src/HOL/Probability/Binary_Product_Measure.thy
author hoelzl
Tue, 29 Mar 2011 14:27:39 +0200
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parent 42067 src/HOL/Probability/Product_Measure.thy@66c8281349ec
child 42950 6e5c2a3c69da
permissions -rw-r--r--
split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
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(*  Title:      HOL/Probability/Binary_Product_Measure.thy
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    Author:     Johannes Hölzl, TU München
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*)
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header {*Binary product measures*}
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theory Binary_Product_Measure
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imports Lebesgue_Integration
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begin
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lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
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  by auto
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lemma Pair_vimage_times[simp]: "\<And>A B x. Pair x -` (A \<times> B) = (if x \<in> A then B else {})"
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  by auto
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lemma rev_Pair_vimage_times[simp]: "\<And>A B y. (\<lambda>x. (x, y)) -` (A \<times> B) = (if y \<in> B then A else {})"
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  by auto
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lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
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  by (cases x) simp
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lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
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  by (auto simp: fun_eq_iff)
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section "Binary products"
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definition
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  "pair_measure_generator A B =
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    \<lparr> space = space A \<times> space B,
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      sets = {a \<times> b | a b. a \<in> sets A \<and> b \<in> sets B},
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      measure = \<lambda>X. \<integral>\<^isup>+x. (\<integral>\<^isup>+y. indicator X (x,y) \<partial>B) \<partial>A \<rparr>"
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definition pair_measure (infixr "\<Otimes>\<^isub>M" 80) where
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  "A \<Otimes>\<^isub>M B = sigma (pair_measure_generator A B)"
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locale pair_sigma_algebra = M1: sigma_algebra M1 + M2: sigma_algebra M2
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  for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
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abbreviation (in pair_sigma_algebra)
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  "E \<equiv> pair_measure_generator M1 M2"
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abbreviation (in pair_sigma_algebra)
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  "P \<equiv> M1 \<Otimes>\<^isub>M M2"
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lemma sigma_algebra_pair_measure:
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  "sets M1 \<subseteq> Pow (space M1) \<Longrightarrow> sets M2 \<subseteq> Pow (space M2) \<Longrightarrow> sigma_algebra (pair_measure M1 M2)"
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  by (force simp: pair_measure_def pair_measure_generator_def intro!: sigma_algebra_sigma)
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sublocale pair_sigma_algebra \<subseteq> sigma_algebra P
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  using M1.space_closed M2.space_closed
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  by (rule sigma_algebra_pair_measure)
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lemma pair_measure_generatorI[intro, simp]:
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  "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (pair_measure_generator A B)"
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  by (auto simp add: pair_measure_generator_def)
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lemma pair_measureI[intro, simp]:
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  "x \<in> sets A \<Longrightarrow> y \<in> sets B \<Longrightarrow> x \<times> y \<in> sets (A \<Otimes>\<^isub>M B)"
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  by (auto simp add: pair_measure_def)
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lemma space_pair_measure:
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  "space (A \<Otimes>\<^isub>M B) = space A \<times> space B"
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  by (simp add: pair_measure_def pair_measure_generator_def)
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lemma sets_pair_measure_generator:
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  "sets (pair_measure_generator N M) = (\<lambda>(x, y). x \<times> y) ` (sets N \<times> sets M)"
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  unfolding pair_measure_generator_def by auto
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lemma pair_measure_generator_sets_into_space:
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  assumes "sets M \<subseteq> Pow (space M)" "sets N \<subseteq> Pow (space N)"
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  shows "sets (pair_measure_generator M N) \<subseteq> Pow (space (pair_measure_generator M N))"
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  using assms by (auto simp: pair_measure_generator_def)
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lemma pair_measure_generator_Int_snd:
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  assumes "sets S1 \<subseteq> Pow (space S1)"
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  shows "sets (pair_measure_generator S1 (algebra.restricted_space S2 A)) =
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         sets (algebra.restricted_space (pair_measure_generator S1 S2) (space S1 \<times> A))"
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  (is "?L = ?R")
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  apply (auto simp: pair_measure_generator_def image_iff)
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  using assms
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  apply (rule_tac x="a \<times> xa" in exI)
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  apply force
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  using assms
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  apply (rule_tac x="a" in exI)
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  apply (rule_tac x="b \<inter> A" in exI)
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  apply auto
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  done
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lemma (in pair_sigma_algebra)
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  shows measurable_fst[intro!, simp]:
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    "fst \<in> measurable P M1" (is ?fst)
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  and measurable_snd[intro!, simp]:
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    "snd \<in> measurable P M2" (is ?snd)
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proof -
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  { fix X assume "X \<in> sets M1"
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    then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. fst -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
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      apply - apply (rule bexI[of _ X]) apply (rule bexI[of _ "space M2"])
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      using M1.sets_into_space by force+ }
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  moreover
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  { fix X assume "X \<in> sets M2"
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    then have "\<exists>X1\<in>sets M1. \<exists>X2\<in>sets M2. snd -` X \<inter> space M1 \<times> space M2 = X1 \<times> X2"
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      apply - apply (rule bexI[of _ "space M1"]) apply (rule bexI[of _ X])
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      using M2.sets_into_space by force+ }
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  ultimately have "?fst \<and> ?snd"
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    by (fastsimp simp: measurable_def sets_sigma space_pair_measure
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                 intro!: sigma_sets.Basic)
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  then show ?fst ?snd by auto
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qed
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lemma (in pair_sigma_algebra) measurable_pair_iff:
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  assumes "sigma_algebra M"
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  shows "f \<in> measurable M P \<longleftrightarrow>
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    (fst \<circ> f) \<in> measurable M M1 \<and> (snd \<circ> f) \<in> measurable M M2"
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proof -
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  interpret M: sigma_algebra M by fact
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  from assms show ?thesis
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  proof (safe intro!: measurable_comp[where b=P])
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    assume f: "(fst \<circ> f) \<in> measurable M M1" and s: "(snd \<circ> f) \<in> measurable M M2"
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    show "f \<in> measurable M P" unfolding pair_measure_def
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    proof (rule M.measurable_sigma)
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      show "sets (pair_measure_generator M1 M2) \<subseteq> Pow (space E)"
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        unfolding pair_measure_generator_def using M1.sets_into_space M2.sets_into_space by auto
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      show "f \<in> space M \<rightarrow> space E"
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        using f s by (auto simp: mem_Times_iff measurable_def comp_def space_sigma pair_measure_generator_def)
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   126
      fix A assume "A \<in> sets E"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   127
      then obtain B C where "B \<in> sets M1" "C \<in> sets M2" "A = B \<times> C"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   128
        unfolding pair_measure_generator_def by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   129
      moreover have "(fst \<circ> f) -` B \<inter> space M \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   130
        using f `B \<in> sets M1` unfolding measurable_def by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   131
      moreover have "(snd \<circ> f) -` C \<inter> space M \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   132
        using s `C \<in> sets M2` unfolding measurable_def by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   133
      moreover have "f -` A \<inter> space M = ((fst \<circ> f) -` B \<inter> space M) \<inter> ((snd \<circ> f) -` C \<inter> space M)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   134
        unfolding `A = B \<times> C` by (auto simp: vimage_Times)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   135
      ultimately show "f -` A \<inter> space M \<in> sets M" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   136
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   137
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   138
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   139
41095
c335d880ff82 cleanup bijectivity btw. product spaces and pairs
hoelzl
parents: 41026
diff changeset
   140
lemma (in pair_sigma_algebra) measurable_pair:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   141
  assumes "sigma_algebra M"
41095
c335d880ff82 cleanup bijectivity btw. product spaces and pairs
hoelzl
parents: 41026
diff changeset
   142
  assumes "(fst \<circ> f) \<in> measurable M M1" "(snd \<circ> f) \<in> measurable M M2"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   143
  shows "f \<in> measurable M P"
41095
c335d880ff82 cleanup bijectivity btw. product spaces and pairs
hoelzl
parents: 41026
diff changeset
   144
  unfolding measurable_pair_iff[OF assms(1)] using assms(2,3) by simp
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   145
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   146
lemma pair_measure_generatorE:
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   147
  assumes "X \<in> sets (pair_measure_generator M1 M2)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   148
  obtains A B where "X = A \<times> B" "A \<in> sets M1" "B \<in> sets M2"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   149
  using assms unfolding pair_measure_generator_def by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   150
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   151
lemma (in pair_sigma_algebra) pair_measure_generator_swap:
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   152
  "(\<lambda>X. (\<lambda>(x,y). (y,x)) -` X \<inter> space M2 \<times> space M1) ` sets E = sets (pair_measure_generator M2 M1)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   153
proof (safe elim!: pair_measure_generatorE)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   154
  fix A B assume "A \<in> sets M1" "B \<in> sets M2"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   155
  moreover then have "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 = B \<times> A"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   156
    using M1.sets_into_space M2.sets_into_space by auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   157
  ultimately show "(\<lambda>(x, y). (y, x)) -` (A \<times> B) \<inter> space M2 \<times> space M1 \<in> sets (pair_measure_generator M2 M1)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   158
    by (auto intro: pair_measure_generatorI)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   159
next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   160
  fix A B assume "A \<in> sets M1" "B \<in> sets M2"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   161
  then show "B \<times> A \<in> (\<lambda>X. (\<lambda>(x, y). (y, x)) -` X \<inter> space M2 \<times> space M1) ` sets E"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   162
    using M1.sets_into_space M2.sets_into_space
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   163
    by (auto intro!: image_eqI[where x="A \<times> B"] pair_measure_generatorI)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   164
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   165
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   166
lemma (in pair_sigma_algebra) sets_pair_sigma_algebra_swap:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   167
  assumes Q: "Q \<in> sets P"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   168
  shows "(\<lambda>(x,y). (y, x)) -` Q \<in> sets (M2 \<Otimes>\<^isub>M M1)" (is "_ \<in> sets ?Q")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   169
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   170
  let "?f Q" = "(\<lambda>(x,y). (y, x)) -` Q \<inter> space M2 \<times> space M1"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   171
  have *: "(\<lambda>(x,y). (y, x)) -` Q = ?f Q"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   172
    using sets_into_space[OF Q] by (auto simp: space_pair_measure)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   173
  have "sets (M2 \<Otimes>\<^isub>M M1) = sets (sigma (pair_measure_generator M2 M1))"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   174
    unfolding pair_measure_def ..
