src/HOL/Probability/Projective_Family.thy
author hoelzl
Mon, 19 Nov 2012 16:09:11 +0100
changeset 50124 4161c834c2fd
parent 50123 69b35a75caf3
child 50244 de72bbe42190
permissions -rw-r--r--
tuned FinMap
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Probability/Projective_Family.thy
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    Author:     Fabian Immler, TU München
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    Author:     Johannes Hölzl, TU München
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*)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
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6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
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header {*Projective Family*}
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theory Projective_Family
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imports Finite_Product_Measure Probability_Measure
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begin
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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50087
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lemma (in product_prob_space) distr_restrict:
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  assumes "J \<noteq> {}" "J \<subseteq> K" "finite K"
635d73673b5e regularity of measures, therefore:
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  shows "(\<Pi>\<^isub>M i\<in>J. M i) = distr (\<Pi>\<^isub>M i\<in>K. M i) (\<Pi>\<^isub>M i\<in>J. M i) (\<lambda>f. restrict f J)" (is "?P = ?D")
635d73673b5e regularity of measures, therefore:
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    15
proof (rule measure_eqI_generator_eq)
635d73673b5e regularity of measures, therefore:
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  have "finite J" using `J \<subseteq> K` `finite K` by (auto simp add: finite_subset)
635d73673b5e regularity of measures, therefore:
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  interpret J: finite_product_prob_space M J proof qed fact
635d73673b5e regularity of measures, therefore:
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  interpret K: finite_product_prob_space M K proof qed fact
635d73673b5e regularity of measures, therefore:
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635d73673b5e regularity of measures, therefore:
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  let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
635d73673b5e regularity of measures, therefore:
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  let ?F = "\<lambda>i. \<Pi>\<^isub>E k\<in>J. space (M k)"
635d73673b5e regularity of measures, therefore:
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  let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
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  show "Int_stable ?J"
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    by (rule Int_stable_PiE)
635d73673b5e regularity of measures, therefore:
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  show "range ?F \<subseteq> ?J" "(\<Union>i. ?F i) = ?\<Omega>"
635d73673b5e regularity of measures, therefore:
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    using `finite J` by (auto intro!: prod_algebraI_finite)
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  { fix i show "emeasure ?P (?F i) \<noteq> \<infinity>" by simp }
635d73673b5e regularity of measures, therefore:
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  show "?J \<subseteq> Pow ?\<Omega>" by (auto simp: Pi_iff dest: sets_into_space)
635d73673b5e regularity of measures, therefore:
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  show "sets (\<Pi>\<^isub>M i\<in>J. M i) = sigma_sets ?\<Omega> ?J" "sets ?D = sigma_sets ?\<Omega> ?J"
635d73673b5e regularity of measures, therefore:
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    using `finite J` by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)
635d73673b5e regularity of measures, therefore:
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635d73673b5e regularity of measures, therefore:
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  fix X assume "X \<in> ?J"
635d73673b5e regularity of measures, therefore:
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  then obtain E where [simp]: "X = Pi\<^isub>E J E" and E: "\<forall>i\<in>J. E i \<in> sets (M i)" by auto
635d73673b5e regularity of measures, therefore:
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  with `finite J` have X: "X \<in> sets (Pi\<^isub>M J M)"
635d73673b5e regularity of measures, therefore:
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    by simp
635d73673b5e regularity of measures, therefore:
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635d73673b5e regularity of measures, therefore:
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  have "emeasure ?P X = (\<Prod> i\<in>J. emeasure (M i) (E i))"
635d73673b5e regularity of measures, therefore:
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    using E by (simp add: J.measure_times)
635d73673b5e regularity of measures, therefore:
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  also have "\<dots> = (\<Prod> i\<in>J. emeasure (M i) (if i \<in> J then E i else space (M i)))"
635d73673b5e regularity of measures, therefore:
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    by simp
635d73673b5e regularity of measures, therefore:
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  also have "\<dots> = (\<Prod> i\<in>K. emeasure (M i) (if i \<in> J then E i else space (M i)))"
