src/HOL/Probability/Complete_Measure.thy
author wenzelm
Thu, 18 Apr 2013 17:07:01 +0200
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permissions -rw-r--r--
simplifier uses proper Proof.context instead of historic type simpset;
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(*  Title:      HOL/Probability/Complete_Measure.thy
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    Author:     Robert Himmelmann, Johannes Hoelzl, TU Muenchen
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*)
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theory Complete_Measure
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imports Lebesgue_Integration
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begin
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definition
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  "split_completion M A p = (if A \<in> sets M then p = (A, {}) else
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   \<exists>N'. A = fst p \<union> snd p \<and> fst p \<inter> snd p = {} \<and> fst p \<in> sets M \<and> snd p \<subseteq> N' \<and> N' \<in> null_sets M)"
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3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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definition
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  "main_part M A = fst (Eps (split_completion M A))"
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definition
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  "null_part M A = snd (Eps (split_completion M A))"
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definition completion :: "'a measure \<Rightarrow> 'a measure" where
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  "completion M = measure_of (space M) { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }
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    (emeasure M \<circ> main_part M)"
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lemma completion_into_space:
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  "{ S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' } \<subseteq> Pow (space M)"
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  using sets.sets_into_space by auto
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lemma space_completion[simp]: "space (completion M) = space M"
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  unfolding completion_def using space_measure_of[OF completion_into_space] by simp
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lemma completionI:
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  assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
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  shows "A \<in> { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
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  using assms by auto
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lemma completionE:
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  assumes "A \<in> { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
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  obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
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  using assms by auto
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lemma sigma_algebra_completion:
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  "sigma_algebra (space M) { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
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    (is "sigma_algebra _ ?A")
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  unfolding sigma_algebra_iff2
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proof (intro conjI ballI allI impI)
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  show "?A \<subseteq> Pow (space M)"
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    using sets.sets_into_space by auto
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next
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  show "{} \<in> ?A" by auto
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next
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  let ?C = "space M"
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  fix A assume "A \<in> ?A" from completionE[OF this] guess S N N' .
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  then show "space M - A \<in> ?A"
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    by (intro completionI[of _ "(?C - S) \<inter> (?C - N')" "(?C - S) \<inter> N' \<inter> (?C - N)"]) auto
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next
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  fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> ?A"
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  then have "\<forall>n. \<exists>S N N'. A n = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'"
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    by (auto simp: image_subset_iff)
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  from choice[OF this] guess S ..
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  from choice[OF this] guess N ..
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  from choice[OF this] guess N' ..
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  then show "UNION UNIV A \<in> ?A"
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    using null_sets_UN[of N']
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    by (intro completionI[of _ "UNION UNIV S" "UNION UNIV N" "UNION UNIV N'"]) auto
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qed
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lemma sets_completion:
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  "sets (completion M) = { S \<union> N |S N N'. S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N' }"
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  using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_completion] by (simp add: completion_def)
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lemma sets_completionE:
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  assumes "A \<in> sets (completion M)"
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  obtains S N N' where "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
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  using assms unfolding sets_completion by auto
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lemma sets_completionI:
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  assumes "A = S \<union> N" "N \<subseteq> N'" "N' \<in> null_sets M" "S \<in> sets M"
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  shows "A \<in> sets (completion M)"
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  using assms unfolding sets_completion by auto
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lemma sets_completionI_sets[intro, simp]:
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  "A \<in> sets M \<Longrightarrow> A \<in> sets (completion M)"
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  unfolding sets_completion by force
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lemma null_sets_completion:
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  assumes "N' \<in> null_sets M" "N \<subseteq> N'" shows "N \<in> sets (completion M)"
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    86
  using assms by (intro sets_completionI[of N "{}" N N']) auto
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lemma split_completion:
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  assumes "A \<in> sets (completion M)"
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  shows "split_completion M A (main_part M A, null_part M A)"
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proof cases
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  assume "A \<in> sets M" then show ?thesis
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    by (simp add: split_completion_def[abs_def] main_part_def null_part_def)
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next
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  assume nA: "A \<notin> sets M"
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  show ?thesis
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    unfolding main_part_def null_part_def if_not_P[OF nA]
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  proof (rule someI2_ex)
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    from assms[THEN sets_completionE] guess S N N' . note A = this
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    let ?P = "(S, N - S)"
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    show "\<exists>p. split_completion M A p"
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   102
      unfolding split_completion_def if_not_P[OF nA] using A
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    proof (intro exI conjI)
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      show "A = fst ?P \<union> snd ?P" using A by auto
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      show "snd ?P \<subseteq> N'" using A by auto
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   qed auto
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  qed auto
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qed
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lemma
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  assumes "S \<in> sets (completion M)"
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   112
  shows main_part_sets[intro, simp]: "main_part M S \<in> sets M"
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    and main_part_null_part_Un[simp]: "main_part M S \<union> null_part M S = S"
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    and main_part_null_part_Int[simp]: "main_part M S \<inter> null_part M S = {}"
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  using split_completion[OF assms]
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  by (auto simp: split_completion_def split: split_if_asm)
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lemma main_part[simp]: "S \<in> sets M \<Longrightarrow> main_part M S = S"
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  using split_completion[of S M]
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  by (auto simp: split_completion_def split: split_if_asm)
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lemma null_part:
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  assumes "S \<in> sets (completion M)" shows "\<exists>N. N\<in>null_sets M \<and> null_part M S \<subseteq> N"
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  using split_completion[OF assms] by (auto simp: split_completion_def split: split_if_asm)
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   125
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lemma null_part_sets[intro, simp]:
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  assumes "S \<in> sets M" shows "null_part M S \<in> sets M" "emeasure M (null_part M S) = 0"
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proof -
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  have S: "S \<in> sets (completion M)" using assms by auto
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  have "S - main_part M S \<in> sets M" using assms by auto
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  moreover
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  from main_part_null_part_Un[OF S] main_part_null_part_Int[OF S]
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   133
  have "S - main_part M S = null_part M S" by auto
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  ultimately show sets: "null_part M S \<in> sets M" by auto
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  from null_part[OF S] guess N ..