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   175
  also have "\<dots> = sigma_sets (space M2 \<times> space M1) (?f ` sets E)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   176
    unfolding sigma_def pair_measure_generator_swap[symmetric]
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   177
    by (simp add: pair_measure_generator_def)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   178
  also have "\<dots> = ?f ` sigma_sets (space M1 \<times> space M2) (sets E)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   179
    using M1.sets_into_space M2.sets_into_space
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   180
    by (intro sigma_sets_vimage) (auto simp: pair_measure_generator_def)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   181
  also have "\<dots> = ?f ` sets P"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   182
    unfolding pair_measure_def pair_measure_generator_def sigma_def by simp
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   183
  finally show ?thesis
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   184
    using Q by (subst *) auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   185
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   186
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   187
lemma (in pair_sigma_algebra) pair_sigma_algebra_swap_measurable:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   188
  shows "(\<lambda>(x,y). (y, x)) \<in> measurable P (M2 \<Otimes>\<^isub>M M1)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   189
    (is "?f \<in> measurable ?P ?Q")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   190
  unfolding measurable_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   191
proof (intro CollectI conjI Pi_I ballI)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   192
  fix x assume "x \<in> space ?P" then show "(case x of (x, y) \<Rightarrow> (y, x)) \<in> space ?Q"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   193
    unfolding pair_measure_generator_def pair_measure_def by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   194
next
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   195
  fix A assume "A \<in> sets (M2 \<Otimes>\<^isub>M M1)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   196
  interpret Q: pair_sigma_algebra M2 M1 by default
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   197
  with Q.sets_pair_sigma_algebra_swap[OF `A \<in> sets (M2 \<Otimes>\<^isub>M M1)`]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   198
  show "?f -` A \<inter> space ?P \<in> sets ?P" by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   199
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   200
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   201
lemma (in pair_sigma_algebra) measurable_cut_fst[simp,intro]:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   202
  assumes "Q \<in> sets P" shows "Pair x -` Q \<in> sets M2"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   203
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   204
  let ?Q' = "{Q. Q \<subseteq> space P \<and> Pair x -` Q \<in> sets M2}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   205
  let ?Q = "\<lparr> space = space P, sets = ?Q' \<rparr>"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   206
  interpret Q: sigma_algebra ?Q
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   207
    proof qed (auto simp: vimage_UN vimage_Diff space_pair_measure)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   208
  have "sets E \<subseteq> sets ?Q"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   209
    using M1.sets_into_space M2.sets_into_space
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   210
    by (auto simp: pair_measure_generator_def space_pair_measure)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   211
  then have "sets P \<subseteq> sets ?Q"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   212
    apply (subst pair_measure_def, intro Q.sets_sigma_subset)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   213
    by (simp add: pair_measure_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   214
  with assms show ?thesis by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   215
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   216
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   217
lemma (in pair_sigma_algebra) measurable_cut_snd:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   218
  assumes Q: "Q \<in> sets P" shows "(\<lambda>x. (x, y)) -` Q \<in> sets M1" (is "?cut Q \<in> sets M1")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   219
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   220
  interpret Q: pair_sigma_algebra M2 M1 by default
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   221
  with Q.measurable_cut_fst[OF sets_pair_sigma_algebra_swap[OF Q], of y]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   222
  show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   223
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   224
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   225
lemma (in pair_sigma_algebra) measurable_pair_image_snd:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   226
  assumes m: "f \<in> measurable P M" and "x \<in> space M1"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   227
  shows "(\<lambda>y. f (x, y)) \<in> measurable M2 M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   228
  unfolding measurable_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   229
proof (intro CollectI conjI Pi_I ballI)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   230
  fix y assume "y \<in> space M2" with `f \<in> measurable P M` `x \<in> space M1`
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   231
  show "f (x, y) \<in> space M"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   232
    unfolding measurable_def pair_measure_generator_def pair_measure_def by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   233
next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   234
  fix A assume "A \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   235
  then have "Pair x -` (f -` A \<inter> space P) \<in> sets M2" (is "?C \<in> _")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   236
    using `f \<in> measurable P M`
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   237
    by (intro measurable_cut_fst) (auto simp: measurable_def)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   238
  also have "?C = (\<lambda>y. f (x, y)) -` A \<inter> space M2"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   239
    using `x \<in> space M1` by (auto simp: pair_measure_generator_def pair_measure_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   240
  finally show "(\<lambda>y. f (x, y)) -` A \<inter> space M2 \<in> sets M2" .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   241
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   242
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   243
lemma (in pair_sigma_algebra) measurable_pair_image_fst:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   244
  assumes m: "f \<in> measurable P M" and "y \<in> space M2"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   245
  shows "(\<lambda>x. f (x, y)) \<in> measurable M1 M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   246
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   247
  interpret Q: pair_sigma_algebra M2 M1 by default
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   248
  from Q.measurable_pair_image_snd[OF measurable_comp `y \<in> space M2`,
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   249
                                      OF Q.pair_sigma_algebra_swap_measurable m]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   250
  show ?thesis by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   251
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   252
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   253
lemma (in pair_sigma_algebra) Int_stable_pair_measure_generator: "Int_stable E"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   254
  unfolding Int_stable_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   255
proof (intro ballI)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   256
  fix A B assume "A \<in> sets E" "B \<in> sets E"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   257
  then obtain A1 A2 B1 B2 where "A = A1 \<times> A2" "B = B1 \<times> B2"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   258
    "A1 \<in> sets M1" "A2 \<in> sets M2" "B1 \<in> sets M1" "B2 \<in> sets M2"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   259
    unfolding pair_measure_generator_def by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   260
  then show "A \<inter> B \<in> sets E"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   261
    by (auto simp add: times_Int_times pair_measure_generator_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   262
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   263
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   264
lemma finite_measure_cut_measurable:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   265
  fixes M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   266
  assumes "sigma_finite_measure M1" "finite_measure M2"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   267
  assumes "Q \<in> sets (M1 \<Otimes>\<^isub>M M2)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   268
  shows "(\<lambda>x. measure M2 (Pair x -` Q)) \<in> borel_measurable M1"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   269
    (is "?s Q \<in> _")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   270
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   271
  interpret M1: sigma_finite_measure M1 by fact
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   272
  interpret M2: finite_measure M2 by fact
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   273
  interpret pair_sigma_algebra M1 M2 by default
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   274
  have [intro]: "sigma_algebra M1" by fact
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   275
  have [intro]: "sigma_algebra M2" by fact
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   276
  let ?D = "\<lparr> space = space P, sets = {A\<in>sets P. ?s A \<in> borel_measurable M1}  \<rparr>"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   277
  note space_pair_measure[simp]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   278
  interpret dynkin_system ?D
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   279
  proof (intro dynkin_systemI)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   280
    fix A assume "A \<in> sets ?D" then show "A \<subseteq> space ?D"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   281
      using sets_into_space by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   282
  next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   283
    from top show "space ?D \<in> sets ?D"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   284
      by (auto simp add: if_distrib intro!: M1.measurable_If)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   285
  next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   286
    fix A assume "A \<in> sets ?D"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   287
    with sets_into_space have "\<And>x. measure M2 (Pair x -` (space M1 \<times> space M2 - A)) =
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   288
        (if x \<in> space M1 then measure M2 (space M2) - ?s A x else 0)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   289
      by (auto intro!: M2.measure_compl simp: vimage_Diff)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   290