635d73673b5e regularity of measures, therefore:
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    using `finite K` `J \<subseteq> K`
635d73673b5e regularity of measures, therefore:
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    by (intro setprod_mono_one_left) (auto simp: M.emeasure_space_1)
635d73673b5e regularity of measures, therefore:
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  also have "\<dots> = emeasure (Pi\<^isub>M K M) (\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i))"
635d73673b5e regularity of measures, therefore:
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    45
    using E by (simp add: K.measure_times)
635d73673b5e regularity of measures, therefore:
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    46
  also have "(\<Pi>\<^isub>E i\<in>K. if i \<in> J then E i else space (M i)) = (\<lambda>f. restrict f J) -` Pi\<^isub>E J E \<inter> (\<Pi>\<^isub>E i\<in>K. space (M i))"
50123
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    using `J \<subseteq> K` sets_into_space E by (force simp: Pi_iff PiE_def split: split_if_asm)
50087
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  finally show "emeasure (Pi\<^isub>M J M) X = emeasure ?D X"
635d73673b5e regularity of measures, therefore:
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    49
    using X `J \<subseteq> K` apply (subst emeasure_distr)
635d73673b5e regularity of measures, therefore:
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    by (auto intro!: measurable_restrict_subset simp: space_PiM)
635d73673b5e regularity of measures, therefore:
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qed
635d73673b5e regularity of measures, therefore:
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635d73673b5e regularity of measures, therefore:
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lemma (in product_prob_space) emeasure_prod_emb[simp]:
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  assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" and X: "X \<in> sets (Pi\<^isub>M J M)"
635d73673b5e regularity of measures, therefore:
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  shows "emeasure (Pi\<^isub>M L M) (prod_emb L M J X) = emeasure (Pi\<^isub>M J M) X"
635d73673b5e regularity of measures, therefore:
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  by (subst distr_restrict[OF L])
635d73673b5e regularity of measures, therefore:
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     (simp add: prod_emb_def space_PiM emeasure_distr measurable_restrict_subset L X)
635d73673b5e regularity of measures, therefore:
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definition
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  limP :: "'i set \<Rightarrow> ('i \<Rightarrow> 'a measure) \<Rightarrow> ('i set \<Rightarrow> ('i \<Rightarrow> 'a) measure) \<Rightarrow> ('i \<Rightarrow> 'a) measure" where
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  "limP I M P = extend_measure (\<Pi>\<^isub>E i\<in>I. space (M i))
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    {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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    (\<lambda>(J, X). prod_emb I M J (\<Pi>\<^isub>E j\<in>J. X j))
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    (\<lambda>(J, X). emeasure (P J) (Pi\<^isub>E J X))"
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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50095
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abbreviation "lim\<^isub>P \<equiv> limP"
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
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94d7dfa9f404 renamed to more appropriate lim_P for projective limit
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lemma space_limP[simp]: "space (limP I M P) = space (PiM I M)"
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
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    69
  by (auto simp add: limP_def space_PiM prod_emb_def intro!: space_extend_measure)
50039
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lemma sets_limP[simp]: "sets (limP I M P) = sets (PiM I M)"
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    72
  by (auto simp add: limP_def sets_PiM prod_algebra_def prod_emb_def intro!: sets_extend_measure)
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lemma measurable_limP1[simp]: "measurable (limP I M P) M' = measurable (\<Pi>\<^isub>M i\<in>I. M i) M'"
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    75
  unfolding measurable_def by auto
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lemma measurable_limP2[simp]: "measurable M' (limP I M P) = measurable M' (\<Pi>\<^isub>M i\<in>I. M i)"
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    78
  unfolding measurable_def by auto
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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locale projective_family =
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  fixes I::"'i set" and P::"'i set \<Rightarrow> ('i \<Rightarrow> 'a) measure" and M::"('i \<Rightarrow> 'a measure)"
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  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>
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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    83
     (P H) (prod_emb H M J X) = (P J) X"
50101
a3bede207a04 renamed prob_space to proj_prob_space as it clashed with Probability_Measure.prob_space
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    84
