) set set" where
    1.43 +abbreviation (in projective_family)
    1.44 +  "emb L K X \<equiv> prod_emb L M K X"
    1.45 +
    1.46 +definition (in projective_family) generator :: "('i \<Rightarrow> 'a) set set" where
    1.47    "generator = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M))"
    1.48  
    1.49 -lemma (in product_prob_space) generatorI':
    1.50 +lemma (in projective_family) generatorI':
    1.51    "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"
    1.52    unfolding generator_def by auto
    1.53  
    1.54 -lemma (in product_prob_space) algebra_generator:
    1.55 +lemma (in projective_family) algebra_generator:
    1.56    assumes "I \<noteq> {}" shows "algebra (\<Pi>\<^isub>E i\<in>I. space (M i)) generator" (is "algebra ?\<Omega> ?G")
    1.57    unfolding algebra_def algebra_axioms_def ring_of_sets_iff
    1.58  proof (intro conjI ballI)
    1.59 @@ -135,7 +142,7 @@
    1.60      unfolding * using XA XB by (safe intro!: generatorI') auto
    1.61  qed
    1.62  
    1.63 -lemma (in product_prob_space) sets_PiM_generator:
    1.64 +lemma (in projective_family) sets_PiM_generator:
    1.65    "sets (PiM I M) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
    1.66  proof cases
    1.67    assume "I = {}" then show ?thesis
    1.68 @@ -154,57 +161,56 @@
    1.69    qed (auto simp: generator_def space_PiM[symmetric] intro!: sigma_sets_subset)
    1.70  qed
    1.71  
    1.72 -
    1.73 -lemma (in product_prob_space) generatorI:
    1.74 +lemma (in projective_family) generatorI:
    1.75    "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"
    1.76    unfolding generator_def by auto
    1.77  
    1.78 -definition (in product_prob_space)
    1.79 +definition (in projective_family)
    1.80    "\<mu>G A =
    1.81 -    (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))"
    1.82 +    (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 (PiP J M P) X))"
    1.83  
    1.84 -lemma (in product_prob_space) \<mu>G_spec:
    1.85 +lemma (in projective_family) \<mu>G_spec:
    1.86    assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
    1.87 -  shows "\<mu>G A = emeasure (Pi\<^isub>M J M) X"
    1.88 +  shows "\<mu>G A = emeasure (PiP J M P) X"
    1.89    unfolding \<mu>G_def
    1.90  proof (intro the_equality allI impI ballI)
    1.91    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)"
    1.92 -  have "emeasure (Pi\<^isub>M K M) Y = emeasure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) K Y)"
    1.93 +  have "emeasure (PiP K M P) Y = emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) K Y)"
    1.94      using K J by simp
    1.95    also have "emb (K \<union> J) K Y = emb (K \<union> J) J X"
    1.96      using K J by (simp add: prod_emb_injective[of "K \<union> J" I])
    1.97 -  also have "emeasure (Pi\<^isub>M (K \<union> J) M) (emb (K \<union> J) J X) = emeasure (Pi\<^isub>M J M) X"
    1.98 +  also have "emeasure (PiP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (PiP J M P) X"
    1.99      using K J by simp
   1.100 -  finally show "emeasure (Pi\<^isub>M J M) X = emeasure (Pi\<^isub>M K M) Y" ..
   1.101 +  finally show "emeasure (PiP J M P) X = emeasure (PiP K M P) Y" ..