    with `A \<in> sets ?D` top show "space ?D - A \<in> sets ?D"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   291
      by (auto intro!: Diff M1.measurable_If M1.borel_measurable_extreal_diff)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   292
  next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   293
    fix F :: "nat \<Rightarrow> ('a\<times>'b) set" assume "disjoint_family F" "range F \<subseteq> sets ?D"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   294
    moreover then have "\<And>x. measure M2 (\<Union>i. Pair x -` F i) = (\<Sum>i. ?s (F i) x)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   295
      by (intro M2.measure_countably_additive[symmetric])
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   296
         (auto simp: disjoint_family_on_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   297
    ultimately show "(\<Union>i. F i) \<in> sets ?D"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   298
      by (auto simp: vimage_UN intro!: M1.borel_measurable_psuminf)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   299
  qed
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   300
  have "sets P = sets ?D" apply (subst pair_measure_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   301
  proof (intro dynkin_lemma)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   302
    show "Int_stable E" by (rule Int_stable_pair_measure_generator)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   303
    from M1.sets_into_space have "\<And>A. A \<in> sets M1 \<Longrightarrow> {x \<in> space M1. x \<in> A} = A"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   304
      by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   305
    then show "sets E \<subseteq> sets ?D"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   306
      by (auto simp: pair_measure_generator_def sets_sigma if_distrib
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   307
               intro: sigma_sets.Basic intro!: M1.measurable_If)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   308
  qed (auto simp: pair_measure_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   309
  with `Q \<in> sets P` have "Q \<in> sets ?D" by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   310
  then show "?s Q \<in> borel_measurable M1" by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   311
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   312
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   313
subsection {* Binary products of $\sigma$-finite measure spaces *}
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   314
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   315
locale pair_sigma_finite = M1: sigma_finite_measure M1 + M2: sigma_finite_measure M2
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   316
  for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   317
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   318
sublocale pair_sigma_finite \<subseteq> pair_sigma_algebra M1 M2
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   319
  by default
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   320
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   321
lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   322
  by auto
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   323
42146
5b52c6a9c627 split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents: 42067
diff changeset
   324
lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
5b52c6a9c627 split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents: 42067
diff changeset
   325
proof
5b52c6a9c627 split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents: 42067
diff changeset
   326
  fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B"
5b52c6a9c627 split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents: 42067
diff changeset
   327
    by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros)
5b52c6a9c627 split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents: 42067
diff changeset
   328
qed
5b52c6a9c627 split Product_Measure into Binary_Product_Measure and Finite_Product_Measure
hoelzl
parents: 42067
diff changeset
   329
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   330
lemma (in pair_sigma_finite) measure_cut_measurable_fst:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   331
  assumes "Q \<in> sets P" shows "(\<lambda>x. measure M2 (Pair x -` Q)) \<in> borel_measurable M1" (is "?s Q \<in> _")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   332
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   333
  have [intro]: "sigma_algebra M1" and [intro]: "sigma_algebra M2" by default+
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   334
  have M1: "sigma_finite_measure M1" by default
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   335
  from M2.disjoint_sigma_finite guess F .. note F = this
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   336
  then have F_sets: "\<And>i. F i \<in> sets M2" by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   337
  let "?C x i" = "F i \<inter> Pair x -` Q"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   338
  { fix i
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   339
    let ?R = "M2.restricted_space (F i)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   340
    have [simp]: "space M1 \<times> F i \<inter> space M1 \<times> space M2 = space M1 \<times> F i"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   341
      using F M2.sets_into_space by auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   342
    let ?R2 = "M2.restricted_space (F i)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   343
    have "(\<lambda>x. measure ?R2 (Pair x -` (space M1 \<times> space ?R2 \<inter> Q))) \<in> borel_measurable M1"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   344
    proof (intro finite_measure_cut_measurable[OF M1])
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   345
      show "finite_measure ?R2"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   346
        using F by (intro M2.restricted_to_finite_measure) auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   347
      have "(space M1 \<times> space ?R2) \<inter> Q \<in> (op \<inter> (space M1 \<times> F i)) ` sets P"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   348
        using `Q \<in> sets P` by (auto simp: image_iff)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   349
      also have "\<dots> = sigma_sets (space M1 \<times> F i) ((op \<inter> (space M1 \<times> F i)) ` sets E)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   350
        unfolding pair_measure_def pair_measure_generator_def sigma_def
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   351
        using `F i \<in> sets M2` M2.sets_into_space
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   352
        by (auto intro!: sigma_sets_Int sigma_sets.Basic)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   353
      also have "\<dots> \<subseteq> sets (M1 \<Otimes>\<^isub>M ?R2)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   354
        using M1.sets_into_space
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   355
        apply (auto simp: times_Int_times pair_measure_def pair_measure_generator_def sigma_def
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   356
                    intro!: sigma_sets_subseteq)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   357
        apply (rule_tac x="a" in exI)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   358
        apply (rule_tac x="b \<inter> F i" in exI)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   359
        by auto
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   360
      finally show "(space M1 \<times> space ?R2) \<inter> Q \<in> sets (M1 \<Otimes>\<^isub>M ?R2)" .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   361
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   362
    moreover have "\<And>x. Pair x -` (space M1 \<times> F i \<inter> Q) = ?C x i"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   363
      using `Q \<in> sets P` sets_into_space by (auto simp: space_pair_measure)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   364
    ultimately have "(\<lambda>x. measure M2 (?C x i)) \<in> borel_measurable M1"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   365
      by simp }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   366
  moreover
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   367
  { fix x
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   368
    have "(\<Sum>i. measure M2 (?C x i)) = measure M2 (\<Union>i. ?C x i)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   369
    proof (intro M2.measure_countably_additive)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   370
      show "range (?C x) \<subseteq> sets M2"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   371
        using F `Q \<in> sets P` by (auto intro!: M2.Int)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   372
      have "disjoint_family F" using F by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   373
      show "disjoint_family (?C x)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   374
        by (rule disjoint_family_on_bisimulation[OF `disjoint_family F`]) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   375
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   376
    also have "(\<Union>i. ?C x i) = Pair x -` Q"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   377
      using F sets_into_space `Q \<in> sets P`
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   378
      by (auto simp: space_pair_measure)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   379
    finally have "measure M2 (Pair x -` Q) = (\<Sum>i. measure M2 (?C x i))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   380
      by simp }
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   381
  ultimately show ?thesis using `Q \<in> sets P` F_sets
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   382
    by (auto intro!: M1.borel_measurable_psuminf M2.Int)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   383
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   384
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   385
lemma (in pair_sigma_finite) measure_cut_measurable_snd:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   386
  assumes "Q \<in> sets P" shows "(\<lambda>y. M1.\<mu> ((\<lambda>x. (x, y)) -` Q)) \<in> borel_measurable M2"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   387
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   388
  interpret Q: pair_sigma_finite M2 M1 by default
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   389
  note sets_pair_sigma_algebra_swap[OF assms]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   390
  from Q.measure_cut_measurable_fst[OF this]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   391
  show ?thesis by (simp add: vimage_compose[symmetric] comp_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   392
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   393
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   394
lemma (in pair_sigma_algebra) pair_sigma_algebra_measurable:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   395
  assumes "f \<in> measurable P M" shows "(\<lambda>(x,y). f (y, x)) \<in> measurable (M2 \<Otimes>\<^isub>M M1) M"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   396
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   397
  interpret Q: pair_sigma_algebra M2 M1 by default
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   398
  have *: "(\<lambda>(x,y). f (y, x)) = f \<circ> (\<lambda>(x,y). (y, x))" by (simp add: fun_eq_iff)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   399