  assumes proj_prob_space: "\<And>J. finite J \<Longrightarrow> prob_space (P J)"
50039
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    85
  assumes proj_space: "\<And>J. finite J \<Longrightarrow> space (P J) = space (PiM J M)"
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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    86
  assumes proj_sets: "\<And>J. finite J \<Longrightarrow> sets (P J) = sets (PiM J M)"
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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begin
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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50095
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    89
lemma emeasure_limP:
50039
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  assumes "finite J"
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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    91
  assumes "J \<subseteq> I"
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    92
  assumes A: "\<And>i. i\<in>J \<Longrightarrow> A i \<in> sets (M i)"
50095
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  shows "emeasure (limP J M P) (Pi\<^isub>E J A) = emeasure (P J) (Pi\<^isub>E J A)"
50039
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    94
proof -
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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    95
  have "Pi\<^isub>E J (restrict A J) \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50101
diff changeset
    96
    using sets_into_space[OF A] by (auto simp: PiE_iff) blast
50095
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immler
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diff changeset
    97
  hence "emeasure (limP J M P) (Pi\<^isub>E J A) =
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
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    98
    emeasure (limP J M P) (prod_emb J M J (Pi\<^isub>E J A))"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50101
diff changeset
    99
    using assms(1-3) sets_into_space by (auto simp add: prod_emb_id PiE_def Pi_def)
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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   100
  also have "\<dots> = emeasure (P J) (Pi\<^isub>E J A)"
50095
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immler
parents: 50087
diff changeset
   101
  proof (rule emeasure_extend_measure_Pair[OF limP_def])
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
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   102
    show "positive (sets (limP J M P)) (P J)" unfolding positive_def by auto
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
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diff changeset
   103
    show "countably_additive (sets (limP J M P)) (P J)" unfolding countably_additive_def
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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   104
      by (auto simp: suminf_emeasure proj_sets[OF `finite J`])
50040
5da32dc55cd8 assume probability spaces; allow empty index set
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diff changeset
   105
    show "(J \<noteq> {} \<or> J = {}) \<and> finite J \<and> J \<subseteq> J \<and> A \<in> (\<Pi> j\<in>J. sets (M j))"
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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diff changeset
   106
      using assms by auto
50040
5da32dc55cd8 assume probability spaces; allow empty index set
immler@in.tum.de
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diff changeset
   107
    fix K and X::"'i \<Rightarrow> 'a set"
5da32dc55cd8 assume probability spaces; allow empty index set
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diff changeset
   108
    show "prod_emb J M K (Pi\<^isub>E K X) \<in> Pow (\<Pi>\<^isub>E i\<in>J. space (M i))"
5da32dc55cd8 assume probability spaces; allow empty index set
immler@in.tum.de
parents: 50039
diff changeset
   109
      by (auto simp: prod_emb_def)
5da32dc55cd8 assume probability spaces; allow empty index set
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diff changeset
   110
    assume JX: "(K \<noteq> {} \<or> J = {}) \<and> finite K \<and> K \<subseteq> J \<and> X \<in> (\<Pi> j\<in>K. sets (M j))"
5da32dc55cd8 assume probability spaces; allow empty index set
immler@in.tum.de
parents: 50039
diff changeset
   111
    thus "emeasure (P J) (prod_emb J M K (Pi\<^isub>E K X)) = emeasure (P K) (Pi\<^isub>E K X)"
5da32dc55cd8 assume probability spaces; allow empty index set
immler@in.tum.de
parents: 50039
diff changeset
   112
      using assms
5da32dc55cd8 assume probability spaces; allow empty index set
immler@in.tum.de
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diff changeset
   113
      apply (cases "J = {}")
5da32dc55cd8 assume probability spaces; allow empty index set
immler@in.tum.de
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diff changeset
   114
      apply (simp add: prod_emb_id)
5da32dc55cd8 assume probability spaces; allow empty index set
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diff changeset
   115
      apply (fastforce simp add: intro!: projective sets_PiM_I_finite)
5da32dc55cd8 assume probability spaces; allow empty index set
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diff changeset
   116
      done
50039
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   117
  qed
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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   118
  finally show ?thesis .