   1.102  qed (insert J, force)
   1.103  
   1.104 -lemma (in product_prob_space) \<mu>G_eq:
   1.105 -  "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"
   1.106 +lemma (in projective_family) \<mu>G_eq:
   1.107 +  "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 (PiP J M P) X"
   1.108    by (intro \<mu>G_spec) auto
   1.109  
   1.110 -lemma (in product_prob_space) generator_Ex:
   1.111 +lemma (in projective_family) generator_Ex:
   1.112    assumes *: "A \<in> generator"
   1.113 -  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"
   1.114 +  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 (PiP J M P) X"
   1.115  proof -
   1.116    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)"
   1.117      unfolding generator_def by auto
   1.118    with \<mu>G_spec[OF this] show ?thesis by auto
   1.119  qed
   1.120  
   1.121 -lemma (in product_prob_space) generatorE:
   1.122 +lemma (in projective_family) generatorE:
   1.123    assumes A: "A \<in> generator"
   1.124 -  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"
   1.125 +  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 (PiP J M P) X"
   1.126  proof -
   1.127    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"
   1.128 -    "\<mu>G A = emeasure (Pi\<^isub>M J M) X" by auto
   1.129 +    "\<mu>G A = emeasure (PiP J M P) X" by auto
   1.130    then show thesis by (intro that) auto
   1.131  qed
   1.132  
   1.133 -lemma (in product_prob_space) merge_sets:
   1.134 +lemma (in projective_family) merge_sets:
   1.135    "J \<inter> K = {} \<Longrightarrow> A \<in> sets (Pi\<^isub>M (J \<union> K) M) \<Longrightarrow> x \<in> space (Pi\<^isub>M J M) \<Longrightarrow> (\<lambda>y. merge J K (x,y)) -` A \<inter> space (Pi\<^isub>M K M) \<in> sets (Pi\<^isub>M K M)"
   1.136    by simp
   1.137  
   1.138 -lemma (in product_prob_space) merge_emb:
   1.139 +lemma (in projective_family) merge_emb:
   1.140    assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)"
   1.141    shows "((\<lambda>x. merge J (I - J) (y, x)) -` emb I K X \<inter> space (Pi\<^isub>M I M)) =
   1.142      emb I (K - J) ((\<lambda>x. merge J (K - J) (y, x)) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M))"
   1.143 @@ -221,7 +227,7 @@
   1.144         auto
   1.145  qed
   1.146  
   1.147 -lemma (in product_prob_space) positive_\<mu>G: 
   1.148 +lemma (in projective_family) positive_\<mu>G:
   1.149    assumes "I \<noteq> {}"
   1.150    shows "positive generator \<mu>G"
   1.151  proof -
   1.152 @@ -241,7 +247,7 @@
   1.153    qed
   1.154  qed
   1.155  
   1.156 -lemma (in product_prob_space) additive_\<mu>G: 
   1.157 +lemma (in projective_family) additive_\<mu>G:
   1.158    assumes "I \<noteq> {}"
   1.159    shows "additive generator \<mu>G"
   1.160  proof -
   1.161 @@ -263,7 +269,7 @@
   1.162        using J K by simp_all
   1.163      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))"
   1.164        by simp
   1.165 -    also have "\<dots> = emeasure (Pi\<^isub>M (J \<union> K) M) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
   1.166 +    also have "\<dots> = emeasure (PiP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
   1.167        using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq Un del: prod_emb_Un)
   1.168      also have "\<dots> = \<mu>G A + \<mu>G B"
   1.169        using J K JK_disj by (simp add: plus_emeasure[symmetric])
   1.170 @@ -271,6 +277,10 @@
   1.171    qed
   1.172  qed
   1.173  
   1.174 +lemma (in product_prob_space) PiP_PiM_finite[simp]:
   1.175 +  assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" shows "PiP J M (\<lambda>J. PiM J M) = PiM J M"
   1.176 +  using assms by (simp add: PiP_finite)
   1.177 +
   1.178  lemma (in product_prob_space) emeasure_PiM_emb_not_empty:
   1.179    assumes X: "J \<noteq> {}" "J \<subseteq> I" "finite J" "\<forall>i\<in>J. X i \<in> sets (M i)"
   1.180    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)"
   1.181 @@ -295,7 +305,6 @@