  show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   400
    using Q.pair_sigma_algebra_swap_measurable assms
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   401
    unfolding * by (rule measurable_comp)
39088
ca17017c10e6 Measurable on product space is equiv. to measurable components
hoelzl
parents: 39082
diff changeset
   402
qed
ca17017c10e6 Measurable on product space is equiv. to measurable components
hoelzl
parents: 39082
diff changeset
   403
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   404
lemma (in pair_sigma_finite) pair_measure_alt:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   405
  assumes "A \<in> sets P"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   406
  shows "measure (M1 \<Otimes>\<^isub>M M2) A = (\<integral>\<^isup>+ x. measure M2 (Pair x -` A) \<partial>M1)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   407
  apply (simp add: pair_measure_def pair_measure_generator_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   408
proof (rule M1.positive_integral_cong)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   409
  fix x assume "x \<in> space M1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   410
  have *: "\<And>y. indicator A (x, y) = (indicator (Pair x -` A) y :: extreal)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   411
    unfolding indicator_def by auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   412
  show "(\<integral>\<^isup>+ y. indicator A (x, y) \<partial>M2) = measure M2 (Pair x -` A)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   413
    unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   414
    apply (subst M2.positive_integral_indicator)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   415
    apply (rule measurable_cut_fst[OF assms])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   416
    by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   417
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   418
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   419
lemma (in pair_sigma_finite) pair_measure_times:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   420
  assumes A: "A \<in> sets M1" and "B \<in> sets M2"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   421
  shows "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = M1.\<mu> A * measure M2 B"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   422
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   423
  have "measure (M1 \<Otimes>\<^isub>M M2) (A \<times> B) = (\<integral>\<^isup>+ x. measure M2 B * indicator A x \<partial>M1)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   424
    using assms by (auto intro!: M1.positive_integral_cong simp: pair_measure_alt)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   425
  with assms show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   426
    by (simp add: M1.positive_integral_cmult_indicator ac_simps)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   427
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   428
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   429
lemma (in measure_space) measure_not_negative[simp,intro]:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   430
  assumes A: "A \<in> sets M" shows "\<mu> A \<noteq> - \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   431
  using positive_measure[OF A] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   432
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   433
lemma (in pair_sigma_finite) sigma_finite_up_in_pair_measure_generator:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   434
  "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets E \<and> incseq F \<and> (\<Union>i. F i) = space E \<and>
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   435
    (\<forall>i. measure (M1 \<Otimes>\<^isub>M M2) (F i) \<noteq> \<infinity>)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   436
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   437
  obtain F1 :: "nat \<Rightarrow> 'a set" and F2 :: "nat \<Rightarrow> 'b set" where
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   438
    F1: "range F1 \<subseteq> sets M1" "incseq F1" "(\<Union>i. F1 i) = space M1" "\<And>i. M1.\<mu> (F1 i) \<noteq> \<infinity>" and
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   439
    F2: "range F2 \<subseteq> sets M2" "incseq F2" "(\<Union>i. F2 i) = space M2" "\<And>i. M2.\<mu> (F2 i) \<noteq> \<infinity>"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   440
    using M1.sigma_finite_up M2.sigma_finite_up by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   441
  then have space: "space M1 = (\<Union>i. F1 i)" "space M2 = (\<Union>i. F2 i)" by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   442
  let ?F = "\<lambda>i. F1 i \<times> F2 i"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   443
  show ?thesis unfolding space_pair_measure
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   444
  proof (intro exI[of _ ?F] conjI allI)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   445
    show "range ?F \<subseteq> sets E" using F1 F2
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   446
      by (fastsimp intro!: pair_measure_generatorI)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   447
  next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   448
    have "space M1 \<times> space M2 \<subseteq> (\<Union>i. ?F i)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   449
    proof (intro subsetI)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   450
      fix x assume "x \<in> space M1 \<times> space M2"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   451
      then obtain i j where "fst x \<in> F1 i" "snd x \<in> F2 j"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   452
        by (auto simp: space)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   453
      then have "fst x \<in> F1 (max i j)" "snd x \<in> F2 (max j i)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   454
        using `incseq F1` `incseq F2` unfolding incseq_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   455
        by (force split: split_max)+
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   456
      then have "(fst x, snd x) \<in> F1 (max i j) \<times> F2 (max i j)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   457
        by (intro SigmaI) (auto simp add: min_max.sup_commute)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   458
      then show "x \<in> (\<Union>i. ?F i)" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   459
    qed
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   460
    then show "(\<Union>i. ?F i) = space E"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   461
      using space by (auto simp: space pair_measure_generator_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   462
  next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   463
    fix i show "incseq (\<lambda>i. F1 i \<times> F2 i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   464
      using `incseq F1` `incseq F2` unfolding incseq_Suc_iff by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   465
  next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   466
    fix i
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   467
    from F1 F2 have "F1 i \<in> sets M1" "F2 i \<in> sets M2" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   468
    with F1 F2 M1.positive_measure[OF this(1)] M2.positive_measure[OF this(2)]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   469
    show "measure P (F1 i \<times> F2 i) \<noteq> \<infinity>"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   470
      by (simp add: pair_measure_times)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   471
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   472
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   473
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   474
sublocale pair_sigma_finite \<subseteq> sigma_finite_measure P
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   475
proof
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   476
  show "positive P (measure P)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   477
    unfolding pair_measure_def pair_measure_generator_def sigma_def positive_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   478
    by (auto intro: M1.positive_integral_positive M2.positive_integral_positive)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   479
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   480
  show "countably_additive P (measure P)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   481
    unfolding countably_additive_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   482
  proof (intro allI impI)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   483
    fix F :: "nat \<Rightarrow> ('a \<times> 'b) set"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   484
    assume F: "range F \<subseteq> sets P" "disjoint_family F"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   485
    from F have *: "\<And>i. F i \<in> sets P" "(\<Union>i. F i) \<in> sets P" by auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   486
    moreover from F have "\<And>i. (\<lambda>x. measure M2 (Pair x -` F i)) \<in> borel_measurable M1"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   487
      by (intro measure_cut_measurable_fst) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   488
    moreover have "\<And>x. disjoint_family (\<lambda>i. Pair x -` F i)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   489
      by (intro disjoint_family_on_bisimulation[OF F(2)]) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   490
    moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> range (\<lambda>i. Pair x -` F i) \<subseteq> sets M2"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   491
      using F by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   492
    ultimately show "(\<Sum>n. measure P (F n)) = measure P (\<Union>i. F i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   493
      by (simp add: pair_measure_alt vimage_UN M1.positive_integral_suminf[symmetric]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   494
                    M2.measure_countably_additive
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   495
               cong: M1.positive_integral_cong)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   496
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   497
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   498
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   499
  show "\<exists>F::nat \<Rightarrow> ('a \<times> 'b) set. range F \<subseteq> sets P \<and> (\<Union>i. F i) = space P \<and> (\<forall>i. measure P (F i) \<noteq> \<infinity>)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   500
  proof (rule exI[of _ F], intro conjI)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   501
    show "range F \<subseteq> sets P" using F by (auto simp: pair_measure_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   502
    show "(\<Union>i. F i) = space P"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   503
      using F by (auto simp: pair_measure_def pair_measure_generator_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   504
    show "\<forall>i. measure P (F i) \<noteq> \<infinity>" using F by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   505
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   506
qed
39088
ca17017c10e6 Measurable on product space is equiv. to measurable components
hoelzl
parents: 39082
diff changeset
   507
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   508
lemma (in pair_sigma_algebra) sets_swap:
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   509
  assumes "A \<in> sets P"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   510