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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   119
qed
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
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diff changeset
   120
50095
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   121
lemma limP_finite:
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   122
  assumes "finite J"
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   123
  assumes "J \<subseteq> I"
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   124
  shows "limP J M P = P J" (is "?P = _")
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   125
proof (rule measure_eqI_generator_eq)
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   126
  let ?J = "{Pi\<^isub>E J E | E. \<forall>i\<in>J. E i \<in> sets (M i)}"
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   127
  let ?\<Omega> = "(\<Pi>\<^isub>E k\<in>J. space (M k))"
50101
a3bede207a04 renamed prob_space to proj_prob_space as it clashed with Probability_Measure.prob_space
hoelzl
parents: 50095
diff changeset
   128
  interpret prob_space "P J" using proj_prob_space `finite J` by simp
50124
4161c834c2fd tuned FinMap
hoelzl
parents: 50123
diff changeset
   129
  show "emeasure ?P (\<Pi>\<^isub>E k\<in>J. space (M k)) \<noteq> \<infinity>" using assms `finite J` by (auto simp: emeasure_limP)
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   130
  show "sets (limP J M P) = sigma_sets ?\<Omega> ?J" "sets (P J) = sigma_sets ?\<Omega> ?J"
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   131
    using `finite J` proj_sets by (simp_all add: sets_PiM prod_algebra_eq_finite Pi_iff)
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   132
  fix X assume "X \<in> ?J"
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   133
  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
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   134
  with `finite J` have "X \<in> sets (limP J M P)" by simp
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   135
  have emb_self: "prod_emb J M J (Pi\<^isub>E J E) = Pi\<^isub>E J E"
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   136
    using E sets_into_space
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   137
    by (auto intro!: prod_emb_PiE_same_index)
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   138
  show "emeasure (limP J M P) X = emeasure (P J) X"
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   139
    unfolding X using E
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   140
    by (intro emeasure_limP assms) simp
50124
4161c834c2fd tuned FinMap
hoelzl
parents: 50123
diff changeset
   141
qed (auto simp: Pi_iff dest: sets_into_space intro: Int_stable_PiE)
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   142
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   143
lemma emeasure_fun_emb[simp]:
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   144
  assumes L: "J \<noteq> {}" "J \<subseteq> L" "finite L" "L \<subseteq> I" and X: "X \<in> sets (PiM J M)"
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   145
  shows "emeasure (limP L M P) (prod_emb L M J X) = emeasure (limP J M P) X"
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   146
  using assms
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   147
  by (subst limP_finite) (auto simp: limP_finite finite_subset projective)
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   148
50124
4161c834c2fd tuned FinMap
hoelzl
parents: 50123
diff changeset
   149
abbreviation
4161c834c2fd tuned FinMap
hoelzl
parents: 50123
diff changeset
   150
  "emb L K X \<equiv> prod_emb L M K X"
4161c834c2fd tuned FinMap
hoelzl
parents: 50123
diff changeset
   151
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   152
lemma prod_emb_injective:
50124
4161c834c2fd tuned FinMap
hoelzl
parents: 50123
diff changeset
   153
  assumes "J \<subseteq> L" and sets: "X \<in> sets (Pi\<^isub>M J M)" "Y \<in> sets (Pi\<^isub>M J M)"
4161c834c2fd tuned FinMap
hoelzl
parents: 50123
diff changeset
   154
  assumes "emb L J X = emb L J Y"
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   155
  shows "X = Y"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   156
proof (rule injective_vimage_restrict)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   157
  show "X \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))" "Y \<subseteq> (\<Pi>\<^isub>E i\<in>J. space (M i))"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   158
    using sets[THEN sets_into_space] by (auto simp: space_PiM)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   159
  have "\<forall>i\<in>L. \<exists>x. x \<in> space (M i)"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   160
  proof
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   161
    fix i assume "i \<in> L"
50101
a3bede207a04 renamed prob_space to proj_prob_space as it clashed with Probability_Measure.prob_space
hoelzl
parents: 50095
diff changeset
   162
    interpret prob_space "P {i}" using proj_prob_space by simp
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   163
    from not_empty show "\<exists>x. x \<in> space (M i)" by (auto simp add: proj_space space_PiM)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   164
  qed
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   165
  from bchoice[OF this]
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50101
diff changeset
   166
  show "(\<Pi>\<^isub>E i\<in>L. space (M i)) \<noteq> {}" by (auto simp: PiE_def)
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   167
  show "(\<lambda>x. restrict x J) -` X \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i)) = (\<lambda>x. restrict x J) -` Y \<inter> (\<Pi>\<^isub>E i\<in>L. space (M i))"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   168
    using `prod_emb L M J X = prod_emb L M J Y` by (simp add: prod_emb_def)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   169
qed fact
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   170
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   171
definition generator :: "('i \<Rightarrow> 'a) set set" where
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   172