   1.182      ultimately have K: "K \<noteq> {}" "finite K" "K \<subseteq> I" "X \<in> sets (Pi\<^isub>M K M)" "Z = emb I K X"
   1.183        "K - J \<noteq> {}" "K - J \<subseteq> I" "\<mu>G Z = emeasure (Pi\<^isub>M K M) X"
   1.184        by (auto simp: subset_insertI)
   1.185 -
   1.186      let ?M = "\<lambda>y. (\<lambda>x. merge J (K - J) (y, x)) -` emb (J \<union> K) K X \<inter> space (Pi\<^isub>M (K - J) M)"
   1.187      { fix y assume y: "y \<in> space (Pi\<^isub>M J M)"
   1.188        note * = merge_emb[OF `K \<subseteq> I` `J \<subseteq> I` y, of X]
   1.189 @@ -360,7 +369,7 @@
   1.190          using A positive_\<mu>G[OF I_not_empty] by (auto intro!: INF_greatest simp: positive_def)
   1.191        ultimately have "0 < ?a" by auto
   1.192  
   1.193 -      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"
   1.194 +      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 (PiP J M (\<lambda>J. (Pi\<^isub>M J M))) X"
   1.195          using A by (intro allI generator_Ex) auto
   1.196        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)"
   1.197          and A': "\<And>n. A n = emb I (J' n) (X' n)"
     2.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     2.2 +++ b/src/HOL/Probability/Projective_Family.thy	Wed Nov 07 11:33:27 2012 +0100
     2.3 @@ -0,0 +1,113 @@
     2.4 +theory Projective_Family
     2.5 +imports Finite_Product_Measure Probability_Measure
     2.6 +begin
     2.7 +
     2.8 +definition
     2.9 +  PiP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
    2.10 +  "PiP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
    2.11 +    {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
    2.12 +    (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
    2.13 +    (\<lambda>(J, X). emeasure (P J) (Pi\<^isub>E J X))"
    2.14 +
    2.15 +lemma space_PiP[simp]: "space (PiP I M P) = space (PiM I M)"
    2.16 +  by (auto simp add: PiP_def space_PiM prod_emb_def intro!: space_extend_measure)
    2.17 +
    2.18 +lemma sets_PiP[simp]: "sets (PiP I M P) = sets (PiM I M)"
    2.19 +  by (auto simp add: PiP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_extend_measure)
    2.20 +
    2.21 +lemma measurable_PiP1[simp]: "measurable (PiP I M P) M' = measurable (\<Pi>\<^isub>M i\<in>I. M i) M'"
    2.22 +  unfolding measurable_def by auto
    2.23 +
    2.24 +lemma measurable_PiP2[simp]: "measurable M' (PiP I M P) = measurable M' (\<Pi>\<^isub>M i\<in>I. M i)"
    2.25 +  unfolding measurable_def by auto
    2.26 +
    2.27 +locale projective_family =
    2.28 +  fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)"
    2.29 +  assumes projective: "\<And>J H X. J \<noteq> {} \<Longrightarrow> J \<subseteq> H \<Longrightarrow> H \<subseteq> I \<Longrightarrow> finite H \<Longrightarrow> X \<in> sets (PiM J M) \<Longrightarrow>
    2.30 +     (P H) (prod_emb H M J X) = (P J) X"
    2.31 +  assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)"
    2.32 +  assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)"
    2.33 +  assumes proj_finite_measure: "\<And>J. finite J \<Longrightarrow> emeasure (P J) (space (PiM J M)) \<noteq> \<infinity>"
    2.34 +  assumes prob_space: "\<And>i. prob_space (M i)"
    2.35 +begin
    2.36 +
    2.37 +lemma emeasure_PiP:
    2.38 +  assumes "J \<noteq> {}"
    2.39 +  assumes "finite J"
    2.40 +  assumes "J \<subseteq> I"
    2.41 +  assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)"
    2.42 +  shows "emeasure (PiP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)"
    2.43 +proof -
    2.44 +  have "Pi\<^isub>E J (restrict A J) \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
    2.45 +  proof safe
    2.46 +    fix x j assume "x \<in> Pi J (restrict A J)" "j \<in> J"
    2.47 +    hence "x j \<in> restrict A J j" by (auto simp: Pi_def)
    2.48 +    also have "\<dots> \<subseteq> space (M j)" using sets_into_space A `j \<in> J` by auto
    2.49 +    finally show "x j \<in> space (M j)" .
