1.1 --- a/src/HOL/Probability/Infinite_Product_Measure.thy	Wed Nov 07 11:33:27 2012 +0100
     1.2 +++ b/src/HOL/Probability/Infinite_Product_Measure.thy	Wed Nov 07 14:41:49 2012 +0100
     1.3 @@ -87,6 +87,11 @@
     1.4    fix J::"'i set" assume "finite J"
     1.5    interpret f: finite_product_prob_space M J proof qed fact
     1.6    show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) \<noteq> \<infinity>" by simp
     1.7 +  show "\<exists>A. range A \<subseteq> sets (Pi\<^isub>M J M) \<and>
     1.8 +            (\<Union>i. A i) = space (Pi\<^isub>M J M) \<and>
     1.9 +            (\<forall>i. emeasure (Pi\<^isub>M J M) (A i) \<noteq> \<infinity>)" using sigma_finite[OF `finite J`]
    1.10 +    by (auto simp add: sigma_finite_measure_def)
    1.11 +  show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) = 1" by (rule f.emeasure_space_1)
    1.12  qed simp_all
    1.13  
    1.14  lemma (in projective_family) prod_emb_injective:

     2.1 --- a/src/HOL/Probability/Projective_Family.thy	Wed Nov 07 11:33:27 2012 +0100
     2.2 +++ b/src/HOL/Probability/Projective_Family.thy	Wed Nov 07 14:41:49 2012 +0100
     2.3 @@ -25,14 +25,14 @@
     2.4    fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)"
     2.5    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>
     2.6       (P H) (prod_emb H M J X) = (P J) X"
     2.7 +  assumes prob_space: "\<And>J. finite J \<Longrightarrow> prob_space (P J)"
     2.8    assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)"
     2.9    assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)"
    2.10    assumes proj_finite_measure: "\<And>J. finite J \<Longrightarrow> emeasure (P J) (space (PiM J M)) \<noteq> \<infinity>"
    2.11 -  assumes prob_space: "\<And>i. prob_space (M i)"
    2.12 +  assumes measure_space: "\<And>i. prob_space (M i)"
    2.13  begin
    2.14  
    2.15  lemma emeasure_PiP:
    2.16 -  assumes "J \<noteq> {}"
    2.17    assumes "finite J"
    2.18    assumes "J \<subseteq> I"
    2.19    assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)"
    2.20 @@ -49,30 +49,27 @@
    2.21      emeasure (PiP J M P) (prod_emb J M J (Pi\<^isub>E J A))"
    2.22      using assms(1-3) sets_into_space by (auto simp add: prod_emb_id Pi_def)
    2.23    also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)"
    2.24 -  proof (rule emeasure_extend_measure[OF PiP_def, where i="(J, A)", simplified,
    2.25 -        of J M "P J" P])
    2.26 -    show "positive (sets (PiM J M)) (P J)" unfolding positive_def by auto
    2.27 -    show "countably_additive (sets (PiM J M)) (P J)" unfolding countably_additive_def
    2.28 +  proof (rule emeasure_extend_measure_Pair[OF PiP_def])
    2.29 +    show "positive (sets (PiP J M P)) (P J)" unfolding positive_def by auto
    2.30 +    show "countably_additive (sets (PiP J M P)) (P J)" unfolding countably_additive_def
    2.31        by (auto simp: suminf_emeasure proj_sets[OF `finite J`])
    2.32 -    show "(\<lambda>(Ja, X). prod_emb J M Ja (Pi\<^isub>E Ja X)) ` {(Ja, X). (Ja = {} \<longrightarrow> J = {}) \<and>
    2.33 -      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>
    2.34 -      (\<lambda>(Ja, X). prod_emb J M Ja (Pi\<^isub>E Ja X)) `
    2.35 -        {(Ja, X). (Ja = {} \<longrightarrow> J = {}) \<and> finite Ja \<and> Ja \<subseteq> J \<and> X \<in> (\<Pi> j\<in>Ja. sets (M j))} \<subseteq>
    2.36 -        Pow (extensional J)" by (auto simp: prod_emb_def)
    2.37 -    show "(J = {} \<longrightarrow> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
    2.38 +    show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
    2.39        using assms by auto
    2.40 -    fix i
    2.41 -    assume
    2.42 -      "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))"
    2.43 -    thus "emeasure (P J) (case i of (Ja, X) \<Rightarrow> prod_emb J M Ja (Pi\<^isub>E Ja X)) =
    2.44 -        (case i of (J, X) \<Rightarrow> emeasure (P J) (Pi\<^isub>E J X))" using assms
    2.45 -      by (cases i) (auto simp add: intro!: projective sets_PiM_I_finite)
    2.46 +    fix K and X::"'i \<Rightarrow> 'a set"
    2.47 +    show "prod_emb J M K (Pi\<^isub>E K X) \<in> Pow (\<Pi>\<^isub>E i\<in>J. space (M i))"
    2.48 +      by (auto simp: prod_emb_def)
    2.49 +    assume JX: "(K \<noteq> {} \<or> J = {}) \<and> finite K \<and> K \<subseteq> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))"
    2.50 +    thus "emeasure (P J) (prod_emb J M K (Pi\<^isub>E K X)) = emeasure (P K) (Pi\<^isub>E K X)"
    2.51 +      using assms
    2.52 +      apply (cases "J = {}")
    2.53 +      apply (simp add: prod_emb_id)
    2.54 +      apply (fastforce simp add: intro!: projective sets_PiM_I_finite)
    2.55 +      done
    2.56    qed
    2.57    finally show ?thesis .
    2.58  qed
    2.59  
    2.60  lemma PiP_finite:
    2.61 -  assumes "J \<noteq> {}"
    2.62    assumes "finite J"
    2.63    assumes "J \<subseteq> I"
    2.64    shows "PiP J M P = P J" (is "?P = _")
    2.65 @@ -108,6 +105,6 @@
    2.66  end
    2.67  
    2.68  sublocale projective_family \<subseteq> M: prob_space "M i" for i
    2.69 -  by (rule prob_space)
    2.70 +  by (rule measure_space)
    2.71  
    2.72  end