  "generator = (\<Union>J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emb I J ` sets (Pi\<^isub>M J M))"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   173
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   174
lemma generatorI':
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   175
  "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"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   176
  unfolding generator_def by auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   177
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   178
lemma algebra_generator:
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   179
  assumes "I \<noteq> {}" shows "algebra (\<Pi>\<^isub>E i\<in>I. space (M i)) generator" (is "algebra ?\<Omega> ?G")
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   180
  unfolding algebra_def algebra_axioms_def ring_of_sets_iff
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   181
proof (intro conjI ballI)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   182
  let ?G = generator
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   183
  show "?G \<subseteq> Pow ?\<Omega>"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   184
    by (auto simp: generator_def prod_emb_def)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   185
  from `I \<noteq> {}` obtain i where "i \<in> I" by auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   186
  then show "{} \<in> ?G"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   187
    by (auto intro!: exI[of _ "{i}"] image_eqI[where x="\<lambda>i. {}"]
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   188
             simp: sigma_sets.Empty generator_def prod_emb_def)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   189
  from `i \<in> I` show "?\<Omega> \<in> ?G"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   190
    by (auto intro!: exI[of _ "{i}"] image_eqI[where x="Pi\<^isub>E {i} (\<lambda>i. space (M i))"]
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   191
             simp: generator_def prod_emb_def)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   192
  fix A assume "A \<in> ?G"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   193
  then obtain JA XA where XA: "JA \<noteq> {}" "finite JA" "JA \<subseteq> I" "XA \<in> sets (Pi\<^isub>M JA M)" and A: "A = emb I JA XA"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   194
    by (auto simp: generator_def)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   195
  fix B assume "B \<in> ?G"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   196
  then obtain JB XB where XB: "JB \<noteq> {}" "finite JB" "JB \<subseteq> I" "XB \<in> sets (Pi\<^isub>M JB M)" and B: "B = emb I JB XB"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   197
    by (auto simp: generator_def)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   198
  let ?RA = "emb (JA \<union> JB) JA XA"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   199
  let ?RB = "emb (JA \<union> JB) JB XB"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   200
  have *: "A - B = emb I (JA \<union> JB) (?RA - ?RB)" "A \<union> B = emb I (JA \<union> JB) (?RA \<union> ?RB)"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   201
    using XA A XB B by auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   202
  show "A - B \<in> ?G" "A \<union> B \<in> ?G"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   203
    unfolding * using XA XB by (safe intro!: generatorI') auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   204
qed
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   205
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   206
lemma sets_PiM_generator:
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   207
  "sets (PiM I M) = sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   208
proof cases
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   209
  assume "I = {}" then show ?thesis
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   210
    unfolding generator_def
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   211
    by (auto simp: sets_PiM_empty sigma_sets_empty_eq cong: conj_cong)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   212
next
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   213
  assume "I \<noteq> {}"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   214
  show ?thesis
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   215
  proof
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   216
    show "sets (Pi\<^isub>M I M) \<subseteq> sigma_sets (\<Pi>\<^isub>E i\<in>I. space (M i)) generator"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   217
      unfolding sets_PiM
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   218
    proof (safe intro!: sigma_sets_subseteq)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   219
      fix A assume "A \<in> prod_algebra I M" with `I \<noteq> {}` show "A \<in> generator"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   220
        by (auto intro!: generatorI' sets_PiM_I_finite elim!: prod_algebraE)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   221
    qed
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   222
  qed (auto simp: generator_def space_PiM[symmetric] intro!: sigma_sets_subset)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   223
qed
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   224
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   225
lemma generatorI:
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   226
  "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"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   227
  unfolding generator_def by auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   228
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   229
definition
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   230
  "\<mu>G A =
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   231
    (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 (limP J M P) X))"
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   232
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   233
lemma \<mu>G_spec:
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   234
