src/HOL/Probability/Projective_Family.thy
changeset 50040 5da32dc55cd8
parent 50039 bfd5198cbe40
child 50041 afe886a04198
equal deleted inserted replaced
50039:bfd5198cbe40 50040:5da32dc55cd8
    23 
    23 
    24 locale projective_family =
    24 locale projective_family =
    25   fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)"
    25   fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)"
    26   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>
    26   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>
    27      (P H) (prod_emb H M J X) = (P J) X"
    27      (P H) (prod_emb H M J X) = (P J) X"
       
    28   assumes prob_space: "\<And>J. finite J \<Longrightarrow> prob_space (P J)"
    28   assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)"
    29   assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)"
    29   assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)"
    30   assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)"
    30   assumes proj_finite_measure: "\<And>J. finite J \<Longrightarrow> emeasure (P J) (space (PiM J M)) \<noteq> \<infinity>"
    31   assumes proj_finite_measure: "\<And>J. finite J \<Longrightarrow> emeasure (P J) (space (PiM J M)) \<noteq> \<infinity>"
    31   assumes prob_space: "\<And>i. prob_space (M i)"
    32   assumes measure_space: "\<And>i. prob_space (M i)"
    32 begin
    33 begin
    33 
    34 
    34 lemma emeasure_PiP:
    35 lemma emeasure_PiP:
    35   assumes "J \<noteq> {}"
       
    36   assumes "finite J"
    36   assumes "finite J"
    37   assumes "J \<subseteq> I"
    37   assumes "J \<subseteq> I"
    38   assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)"
    38   assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)"
    39   shows "emeasure (PiP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)"
    39   shows "emeasure (PiP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)"
    40 proof -
    40 proof -
    47   qed
    47   qed
    48   hence "emeasure (PiP J M P) (Pi\<^isub>E J A) =
    48   hence "emeasure (PiP J M P) (Pi\<^isub>E J A) =
    49     emeasure (PiP J M P) (prod_emb J M J (Pi\<^isub>E J A))"
    49     emeasure (PiP J M P) (prod_emb J M J (Pi\<^isub>E J A))"
    50     using assms(1-3) sets_into_space by (auto simp add: prod_emb_id Pi_def)
    50     using assms(1-3) sets_into_space by (auto simp add: prod_emb_id Pi_def)
    51   also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)"
    51   also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)"
    52   proof (rule emeasure_extend_measure[OF PiP_def, where i="(J, A)", simplified,
    52   proof (rule emeasure_extend_measure_Pair[OF PiP_def])
    53         of J M "P J" P])
    53     show "positive (sets (PiP J M P)) (P J)" unfolding positive_def by auto
    54     show "positive (sets (PiM J M)) (P J)" unfolding positive_def by auto
    54     show "countably_additive (sets (PiP J M P)) (P J)" unfolding countably_additive_def
    55     show "countably_additive (sets (PiM J M)) (P J)" unfolding countably_additive_def
       
    56       by (auto simp: suminf_emeasure proj_sets[OF `finite J`])
    55       by (auto simp: suminf_emeasure proj_sets[OF `finite J`])
    57     show "(\<lambda>(Ja, X). prod_emb J M Ja (Pi\<^isub>E Ja X)) ` {(Ja, X). (Ja = {} \<longrightarrow> J = {}) \<and>
    56     show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
    58       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>
       
    59       (\<lambda>(Ja, X). prod_emb J M Ja (Pi\<^isub>E Ja X)) `
       
    60         {(Ja, X). (Ja = {} \<longrightarrow> J = {}) \<and> finite Ja \<and> Ja \<subseteq> J \<and> X \<in> (\<Pi> j\<in>Ja. sets (M j))} \<subseteq>
       
    61         Pow (extensional J)" by (auto simp: prod_emb_def)
       
    62     show "(J = {} \<longrightarrow> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
       
    63       using assms by auto
    57       using assms by auto
    64     fix i
    58     fix K and X::"'i \<Rightarrow> 'a set"
    65     assume
    59     show "prod_emb J M K (Pi\<^isub>E K X) \<in> Pow (\<Pi>\<^isub>E i\<in>J. space (M i))"
    66       "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))"
    60       by (auto simp: prod_emb_def)
    67     thus "emeasure (P J) (case i of (Ja, X) \<Rightarrow> prod_emb J M Ja (Pi\<^isub>E Ja X)) =
    61     assume JX: "(K \<noteq> {} \<or> J = {}) \<and> finite K \<and> K \<subseteq> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))"
    68         (case i of (J, X) \<Rightarrow> emeasure (P J) (Pi\<^isub>E J X))" using assms
    62     thus "emeasure (P J) (prod_emb J M K (Pi\<^isub>E K X)) = emeasure (P K) (Pi\<^isub>E K X)"
    69       by (cases i) (auto simp add: intro!: projective sets_PiM_I_finite)
    63       using assms
       
    64       apply (cases "J = {}")
       
    65       apply (simp add: prod_emb_id)
       
    66       apply (fastforce simp add: intro!: projective sets_PiM_I_finite)
       
    67       done
    70   qed
    68   qed
    71   finally show ?thesis .
    69   finally show ?thesis .
    72 qed
    70 qed
    73 
    71 
    74 lemma PiP_finite:
    72 lemma PiP_finite:
    75   assumes "J \<noteq> {}"
       
    76   assumes "finite J"
    73   assumes "finite J"
    77   assumes "J \<subseteq> I"
    74   assumes "J \<subseteq> I"
    78   shows "PiP J M P = P J" (is "?P = _")
    75   shows "PiP J M P = P J" (is "?P = _")
    79 proof (rule measure_eqI_generator_eq)
    76 proof (rule measure_eqI_generator_eq)
    80   let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
    77   let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
   106   by (subst PiP_finite) (auto simp: PiP_finite finite_subset projective)
   103   by (subst PiP_finite) (auto simp: PiP_finite finite_subset projective)
   107 
   104 
   108 end
   105 end
   109 
   106 
   110 sublocale projective_family \<subseteq> M: prob_space "M i" for i
   107 sublocale projective_family \<subseteq> M: prob_space "M i" for i
   111   by (rule prob_space)
   108   by (rule measure_space)
   112 
   109 
   113 end
   110 end