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   136
  with emeasure_eq_0[of N _ "null_part M S"] sets
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   137
  show "emeasure M (null_part M S) = 0" by auto
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qed
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lemma emeasure_main_part_UN:
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  fixes S :: "nat \<Rightarrow> 'a set"
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  assumes "range S \<subseteq> sets (completion M)"
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   143
  shows "emeasure M (main_part M (\<Union>i. (S i))) = emeasure M (\<Union>i. main_part M (S i))"
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   144
proof -
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   145
  have S: "\<And>i. S i \<in> sets (completion M)" using assms by auto
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   146
  then have UN: "(\<Union>i. S i) \<in> sets (completion M)" by auto
05663f75964c reworked Probability theory
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   147
  have "\<forall>i. \<exists>N. N \<in> null_sets M \<and> null_part M (S i) \<subseteq> N"
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   148
    using null_part[OF S] by auto
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   149
  from choice[OF this] guess N .. note N = this
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   150
  then have UN_N: "(\<Union>i. N i) \<in> null_sets M" by (intro null_sets_UN) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   151
  have "(\<Union>i. S i) \<in> sets (completion M)" using S by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   152
  from null_part[OF this] guess N' .. note N' = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   153
  let ?N = "(\<Union>i. N i) \<union> N'"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   154
  have null_set: "?N \<in> null_sets M" using N' UN_N by (intro null_sets.Un) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   155
  have "main_part M (\<Union>i. S i) \<union> ?N = (main_part M (\<Union>i. S i) \<union> null_part M (\<Union>i. S i)) \<union> ?N"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   156
    using N' by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   157
  also have "\<dots> = (\<Union>i. main_part M (S i) \<union> null_part M (S i)) \<union> ?N"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   158
    unfolding main_part_null_part_Un[OF S] main_part_null_part_Un[OF UN] by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   159
  also have "\<dots> = (\<Union>i. main_part M (S i)) \<union> ?N"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   160
    using N by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   161
  finally have *: "main_part M (\<Union>i. S i) \<union> ?N = (\<Union>i. main_part M (S i)) \<union> ?N" .
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   162
  have "emeasure M (main_part M (\<Union>i. S i)) = emeasure M (main_part M (\<Union>i. S i) \<union> ?N)"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   163
    using null_set UN by (intro emeasure_Un_null_set[symmetric]) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   164
  also have "\<dots> = emeasure M ((\<Union>i. main_part M (S i)) \<union> ?N)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   165
    unfolding * ..