  assumes J: "J \<noteq> {}" "finite J" "J \<subseteq> I" "A = emb I J X" "X \<in> sets (Pi\<^isub>M J M)"
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   235
  shows "\<mu>G A = emeasure (limP J M P) X"
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   236
  unfolding \<mu>G_def
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   237
proof (intro the_equality allI impI ballI)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   238
  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)"
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   239
  have "emeasure (limP K M P) Y = emeasure (limP (K \<union> J) M P) (emb (K \<union> J) K Y)"
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   240
    using K J by simp
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   241
  also have "emb (K \<union> J) K Y = emb (K \<union> J) J X"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   242
    using K J by (simp add: prod_emb_injective[of "K \<union> J" I])
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   243
  also have "emeasure (limP (K \<union> J) M P) (emb (K \<union> J) J X) = emeasure (limP J M P) X"
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   244
    using K J by simp
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   245
  finally show "emeasure (limP J M P) X = emeasure (limP K M P) Y" ..
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   246
qed (insert J, force)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   247
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   248
lemma \<mu>G_eq:
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   249
  "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 (limP J M P) X"
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   250
  by (intro \<mu>G_spec) auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   251
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   252
lemma generator_Ex:
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   253
  assumes *: "A \<in> generator"
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   254
  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 (limP J M P) X"
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   255
proof -
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   256
  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)"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   257
    unfolding generator_def by auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   258
  with \<mu>G_spec[OF this] show ?thesis by auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   259
qed
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   260
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   261
lemma generatorE:
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   262
  assumes A: "A \<in> generator"
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   263
  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 (limP J M P) X"
50124
4161c834c2fd tuned FinMap
hoelzl
parents: 50123
diff changeset
   264
  using generator_Ex[OF A] by atomize_elim auto
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   265
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   266
lemma merge_sets:
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   267
  "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)"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   268
  by simp
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   269
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   270
lemma merge_emb:
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   271
  assumes "K \<subseteq> I" "J \<subseteq> I" and y: "y \<in> space (Pi\<^isub>M J M)"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   272
  shows "((\<lambda>x. merge J (I - J) (y, x)) -` emb I K X \<inter> space (Pi\<^isub>M I M)) =
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   273
    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))"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   274
proof -
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   275
  have [simp]: "\<And>x J K L. merge J K (y, restrict x L) = merge J (K \<inter> L) (y, x)"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   276
    by (auto simp: restrict_def merge_def)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   277
  have [simp]: "\<And>x J K L. restrict (merge J K (y, x)) L = merge (J \<inter> L) (K \<inter> L) (y, x)"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   278
    by (auto simp: restrict_def merge_def)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   279
  have [simp]: "(I - J) \<inter> K = K - J" using `K \<subseteq> I` `J \<subseteq> I` by auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   280
  have [simp]: "(K - J) \<inter> (K \<union> J) = K - J" by auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   281
  have [simp]: "(K - J) \<inter> K = K - J" by auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   282
  from y `K \<subseteq> I` `J \<subseteq> I` show ?thesis
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50101
diff changeset
   283
    by (simp split: split_merge add: prod_emb_def Pi_iff PiE_def extensional_merge_sub set_eq_iff space_PiM)
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   284
       auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   285
qed
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   286
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   287
lemma positive_\<mu>G:
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   288
  assumes "I \<noteq> {}"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   289
  shows "positive generator \<mu>G"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   290
proof -
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   291
  interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   292
  show ?thesis
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   293
  proof (intro positive_def[THEN iffD2] conjI ballI)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   294
    from generatorE[OF G.empty_sets] guess J X . note this[simp]
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   295
    have "X = {}"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   296
