src/HOL/Probability/Projective_Family.thy
 changeset 50040 5da32dc55cd8 parent 50039 bfd5198cbe40 child 50041 afe886a04198
equal 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:`
`    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:`
`    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`