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   166
  also have "\<dots> = emeasure M (\<Union>i. main_part M (S i))"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   167
    using null_set S by (intro emeasure_Un_null_set) auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41097
diff changeset
   168
  finally show ?thesis .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   169
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   170
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   171
lemma emeasure_completion[simp]:
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   172
  assumes S: "S \<in> sets (completion M)" shows "emeasure (completion M) S = emeasure M (main_part M S)"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   173
proof (subst emeasure_measure_of[OF completion_def completion_into_space])
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   174
  let ?\<mu> = "emeasure M \<circ> main_part M"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   175
  show "S \<in> sets (completion M)" "?\<mu> S = emeasure M (main_part M S) " using S by simp_all
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   176
  show "positive (sets (completion M)) ?\<mu>"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   177
    by (simp add: positive_def emeasure_nonneg)
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   178
  show "countably_additive (sets (completion M)) ?\<mu>"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   179
  proof (intro countably_additiveI)
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   180
    fix A :: "nat \<Rightarrow> 'a set" assume A: "range A \<subseteq> sets (completion M)" "disjoint_family A"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   181
    have "disjoint_family (\<lambda>i. main_part M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   182
    proof (intro disjoint_family_on_bisimulation[OF A(2)])
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   183
      fix n m assume "A n \<inter> A m = {}"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   184
      then have "(main_part M (A n) \<union> null_part M (A n)) \<inter> (main_part M (A m) \<union> null_part M (A m)) = {}"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   185
        using A by (subst (1 2) main_part_null_part_Un) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   186
      then show "main_part M (A n) \<inter> main_part M (A m) = {}" by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   187
    qed
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   188
    then have "(\<Sum>n. emeasure M (main_part M (A n))) = emeasure M (\<Union>i. main_part M (A i))"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   189
      using A by (auto intro!: suminf_emeasure)
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   190
    then show "(\<Sum>n. ?\<mu> (A n)) = ?\<mu> (UNION UNIV A)"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   191
      by (simp add: completion_def emeasure_main_part_UN[OF A(1)])
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   192
  qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   193
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   194
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   195
lemma emeasure_completion_UN:
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   196
  "range S \<subseteq> sets (completion M) \<Longrightarrow>
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   197
    emeasure (completion M) (\<Union>i::nat. (S i)) = emeasure M (\<Union>i. main_part M (S i))"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   198
  by (subst emeasure_completion) (auto simp add: emeasure_main_part_UN)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   199
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   200
lemma emeasure_completion_Un:
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   201
  assumes S: "S \<in> sets (completion M)" and T: "T \<in> sets (completion M)"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   202
  shows "emeasure (completion M) (S \<union> T) = emeasure M (main_part M S \<union> main_part M T)"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   203
proof (subst emeasure_completion)
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   204
  have UN: "(\<Union>i. binary (main_part M S) (main_part M T) i) = (\<Union>i. main_part M (binary S T i))"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   205
    unfolding binary_def by (auto split: split_if_asm)
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   206
  show "emeasure M (main_part M (S \<union> T)) = emeasure M (main_part M S \<union> main_part M T)"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   207
    using emeasure_main_part_UN[of "binary S T" M] assms
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   208
    unfolding range_binary_eq Un_range_binary UN by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   209
qed (auto intro: S T)
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   210
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   211
lemma sets_completionI_sub:
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   212
  assumes N: "N' \<in> null_sets M" "N \<subseteq> N'"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   213
  shows "N \<in> sets (completion M)"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   214
  using assms by (intro sets_completionI[of _ "{}" N N']) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   215
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   216
lemma completion_ex_simple_function:
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   217
  assumes f: "simple_function (completion M) f"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   218
  shows "\<exists>f'. simple_function M f' \<and> (AE x in M. f x = f' x)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   219
proof -
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 43920
diff changeset
   220
  let ?F = "\<lambda>x. f -` {x} \<inter> space M"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   221
  have F: "\<And>x. ?F x \<in> sets (completion M)" and fin: "finite (f`space M)"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   222
    using simple_functionD[OF f] simple_functionD[OF f] by simp_all
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   223
  have "\<forall>x. \<exists>N. N \<in> null_sets M \<and> null_part M (?F x) \<subseteq> N"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   224
    using F null_part by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   225
  from choice[OF this] obtain N where
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   226
    N: "\<And>x. null_part M (?F x) \<subseteq> N x" "\<And>x. N x \<in> null_sets M" by auto
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 43920
diff changeset
   227
  let ?N = "\<Union>x\<in>f`space M. N x"
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 43920
diff changeset
   228
  let ?f' = "\<lambda>x. if x \<in> ?N then undefined else f x"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   229
  have sets: "?N \<in> null_sets M" using N fin by (intro null_sets.finite_UN) auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   230
  show ?thesis unfolding simple_function_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   231
  proof (safe intro!: exI[of _ ?f'])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   232
    have "?f' ` space M \<subseteq> f`space M \<union> {undefined}" by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   233
    from finite_subset[OF this] simple_functionD(1)[OF f]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   234
    show "finite (?f' ` space M)" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   235
  next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   236
    fix x assume "x \<in> space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   237
    have "?f' -` {?f' x} \<inter> space M =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   238
      (if x \<in> ?N then ?F undefined \<union> ?N
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   239
       else if f x = undefined then ?F (f x) \<union> ?N
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   240
       else ?F (f x) - ?N)"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 47694
diff changeset
   241
      using N(2) sets.sets_into_space by (auto split: split_if_asm simp: null_sets_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   242
    moreover { fix y have "?F y \<union> ?N \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   243