      by (rule prod_emb_injective[of J I]) simp_all
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   297
    then show "\<mu>G {} = 0" by simp
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   298
  next
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   299
    fix A assume "A \<in> generator"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   300
    from generatorE[OF this] guess J X . note this[simp]
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   301
    show "0 \<le> \<mu>G A" by (simp add: emeasure_nonneg)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   302
  qed
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   303
qed
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   304
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   305
lemma additive_\<mu>G:
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   306
  assumes "I \<noteq> {}"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   307
  shows "additive generator \<mu>G"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   308
proof -
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   309
  interpret G!: algebra "\<Pi>\<^isub>E i\<in>I. space (M i)" generator by (rule algebra_generator) fact
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   310
  show ?thesis
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   311
  proof (intro additive_def[THEN iffD2] ballI impI)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   312
    fix A assume "A \<in> generator" with generatorE guess J X . note J = this
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   313
    fix B assume "B \<in> generator" with generatorE guess K Y . note K = this
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   314
    assume "A \<inter> B = {}"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   315
    have JK: "J \<union> K \<noteq> {}" "J \<union> K \<subseteq> I" "finite (J \<union> K)"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   316
      using J K by auto
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   317
    have JK_disj: "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   318
      apply (rule prod_emb_injective[of "J \<union> K" I])
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   319
      apply (insert `A \<inter> B = {}` JK J K)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   320
      apply (simp_all add: Int prod_emb_Int)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   321
      done
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   322
    have AB: "A = emb I (J \<union> K) (emb (J \<union> K) J X)" "B = emb I (J \<union> K) (emb (J \<union> K) K Y)"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   323
      using J K by simp_all
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   324
    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))"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   325
      by simp
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   326
    also have "\<dots> = emeasure (limP (J \<union> K) M P) (emb (J \<union> K) J X \<union> emb (J \<union> K) K Y)"
50042
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   327
      using JK J(1, 4) K(1, 4) by (simp add: \<mu>G_eq Un del: prod_emb_Un)
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   328
    also have "\<dots> = \<mu>G A + \<mu>G B"
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   329
      using J K JK_disj by (simp add: plus_emeasure[symmetric])
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   330
    finally show "\<mu>G (A \<union> B) = \<mu>G A + \<mu>G B" .
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   331
  qed
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   332
qed
6fe18351e9dd moved lemmas into projective_family; added header for theory Projective_Family
immler@in.tum.de
parents: 50041
diff changeset
   333
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   334
end
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
diff changeset
   335
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   336
sublocale product_prob_space \<subseteq> projective_family I "\<lambda>J. PiM J M" M
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   337
proof
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   338
  fix J::"'i set" assume "finite J"
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   339
  interpret f: finite_product_prob_space M J proof qed fact
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   340
  show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) \<noteq> \<infinity>" by simp
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   341
  show "\<exists>A. range A \<subseteq> sets (Pi\<^isub>M J M) \<and>
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   342
            (\<Union>i. A i) = space (Pi\<^isub>M J M) \<and>
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   343
            (\<forall>i. emeasure (Pi\<^isub>M J M) (A i) \<noteq> \<infinity>)" using sigma_finite[OF `finite J`]
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   344
    by (auto simp add: sigma_finite_measure_def)
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   345
  show "emeasure (Pi\<^isub>M J M) (space (Pi\<^isub>M J M)) = 1" by (rule f.emeasure_space_1)
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   346
qed simp_all
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
diff changeset
   347
50095
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   348
lemma (in product_prob_space) limP_PiM_finite[simp]:
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   349
  assumes "J \<noteq> {}" "finite J" "J \<subseteq> I" shows "limP J M (\<lambda>J. PiM J M) = PiM J M"
94d7dfa9f404 renamed to more appropriate lim_P for projective limit
immler
parents: 50087
diff changeset
   350
  using assms by (simp add: limP_finite)
50087
635d73673b5e regularity of measures, therefore:
immler
parents: 50042
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   351
50039
bfd5198cbe40 added projective_family; generalized generator in product_prob_space to projective_family
immler@in.tum.de
parents:
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   352
end