    2.50 +  qed
    2.51 +  hence "emeasure (PiP J M P) (Pi\<^isub>E J A) =
    2.52 +    emeasure (PiP J M P) (prod_emb J M J (Pi\<^isub>E J A))"
    2.53 +    using assms(1-3) sets_into_space by (auto simp add: prod_emb_id Pi_def)
    2.54 +  also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)"
    2.55 +  proof (rule emeasure_extend_measure[OF PiP_def, where i="(J, A)", simplified,
    2.56 +        of J M "P J" P])
    2.57 +    show "positive (sets (PiM J M)) (P J)" unfolding positive_def by auto
    2.58 +    show "countably_additive (sets (PiM J M)) (P J)" unfolding countably_additive_def
    2.59 +      by (auto simp: suminf_emeasure proj_sets[OF `finite J`])
    2.60 +    show "(\<lambda>(Ja, X). prod_emb J M Ja (Pi\<^isub>E Ja X)) ` {(Ja, X). (Ja = {} \<longrightarrow> J = {}) \<and>
    2.61 +      finite Ja \<and> Ja \<subseteq> J \<and> X \<in> (\<Pi> j\<in>Ja. sets (M j))} \<subseteq> Pow (\<Pi> i\<in>J. space (M i)) \<and>
    2.62 +      (\<lambda>(Ja, X). prod_emb J M Ja (Pi\<^isub>E Ja X)) `
    2.63 +        {(Ja, X). (Ja = {} \<longrightarrow> J = {}) \<and> finite Ja \<and> Ja \<subseteq> J \<and> X \<in> (\<Pi> j\<in>Ja. sets (M j))} \<subseteq>
    2.64 +        Pow (extensional J)" by (auto simp: prod_emb_def)
    2.65 +    show "(J = {} \<longrightarrow> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
    2.66 +      using assms by auto
    2.67 +    fix i
    2.68 +    assume
    2.69 +      "case i of (Ja, X) \<Rightarrow> (Ja = {} \<longrightarrow> J = {}) \<and> finite Ja \<and> Ja \<subseteq> J \<and> X \<in> (\<Pi> j\<in>Ja. sets (M j))"
    2.70 +    thus "emeasure (P J) (case i of (Ja, X) \<Rightarrow> prod_emb J M Ja (Pi\<^isub>E Ja X)) =
    2.71 +        (case i of (J, X) \<Rightarrow> emeasure (P J) (Pi\<^isub>E J X))" using assms
    2.72 +      by (cases i) (auto simp add: intro!: projective sets_PiM_I_finite)
    2.73 +  qed
    2.74 +  finally show ?thesis .
    2.75 +qed
    2.76 +
    2.77 +lemma PiP_finite:
    2.78 +  assumes "J \<noteq> {}"
    2.79 +  assumes "finite J"
    2.80 +  assumes "J \<subseteq> I"
    2.81 +  shows "PiP J M P = P J" (is "?P = _")
    2.82 +proof (rule measure_eqI_generator_eq)
    2.83 +  let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
    2.84 +  let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)"
    2.85 +  let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
    2.86 +  show "Int_stable ?J"
    2.87 +    by (rule Int_stable_PiE)
    2.88 +  interpret finite_measure "P J" using proj_finite_measure `finite J`
    2.89 +    by (intro finite_measureI) (simp add: proj_space)
    2.90 +  show "emeasure ?P (?F _) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_PiP)
    2.91 +  show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space)
    2.92 +  show "sets (PiP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J"
    2.93 +    using `finite J` proj_sets by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)
    2.94 +  fix X assume "X \<in> ?J"
    2.95 +  then obtain E where X: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto
    2.96 +  with `finite J` have "X \<in> sets (PiP J M P)" by simp
    2.97 +  have emb_self: "prod_emb J M J (Pi\<^isub>E J E) = Pi\<^isub>E J E"
    2.98 +    using E sets_into_space
    2.99 +    by (auto intro!: prod_emb_PiE_same_index)
   2.100 +  show "emeasure (PiP J M P) X = emeasure (P J) X"
   2.101 +    unfolding X using E
   2.102 +    by (intro emeasure_PiP assms) simp
   2.103 +qed (insert `finite J`, auto intro!: prod_algebraI_finite)
   2.104 +
   2.105 +lemma emeasure_fun_emb[simp]:
   2.106 +  assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" "L \<subseteq> I" and X: "X \<in> sets (PiM J M)"
   2.107 +  shows "emeasure (PiP L M P) (prod_emb L M J X) = emeasure (PiP J M P) X"
   2.108 +  using assms
   2.109 +  by (subst PiP_finite) (auto simp: PiP_finite finite_subset projective)
   2.110 +
   2.111 +end
   2.112 +
   2.113 +sublocale projective_family \<subseteq> M: prob_space "M i" for i
   2.114 +  by (rule prob_space)
   2.115 +
   2.116 +end