src/HOL/Probability/Projective_Family.thy
 author immler@in.tum.de Wed Nov 07 14:41:49 2012 +0100 (2012-11-07) changeset 50040 5da32dc55cd8 parent 50039 bfd5198cbe40 child 50041 afe886a04198 permissions -rw-r--r--
assume probability spaces; allow empty index set
```     1 theory Projective_Family
```
```     2 imports Finite_Product_Measure Probability_Measure
```
```     3 begin
```
```     4
```
```     5 definition
```
```     6   PiP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
```
```     7   "PiP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
```
```     8     {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
```
```     9     (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
```
```    10     (\<lambda>(J, X). emeasure (P J) (Pi\<^isub>E J X))"
```
```    11
```
```    12 lemma space_PiP[simp]: "space (PiP I M P) = space (PiM I M)"
```
```    13   by (auto simp add: PiP_def space_PiM prod_emb_def intro!: space_extend_measure)
```
```    14
```
```    15 lemma sets_PiP[simp]: "sets (PiP I M P) = sets (PiM I M)"
```
```    16   by (auto simp add: PiP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_extend_measure)
```
```    17
```
```    18 lemma measurable_PiP1[simp]: "measurable (PiP I M P) M' = measurable (\<Pi>\<^isub>M i\<in>I. M i) M'"
```
```    19   unfolding measurable_def by auto
```
```    20
```
```    21 lemma measurable_PiP2[simp]: "measurable M' (PiP I M P) = measurable M' (\<Pi>\<^isub>M i\<in>I. M i)"
```
```    22   unfolding measurable_def by auto
```
```    23
```
```    24 locale projective_family =
```
```    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>
```
```    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)"
```
```    29   assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)"
```
```    30   assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)"
```
```    31   assumes proj_finite_measure: "\<And>J. finite J \<Longrightarrow> emeasure (P J) (space (PiM J M)) \<noteq> \<infinity>"
```
```    32   assumes measure_space: "\<And>i. prob_space (M i)"
```
```    33 begin
```
```    34
```
```    35 lemma emeasure_PiP:
```
```    36   assumes "finite J"
```
```    37   assumes "J \<subseteq> 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)"
```
```    40 proof -
```
```    41   have "Pi\<^isub>E J (restrict A J) \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
```
```    42   proof safe
```
```    43     fix x j assume "x \<in> Pi J (restrict A J)" "j \<in> J"
```
```    44     hence "x j \<in> restrict A J j" by (auto simp: Pi_def)
```
```    45     also have "\<dots> \<subseteq> space (M j)" using sets_into_space A `j \<in> J` by auto
```
```    46     finally show "x j \<in> space (M j)" .
```
```    47   qed
```
```    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))"
```
```    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)"
```
```    52   proof (rule emeasure_extend_measure_Pair[OF PiP_def])
```
```    53     show "positive (sets (PiP J M P)) (P J)" unfolding positive_def by auto
```
```    54     show "countably_additive (sets (PiP J M P)) (P J)" unfolding countably_additive_def
```
```    55       by (auto simp: suminf_emeasure proj_sets[OF `finite J`])
```
```    56     show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
```
```    57       using assms by auto
```
```    58     fix K and X::"'i \<Rightarrow> 'a set"
```
```    59     show "prod_emb J M K (Pi\<^isub>E K X) \<in> Pow (\<Pi>\<^isub>E i\<in>J. space (M i))"
```
```    60       by (auto simp: prod_emb_def)
```
```    61     assume JX: "(K \<noteq> {} \<or> J = {}) \<and> finite K \<and> K \<subseteq> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))"
```
```    62     thus "emeasure (P J) (prod_emb J M K (Pi\<^isub>E K X)) = emeasure (P K) (Pi\<^isub>E K X)"
```
```    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
```
```    68   qed
```
```    69   finally show ?thesis .
```
```    70 qed
```
```    71
```
```    72 lemma PiP_finite:
```
```    73   assumes "finite J"
```
```    74   assumes "J \<subseteq> I"
```
```    75   shows "PiP J M P = P J" (is "?P = _")
```
```    76 proof (rule measure_eqI_generator_eq)
```
```    77   let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
```
```    78   let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)"
```
```    79   let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
```
```    80   show "Int_stable ?J"
```
```    81     by (rule Int_stable_PiE)
```
```    82   interpret finite_measure "P J" using proj_finite_measure `finite J`
```
```    83     by (intro finite_measureI) (simp add: proj_space)
```
```    84   show "emeasure ?P (?F _) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_PiP)
```
```    85   show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space)
```
```    86   show "sets (PiP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J"
```
```    87     using `finite J` proj_sets by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)
```
```    88   fix X assume "X \<in> ?J"
```
```    89   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
```
```    90   with `finite J` have "X \<in> sets (PiP J M P)" by simp
```
```    91   have emb_self: "prod_emb J M J (Pi\<^isub>E J E) = Pi\<^isub>E J E"
```
```    92     using E sets_into_space
```
```    93     by (auto intro!: prod_emb_PiE_same_index)
```
```    94   show "emeasure (PiP J M P) X = emeasure (P J) X"
```
```    95     unfolding X using E
```
```    96     by (intro emeasure_PiP assms) simp
```
```    97 qed (insert `finite J`, auto intro!: prod_algebraI_finite)
```
```    98
```
```    99 lemma emeasure_fun_emb[simp]:
```
```   100   assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" "L \<subseteq> I" and X: "X \<in> sets (PiM J M)"
```
```   101   shows "emeasure (PiP L M P) (prod_emb L M J X) = emeasure (PiP J M P) X"
```
```   102   using assms
```
```   103   by (subst PiP_finite) (auto simp: PiP_finite finite_subset projective)
```
```   104
```
```   105 end
```
```   106
```
```   107 sublocale projective_family \<subseteq> M: prob_space "M i" for i
```
```   108   by (rule measure_space)
```
```   109
```
```   110 end
```