      proof cases
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   244
        assume y: "y \<in> f`space M"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   245
        have "?F y \<union> ?N = (main_part M (?F y) \<union> null_part M (?F y)) \<union> ?N"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   246
          using main_part_null_part_Un[OF F] by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   247
        also have "\<dots> = main_part M (?F y) \<union> ?N"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   248
          using y N by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   249
        finally show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   250
          using F sets by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   251
      next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   252
        assume "y \<notin> f`space M" then have "?F y = {}" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   253
        then show ?thesis using sets by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   254
      qed }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   255
    moreover {
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   256
      have "?F (f x) - ?N = main_part M (?F (f x)) \<union> null_part M (?F (f x)) - ?N"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   257
        using main_part_null_part_Un[OF F] by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   258
      also have "\<dots> = main_part M (?F (f x)) - ?N"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   259
        using N `x \<in> space M` by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   260
      finally have "?F (f x) - ?N \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   261
        using F sets by auto }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   262
    ultimately show "?f' -` {?f' x} \<inter> space M \<in> sets M" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   263
  next
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   264
    show "AE x in M. f x = ?f' x"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   265
      by (rule AE_I', rule sets) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   266
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   267
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   268
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   269
lemma completion_ex_borel_measurable_pos:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42866
diff changeset
   270
  fixes g :: "'a \<Rightarrow> ereal"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   271
  assumes g: "g \<in> borel_measurable (completion M)" and "\<And>x. 0 \<le> g x"
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   272
  shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   273
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   274
  from g[THEN borel_measurable_implies_simple_function_sequence'] guess f . note f = this
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41959
diff changeset
   275
  from this(1)[THEN completion_ex_simple_function]
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   276
  have "\<forall>i. \<exists>f'. simple_function M f' \<and> (AE x in M. f i x = f' x)" ..
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   277
  from this[THEN choice] obtain f' where
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41097
diff changeset
   278
    sf: "\<And>i. simple_function M (f' i)" and
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   279
    AE: "\<forall>i. AE x in M. f i x = f' i x" by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   280
  show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   281
  proof (intro bexI)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41959
diff changeset
   282
    from AE[unfolded AE_all_countable[symmetric]]
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46731
diff changeset
   283
    show "AE x in M. g x = (SUP i. f' i x)" (is "AE x in M. g x = ?f x")
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   284
    proof (elim AE_mp, safe intro!: AE_I2)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents:
diff changeset
   285
      fix x assume eq: "\<forall>i. f i x = f' i x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41959
diff changeset
   286
      moreover have "g x = (SUP i. f i x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41959
diff changeset
   287
        unfolding f using `0 \<le> g x` by (auto split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41959
diff changeset
   288
      ultimately show "g x = ?f x" by auto
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    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
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    show "?f \<in> borel_measurable M"
41097
a1abfa4e2b44 use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
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      using sf by (auto intro: borel_measurable_simple_function)
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  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
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qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
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05663f75964c reworked Probability theory
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lemma completion_ex_borel_measurable:
43920
cedb5cb948fd Rename extreal => ereal
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  fixes g :: "'a \<Rightarrow> ereal"
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  assumes g: "g \<in> borel_measurable (completion M)"
05663f75964c reworked Probability theory
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  shows "\<exists>g'\<in>borel_measurable M. (AE x in M. g x = g' x)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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proof -
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05663f75964c reworked Probability theory
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  have "(\<lambda>x. max 0 (g x)) \<in> borel_measurable (completion M)" "\<And>x. 0 \<le> max 0 (g x)" using g by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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diff changeset
   301
  from completion_ex_borel_measurable_pos[OF this] guess g_pos ..
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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   302
  moreover
47694
05663f75964c reworked Probability theory
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   303
  have "(\<lambda>x. max 0 (- g x)) \<in> borel_measurable (completion M)" "\<And>x. 0 \<le> max 0 (- g x)" using g by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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  from completion_ex_borel_measurable_pos[OF this] guess g_neg ..
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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  ultimately
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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   306
  show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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   307
  proof (safe intro!: bexI[of _ "\<lambda>x. g_pos x - g_neg x"])
47694
05663f75964c reworked Probability theory
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diff changeset
   308
    show "AE x in M. max 0 (- g x) = g_neg x \<longrightarrow> max 0 (g x) = g_pos x \<longrightarrow> g x = g_pos x - g_neg x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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diff changeset
   309
    proof (intro AE_I2 impI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff changeset
   310
      fix x assume g: "max 0 (- g x) = g_neg x" "max 0 (g x) = g_pos x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff changeset
   311
      show "g x = g_pos x - g_neg x" unfolding g[symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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   312
        by (cases "g x") (auto split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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   313
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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   314
  qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff changeset
   315
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41959
diff changeset
   316
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   317
end