  shows "(\<lambda>(x, y). (y, x)) -` A \<inter> space (M2 \<Otimes>\<^isub>M M1) \<in> sets (M2 \<Otimes>\<^isub>M M1)"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   511
    (is "_ -` A \<inter> space ?Q \<in> sets ?Q")
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   512
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   513
  have *: "(\<lambda>(x, y). (y, x)) -` A \<inter> space ?Q = (\<lambda>(x, y). (y, x)) -` A"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   514
    using `A \<in> sets P` sets_into_space by (auto simp: space_pair_measure)
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   515
  show ?thesis
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   516
    unfolding * using assms by (rule sets_pair_sigma_algebra_swap)
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   517
qed
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   518
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   519
lemma (in pair_sigma_finite) pair_measure_alt2:
41706
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   520
  assumes A: "A \<in> sets P"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   521
  shows "\<mu> A = (\<integral>\<^isup>+y. M1.\<mu> ((\<lambda>x. (x, y)) -` A) \<partial>M2)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   522
    (is "_ = ?\<nu> A")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   523
proof -
41706
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   524
  interpret Q: pair_sigma_finite M2 M1 by default
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   525
  from sigma_finite_up_in_pair_measure_generator guess F :: "nat \<Rightarrow> ('a \<times> 'b) set" .. note F = this
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   526
  have [simp]: "\<And>m. \<lparr> space = space E, sets = sets (sigma E), measure = m \<rparr> = P\<lparr> measure := m \<rparr>"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   527
    unfolding pair_measure_def by simp
41706
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   528
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   529
  have "\<mu> A = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` A \<inter> space Q.P)"
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   530
  proof (rule measure_unique_Int_stable_vimage[OF Int_stable_pair_measure_generator])
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   531
    show "measure_space P" "measure_space Q.P" by default
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   532
    show "(\<lambda>(y, x). (x, y)) \<in> measurable Q.P P" by (rule Q.pair_sigma_algebra_swap_measurable)
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   533
    show "sets (sigma E) = sets P" "space E = space P" "A \<in> sets (sigma E)"
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   534
      using assms unfolding pair_measure_def by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   535
    show "range F \<subseteq> sets E" "incseq F" "(\<Union>i. F i) = space E" "\<And>i. \<mu> (F i) \<noteq> \<infinity>"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   536
      using F `A \<in> sets P` by (auto simp: pair_measure_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   537
    fix X assume "X \<in> sets E"
41706
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   538
    then obtain A B where X[simp]: "X = A \<times> B" and AB: "A \<in> sets M1" "B \<in> sets M2"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   539
      unfolding pair_measure_def pair_measure_generator_def by auto
41706
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   540
    then have "(\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P = B \<times> A"
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   541
      using M1.sets_into_space M2.sets_into_space by (auto simp: space_pair_measure)
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   542
    then show "\<mu> X = Q.\<mu> ((\<lambda>(y, x). (x, y)) -` X \<inter> space Q.P)"
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   543
      using AB by (simp add: pair_measure_times Q.pair_measure_times ac_simps)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   544
  qed
41706
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   545
  then show ?thesis
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   546
    using sets_into_space[OF A] Q.pair_measure_alt[OF sets_swap[OF A]]
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   547
    by (auto simp add: Q.pair_measure_alt space_pair_measure
a207a858d1f6 prefer p2e before e2p; use measure_unique_Int_stable_vimage;
hoelzl
parents: 41705
diff changeset
   548
             intro!: M2.positive_integral_cong arg_cong[where f="M1.\<mu>"])
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   549
qed
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   550
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   551
lemma pair_sigma_algebra_sigma:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   552
  assumes 1: "incseq S1" "(\<Union>i. S1 i) = space E1" "range S1 \<subseteq> sets E1" and E1: "sets E1 \<subseteq> Pow (space E1)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   553
  assumes 2: "decseq S2" "(\<Union>i. S2 i) = space E2" "range S2 \<subseteq> sets E2" and E2: "sets E2 \<subseteq> Pow (space E2)"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   554
  shows "sets (sigma (pair_measure_generator (sigma E1) (sigma E2))) = sets (sigma (pair_measure_generator E1 E2))"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   555
    (is "sets ?S = sets ?E")
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   556
proof -
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   557
  interpret M1: sigma_algebra "sigma E1" using E1 by (rule sigma_algebra_sigma)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   558
  interpret M2: sigma_algebra "sigma E2" using E2 by (rule sigma_algebra_sigma)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   559
  have P: "sets (pair_measure_generator E1 E2) \<subseteq> Pow (space E1 \<times> space E2)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   560
    using E1 E2 by (auto simp add: pair_measure_generator_def)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   561
  interpret E: sigma_algebra ?E unfolding pair_measure_generator_def
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   562
    using E1 E2 by (intro sigma_algebra_sigma) auto
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   563
  { fix A assume "A \<in> sets E1"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   564
    then have "fst -` A \<inter> space ?E = A \<times> (\<Union>i. S2 i)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   565
      using E1 2 unfolding pair_measure_generator_def by auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   566
    also have "\<dots> = (\<Union>i. A \<times> S2 i)" by auto
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   567
    also have "\<dots> \<in> sets ?E" unfolding pair_measure_generator_def sets_sigma
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   568
      using 2 `A \<in> sets E1`
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   569
      by (intro sigma_sets.Union)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   570
         (force simp: image_subset_iff intro!: sigma_sets.Basic)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   571
    finally have "fst -` A \<inter> space ?E \<in> sets ?E" . }
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   572
  moreover
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   573
  { fix B assume "B \<in> sets E2"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   574
    then have "snd -` B \<inter> space ?E = (\<Union>i. S1 i) \<times> B"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   575
      using E2 1 unfolding pair_measure_generator_def by auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   576
    also have "\<dots> = (\<Union>i. S1 i \<times> B)" by auto
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   577
    also have "\<dots> \<in> sets ?E"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   578
      using 1 `B \<in> sets E2` unfolding pair_measure_generator_def sets_sigma
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   579
      by (intro sigma_sets.Union)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   580
         (force simp: image_subset_iff intro!: sigma_sets.Basic)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   581
    finally have "snd -` B \<inter> space ?E \<in> sets ?E" . }
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   582
  ultimately have proj:
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   583
    "fst \<in> measurable ?E (sigma E1) \<and> snd \<in> measurable ?E (sigma E2)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   584
    using E1 E2 by (subst (1 2) E.measurable_iff_sigma)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   585
                   (auto simp: pair_measure_generator_def sets_sigma)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   586
  { fix A B assume A: "A \<in> sets (sigma E1)" and B: "B \<in> sets (sigma E2)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   587
    with proj have "fst -` A \<inter> space ?E \<in> sets ?E" "snd -` B \<inter> space ?E \<in> sets ?E"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   588
      unfolding measurable_def by simp_all
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   589
    moreover have "A \<times> B = (fst -` A \<inter> space ?E) \<inter> (snd -` B \<inter> space ?E)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   590
      using A B M1.sets_into_space M2.sets_into_space
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   591
      by (auto simp: pair_measure_generator_def)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   592
    ultimately have "A \<times> B \<in> sets ?E" by auto }
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   593
  then have "sigma_sets (space ?E) (sets (pair_measure_generator (sigma E1) (sigma E2))) \<subseteq> sets ?E"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   594
    by (intro E.sigma_sets_subset) (auto simp add: pair_measure_generator_def sets_sigma)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   595
  then have subset: "sets ?S \<subseteq> sets ?E"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   596
    by (simp add: sets_sigma pair_measure_generator_def)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   597
  show "sets ?S = sets ?E"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   598
  proof (intro set_eqI iffI)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   599
    fix A assume "A \<in> sets ?E" then show "A \<in> sets ?S"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   600
      unfolding sets_sigma
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   601
    proof induct
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   602
      case (Basic A) then show ?case
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   603
        by (auto simp: pair_measure_generator_def sets_sigma intro: sigma_sets.Basic)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   604
    qed (auto intro: sigma_sets.intros simp: pair_measure_generator_def)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   605
  next
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   606
    fix A assume "A \<in> sets ?S" then show "A \<in> sets ?E" using subset by auto
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   607
  qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   608
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   609
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   610
section "Fubinis theorem"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   611
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   612
lemma (in pair_sigma_finite) simple_function_cut:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   613
  assumes f: "simple_function P f" "\<And>x. 0 \<le> f x"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   614
  shows "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   615
    and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   616
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   617
  have f_borel: "f \<in> borel_measurable P"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   618
    using f(1) by (rule borel_measurable_simple_function)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   619
  let "?F z" = "f -` {z} \<inter> space P"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   620
  let "?F' x z" = "Pair x -` ?F z"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   621
  { fix x assume "x \<in> space M1"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   622
    have [simp]: "\<And>z y. indicator (?F z) (x, y) = indicator (?F' x z) y"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   623
      by (auto simp: indicator_def)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   624
    have "\<And>y. y \<in> space M2 \<Longrightarrow> (x, y) \<in> space P" using `x \<in> space M1`
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   625
      by (simp add: space_pair_measure)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   626
    moreover have "\<And>x z. ?F' x z \<in> sets M2" using f_borel
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   627
      by (intro borel_measurable_vimage measurable_cut_fst)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   628
    ultimately have "simple_function M2 (\<lambda> y. f (x, y))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   629
      apply (rule_tac M2.simple_function_cong[THEN iffD2, OF _])
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   630
      apply (rule simple_function_indicator_representation[OF f(1)])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   631
      using `x \<in> space M1` by (auto simp del: space_sigma) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   632
  note M2_sf = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   633
  { fix x assume x: "x \<in> space M1"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   634
    then have "(\<integral>\<^isup>+y. f (x, y) \<partial>M2) = (\<Sum>z\<in>f ` space P. z * M2.\<mu> (?F' x z))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   635
      unfolding M2.positive_integral_eq_simple_integral[OF M2_sf[OF x] f(2)]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   636
      unfolding simple_integral_def
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   637
    proof (safe intro!: setsum_mono_zero_cong_left)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   638
      from f(1) show "finite (f ` space P)" by (rule simple_functionD)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   639
    next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   640
      fix y assume "y \<in> space M2" then show "f (x, y) \<in> f ` space P"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   641
        using `x \<in> space M1` by (auto simp: space_pair_measure)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   642
    next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   643
      fix x' y assume "(x', y) \<in> space P"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   644
        "f (x', y) \<notin> (\<lambda>y. f (x, y)) ` space M2"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   645
      then have *: "?F' x (f (x', y)) = {}"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   646
        by (force simp: space_pair_measure)
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   647
      show  "f (x', y) * M2.\<mu> (?F' x (f (x', y))) = 0"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   648
        unfolding * by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   649
    qed (simp add: vimage_compose[symmetric] comp_def
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   650
                   space_pair_measure) }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   651
  note eq = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   652
  moreover have "\<And>z. ?F z \<in> sets P"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   653
    by (auto intro!: f_borel borel_measurable_vimage simp del: space_sigma)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   654
  moreover then have "\<And>z. (\<lambda>x. M2.\<mu> (?F' x z)) \<in> borel_measurable M1"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   655
    by (auto intro!: measure_cut_measurable_fst simp del: vimage_Int)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   656
  moreover have *: "\<And>i x. 0 \<le> M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   657
    using f(1)[THEN simple_functionD(2)] f(2) by (intro M2.positive_measure measurable_cut_fst)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   658
  moreover { fix i assume "i \<in> f`space P"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   659
    with * have "\<And>x. 0 \<le> i * M2.\<mu> (Pair x -` (f -` {i} \<inter> space P))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   660
      using f(2) by auto }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   661
  ultimately
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   662
  show "(\<lambda>x. \<integral>\<^isup>+y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   663
    and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f" using f(2)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   664
    by (auto simp del: vimage_Int cong: measurable_cong
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   665
             intro!: M1.borel_measurable_extreal_setsum setsum_cong
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   666
             simp add: M1.positive_integral_setsum simple_integral_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   667
                       M1.positive_integral_cmult
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   668
                       M1.positive_integral_cong[OF eq]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   669
                       positive_integral_eq_simple_integral[OF f]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   670
                       pair_measure_alt[symmetric])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   671
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   672
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   673
lemma (in pair_sigma_finite) positive_integral_fst_measurable:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   674
  assumes f: "f \<in> borel_measurable P"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   675
  shows "(\<lambda>x. \<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   676
      (is "?C f \<in> borel_measurable M1")
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   677
    and "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   678
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   679
  from borel_measurable_implies_simple_function_sequence'[OF f] guess F . note F = this
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   680
  then have F_borel: "\<And>i. F i \<in> borel_measurable P"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   681
    by (auto intro: borel_measurable_simple_function)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   682
  note sf = simple_function_cut[OF F(1,5)]
41097
a1abfa4e2b44 use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents: 41096
diff changeset
   683
  then have "(\<lambda>x. SUP i. ?C (F i) x) \<in> borel_measurable M1"
a1abfa4e2b44 use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents: 41096
diff changeset
   684
    using F(1) by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   685
  moreover
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   686
  { fix x assume "x \<in> space M1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   687
    from F measurable_pair_image_snd[OF F_borel`x \<in> space M1`]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   688
    have "(\<integral>\<^isup>+y. (SUP i. F i (x, y)) \<partial>M2) = (SUP i. ?C (F i) x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   689
      by (intro M2.positive_integral_monotone_convergence_SUP)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   690
         (auto simp: incseq_Suc_iff le_fun_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   691
    then have "(SUP i. ?C (F i) x) = ?C f x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   692
      unfolding F(4) positive_integral_max_0 by simp }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   693
  note SUPR_C = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   694
  ultimately show "?C f \<in> borel_measurable M1"
41097
a1abfa4e2b44 use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents: 41096
diff changeset
   695
    by (simp cong: measurable_cong)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   696
  have "(\<integral>\<^isup>+x. (SUP i. F i x) \<partial>P) = (SUP i. integral\<^isup>P P (F i))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   697
    using F_borel F
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   698
    by (intro positive_integral_monotone_convergence_SUP) auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   699
  also have "(SUP i. integral\<^isup>P P (F i)) = (SUP i. \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   700
    unfolding sf(2) by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   701
  also have "\<dots> = \<integral>\<^isup>+ x. (SUP i. \<integral>\<^isup>+ y. F i (x, y) \<partial>M2) \<partial>M1" using F sf(1)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   702
    by (intro M1.positive_integral_monotone_convergence_SUP[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   703
       (auto intro!: M2.positive_integral_mono M2.positive_integral_positive
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   704
                simp: incseq_Suc_iff le_fun_def)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   705
  also have "\<dots> = \<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. (SUP i. F i (x, y)) \<partial>M2) \<partial>M1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   706
    using F_borel F(2,5)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   707
    by (auto intro!: M1.positive_integral_cong M2.positive_integral_monotone_convergence_SUP[symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   708
             simp: incseq_Suc_iff le_fun_def measurable_pair_image_snd)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   709
  finally show "(\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>P P f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   710
    using F by (simp add: positive_integral_max_0)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   711
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   712
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   713
lemma (in pair_sigma_finite) measure_preserving_swap:
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   714
  "(\<lambda>(x,y). (y, x)) \<in> measure_preserving (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   715
proof
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   716
  interpret Q: pair_sigma_finite M2 M1 by default
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   717
  show *: "(\<lambda>(x,y). (y, x)) \<in> measurable (M1 \<Otimes>\<^isub>M M2) (M2 \<Otimes>\<^isub>M M1)"
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   718
    using pair_sigma_algebra_swap_measurable .
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   719
  fix X assume "X \<in> sets (M2 \<Otimes>\<^isub>M M1)"
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   720
  from measurable_sets[OF * this] this Q.sets_into_space[OF this]
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   721
  show "measure (M1 \<Otimes>\<^isub>M M2) ((\<lambda>(x, y). (y, x)) -` X \<inter> space P) = measure (M2 \<Otimes>\<^isub>M M1) X"
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   722
    by (auto intro!: M1.positive_integral_cong arg_cong[where f="M2.\<mu>"]
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   723
      simp: pair_measure_alt Q.pair_measure_alt2 space_pair_measure)
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   724
qed
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   725
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   726
lemma (in pair_sigma_finite) positive_integral_product_swap:
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   727
  assumes f: "f \<in> borel_measurable P"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   728
  shows "(\<integral>\<^isup>+x. f (case x of (x,y)\<Rightarrow>(y,x)) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>P P f"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   729
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   730
  interpret Q: pair_sigma_finite M2 M1 by default
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   731
  have "sigma_algebra P" by default
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   732
  with f show ?thesis
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41706
diff changeset
   733
    by (subst Q.positive_integral_vimage[OF _ Q.measure_preserving_swap]) auto
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   734
qed
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   735
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   736
lemma (in pair_sigma_finite) positive_integral_snd_measurable:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   737
  assumes f: "f \<in> borel_measurable P"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   738
  shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>P P f"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   739
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   740
  interpret Q: pair_sigma_finite M2 M1 by default
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   741
  note pair_sigma_algebra_measurable[OF f]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   742
  from Q.positive_integral_fst_measurable[OF this]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   743
  have "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   744
    by simp
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   745
  also have "(\<integral>\<^isup>+ (x, y). f (y, x) \<partial>Q.P) = integral\<^isup>P P f"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   746
    unfolding positive_integral_product_swap[OF f, symmetric]
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   747
    by (auto intro!: Q.positive_integral_cong)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   748
  finally show ?thesis .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   749
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   750
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   751
lemma (in pair_sigma_finite) Fubini:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   752
  assumes f: "f \<in> borel_measurable P"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   753
  shows "(\<integral>\<^isup>+ y. (\<integral>\<^isup>+ x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>\<^isup>+ x. (\<integral>\<^isup>+ y. f (x, y) \<partial>M2) \<partial>M1)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   754
  unfolding positive_integral_snd_measurable[OF assms]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   755
  unfolding positive_integral_fst_measurable[OF assms] ..
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   756
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   757
lemma (in pair_sigma_finite) AE_pair:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   758
  assumes "AE x in P. Q x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   759
  shows "AE x in M1. (AE y in M2. Q (x, y))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   760
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   761
  obtain N where N: "N \<in> sets P" "\<mu> N = 0" "{x\<in>space P. \<not> Q x} \<subseteq> N"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   762
    using assms unfolding almost_everywhere_def by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   763
  show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   764
  proof (rule M1.AE_I)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   765
    from N measure_cut_measurable_fst[OF `N \<in> sets P`]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   766
    show "M1.\<mu> {x\<in>space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} = 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   767
      by (auto simp: pair_measure_alt M1.positive_integral_0_iff)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   768
    show "{x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0} \<in> sets M1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   769
      by (intro M1.borel_measurable_extreal_neq_const measure_cut_measurable_fst N)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   770
    { fix x assume "x \<in> space M1" "M2.\<mu> (Pair x -` N) = 0"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   771
      have "M2.almost_everywhere (\<lambda>y. Q (x, y))"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   772
      proof (rule M2.AE_I)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   773
        show "M2.\<mu> (Pair x -` N) = 0" by fact
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   774
        show "Pair x -` N \<in> sets M2" by (intro measurable_cut_fst N)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   775
        show "{y \<in> space M2. \<not> Q (x, y)} \<subseteq> Pair x -` N"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   776
          using N `x \<in> space M1` unfolding space_sigma space_pair_measure by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   777
      qed }
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   778
    then show "{x \<in> space M1. \<not> M2.almost_everywhere (\<lambda>y. Q (x, y))} \<subseteq> {x \<in> space M1. M2.\<mu> (Pair x -` N) \<noteq> 0}"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   779
      by auto
39088
ca17017c10e6 Measurable on product space is equiv. to measurable components
hoelzl
parents: 39082
diff changeset
   780
  qed
ca17017c10e6 Measurable on product space is equiv. to measurable components
hoelzl
parents: 39082
diff changeset
   781
qed
35833
7b7ae5aa396d Added product measure space
hoelzl
parents:
diff changeset
   782
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   783
lemma (in pair_sigma_algebra) measurable_product_swap:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   784
  "f \<in> measurable (M2 \<Otimes>\<^isub>M M1) M \<longleftrightarrow> (\<lambda>(x,y). f (y,x)) \<in> measurable P M"
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   785
proof -
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   786
  interpret Q: pair_sigma_algebra M2 M1 by default
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   787
  show ?thesis
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   788
    using pair_sigma_algebra_measurable[of "\<lambda>(x,y). f (y, x)"]
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   789
    by (auto intro!: pair_sigma_algebra_measurable Q.pair_sigma_algebra_measurable iffI)
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   790
qed
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   791
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   792
lemma (in pair_sigma_finite) integrable_product_swap:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   793
  assumes "integrable P f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   794
  shows "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x))"
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   795
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   796
  interpret Q: pair_sigma_finite M2 M1 by default
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   797
  have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   798
  show ?thesis unfolding *
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   799
    using assms unfolding integrable_def
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   800
    apply (subst (1 2) positive_integral_product_swap)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   801
    using `integrable P f` unfolding integrable_def
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   802
    by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   803
qed
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   804
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   805
lemma (in pair_sigma_finite) integrable_product_swap_iff:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   806
  "integrable (M2 \<Otimes>\<^isub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable P f"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   807
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   808
  interpret Q: pair_sigma_finite M2 M1 by default
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   809
  from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   810
  show ?thesis by auto
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   811
qed
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   812
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   813
lemma (in pair_sigma_finite) integral_product_swap:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   814
  assumes "integrable P f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   815
  shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^isub>M M1)) = integral\<^isup>L P f"
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   816
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   817
  interpret Q: pair_sigma_finite M2 M1 by default
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   818
  have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   819
  show ?thesis
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   820
    unfolding lebesgue_integral_def *
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   821
    apply (subst (1 2) positive_integral_product_swap)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   822
    using `integrable P f` unfolding integrable_def
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   823
    by (auto simp: *[symmetric] Q.measurable_product_swap[symmetric])
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   824
qed
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   825
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   826
lemma (in pair_sigma_finite) integrable_fst_measurable:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   827
  assumes f: "integrable P f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   828
  shows "M1.almost_everywhere (\<lambda>x. integrable M2 (\<lambda> y. f (x, y)))" (is "?AE")
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   829
    and "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^isup>L P f" (is "?INT")
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   830
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   831
  let "?pf x" = "extreal (f x)" and "?nf x" = "extreal (- f x)"
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   832
  have
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   833
    borel: "?nf \<in> borel_measurable P""?pf \<in> borel_measurable P" and
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   834
    int: "integral\<^isup>P P ?nf \<noteq> \<infinity>" "integral\<^isup>P P ?pf \<noteq> \<infinity>"
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   835
    using assms by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   836
  have "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   837
     "(\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<partial>M1) \<noteq> \<infinity>"
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   838
    using borel[THEN positive_integral_fst_measurable(1)] int
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   839
    unfolding borel[THEN positive_integral_fst_measurable(2)] by simp_all
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   840
  with borel[THEN positive_integral_fst_measurable(1)]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   841
  have AE_pos: "AE x in M1. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   842
    "AE x in M1. (\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2) \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   843
    by (auto intro!: M1.positive_integral_PInf_AE )
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   844
  then have AE: "AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   845
    "AE x in M1. \<bar>\<integral>\<^isup>+y. extreal (- f (x, y)) \<partial>M2\<bar> \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   846
    by (auto simp: M2.positive_integral_positive)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   847
  from AE_pos show ?AE using assms
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   848
    by (simp add: measurable_pair_image_snd integrable_def)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   849
  { fix f have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   850
      using M2.positive_integral_positive
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   851
      by (intro M1.positive_integral_cong_pos) (auto simp: extreal_uminus_le_reorder)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   852
    then have "(\<integral>\<^isup>+ x. - \<integral>\<^isup>+ y. extreal (f x y) \<partial>M2 \<partial>M1) = 0" by simp }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   853
  note this[simp]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   854
  { fix f assume borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable P"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   855
      and int: "integral\<^isup>P P (\<lambda>x. extreal (f x)) \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   856
      and AE: "M1.almost_everywhere (\<lambda>x. (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2) \<noteq> \<infinity>)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   857
    have "integrable M1 (\<lambda>x. real (\<integral>\<^isup>+y. extreal (f (x, y)) \<partial>M2))" (is "integrable M1 ?f")
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   858
    proof (intro integrable_def[THEN iffD2] conjI)
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   859
      show "?f \<in> borel_measurable M1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   860
        using borel by (auto intro!: M1.borel_measurable_real_of_extreal positive_integral_fst_measurable)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   861
      have "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) = (\<integral>\<^isup>+x. (\<integral>\<^isup>+y. extreal (f (x, y))  \<partial>M2) \<partial>M1)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   862
        using AE M2.positive_integral_positive
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   863
        by (auto intro!: M1.positive_integral_cong_AE simp: extreal_real)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   864
      then show "(\<integral>\<^isup>+x. extreal (?f x) \<partial>M1) \<noteq> \<infinity>"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   865
        using positive_integral_fst_measurable[OF borel] int by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   866
      have "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) = (\<integral>\<^isup>+x. 0 \<partial>M1)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   867
        by (intro M1.positive_integral_cong_pos)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   868
           (simp add: M2.positive_integral_positive real_of_extreal_pos)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   869
      then show "(\<integral>\<^isup>+x. extreal (- ?f x) \<partial>M1) \<noteq> \<infinity>" by simp
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   870
    qed }
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   871
  with this[OF borel(1) int(1) AE_pos(2)] this[OF borel(2) int(2) AE_pos(1)]
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   872
  show ?INT
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   873
    unfolding lebesgue_integral_def[of P] lebesgue_integral_def[of M2]
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   874
      borel[THEN positive_integral_fst_measurable(2), symmetric]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   875
    using AE[THEN M1.integral_real]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   876
    by simp
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   877
qed
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   878
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   879
lemma (in pair_sigma_finite) integrable_snd_measurable:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   880
  assumes f: "integrable P f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   881
  shows "M2.almost_everywhere (\<lambda>y. integrable M1 (\<lambda>x. f (x, y)))" (is "?AE")
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   882
    and "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = integral\<^isup>L P f" (is "?INT")
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   883
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   884
  interpret Q: pair_sigma_finite M2 M1 by default
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   885
  have Q_int: "integrable Q.P (\<lambda>(x, y). f (y, x))"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   886
    using f unfolding integrable_product_swap_iff .
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   887
  show ?INT
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   888
    using Q.integrable_fst_measurable(2)[OF Q_int]
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41659
diff changeset
   889
    using integral_product_swap[OF f] by simp
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   890
  show ?AE
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   891
    using Q.integrable_fst_measurable(1)[OF Q_int]
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   892
    by simp
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   893
qed
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   894
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   895
lemma (in pair_sigma_finite) Fubini_integral:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   896
  assumes f: "integrable P f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   897
  shows "(\<integral>y. (\<integral>x. f (x, y) \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1)"
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   898
  unfolding integrable_snd_measurable[OF assms]
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   899
  unfolding integrable_fst_measurable[OF assms] ..
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
   900
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   901
section "Products on finite spaces"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   902
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   903
lemma sigma_sets_pair_measure_generator_finite:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   904
  assumes "finite A" and "finite B"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   905
  shows "sigma_sets (A \<times> B) { a \<times> b | a b. a \<in> Pow A \<and> b \<in> Pow B} = Pow (A \<times> B)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   906
  (is "sigma_sets ?prod ?sets = _")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   907
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   908
  have fin: "finite (A \<times> B)" using assms by (rule finite_cartesian_product)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   909
  fix x assume subset: "x \<subseteq> A \<times> B"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   910
  hence "finite x" using fin by (rule finite_subset)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   911
  from this subset show "x \<in> sigma_sets ?prod ?sets"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   912
  proof (induct x)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   913
    case empty show ?case by (rule sigma_sets.Empty)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   914
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   915
    case (insert a x)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   916
    hence "{a} \<in> sigma_sets ?prod ?sets"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   917
      by (auto simp: pair_measure_generator_def intro!: sigma_sets.Basic)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   918
    moreover have "x \<in> sigma_sets ?prod ?sets" using insert by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   919
    ultimately show ?case unfolding insert_is_Un[of a x] by (rule sigma_sets_Un)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   920
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   921
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   922
  fix x a b
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   923
  assume "x \<in> sigma_sets ?prod ?sets" and "(a, b) \<in> x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   924
  from sigma_sets_into_sp[OF _ this(1)] this(2)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   925
  show "a \<in> A" and "b \<in> B" by auto
35833
7b7ae5aa396d Added product measure space
hoelzl
parents:
diff changeset
   926
qed
7b7ae5aa396d Added product measure space
hoelzl
parents:
diff changeset
   927
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   928
locale pair_finite_sigma_algebra = M1: finite_sigma_algebra M1 + M2: finite_sigma_algebra M2
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   929
  for M1 :: "('a, 'c) measure_space_scheme" and M2 :: "('b, 'd) measure_space_scheme"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   930
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   931
sublocale pair_finite_sigma_algebra \<subseteq> pair_sigma_algebra by default
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   932
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   933
lemma (in pair_finite_sigma_algebra) finite_pair_sigma_algebra:
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   934
  shows "P = \<lparr> space = space M1 \<times> space M2, sets = Pow (space M1 \<times> space M2), \<dots> = algebra.more P \<rparr>"
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35833
diff changeset
   935
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   936
  show ?thesis
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   937
    using sigma_sets_pair_measure_generator_finite[OF M1.finite_space M2.finite_space]
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   938
    by (intro algebra.equality) (simp_all add: pair_measure_def pair_measure_generator_def sigma_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   939
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   940
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   941
sublocale pair_finite_sigma_algebra \<subseteq> finite_sigma_algebra P
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   942
  apply default
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   943
  using M1.finite_space M2.finite_space
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   944
  apply (subst finite_pair_sigma_algebra) apply simp
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   945
  apply (subst (1 2) finite_pair_sigma_algebra) apply simp
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   946
  done
35833
7b7ae5aa396d Added product measure space
hoelzl
parents:
diff changeset
   947
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   948
locale pair_finite_space = M1: finite_measure_space M1 + M2: finite_measure_space M2
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   949
  for M1 M2
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   950
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   951
sublocale pair_finite_space \<subseteq> pair_finite_sigma_algebra
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   952
  by default
35833
7b7ae5aa396d Added product measure space
hoelzl
parents:
diff changeset
   953
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   954
sublocale pair_finite_space \<subseteq> pair_sigma_finite
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   955
  by default
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   956
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   957
lemma (in pair_finite_space) pair_measure_Pair[simp]:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   958
  assumes "a \<in> space M1" "b \<in> space M2"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   959
  shows "\<mu> {(a, b)} = M1.\<mu> {a} * M2.\<mu> {b}"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   960
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   961
  have "\<mu> ({a}\<times>{b}) = M1.\<mu> {a} * M2.\<mu> {b}"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   962
    using M1.sets_eq_Pow M2.sets_eq_Pow assms
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   963
    by (subst pair_measure_times) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   964
  then show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   965
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   966
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   967
lemma (in pair_finite_space) pair_measure_singleton[simp]:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   968
  assumes "x \<in> space M1 \<times> space M2"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   969
  shows "\<mu> {x} = M1.\<mu> {fst x} * M2.\<mu> {snd x}"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   970
  using pair_measure_Pair assms by (cases x) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36649
diff changeset
   971
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   972
sublocale pair_finite_space \<subseteq> finite_measure_space P
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   973
  by default (auto simp: space_pair_measure)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   974
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39098
diff changeset
   975
end