src/HOL/Probability/Borel_Space.thy
author hoelzl
Tue, 22 Mar 2011 20:06:10 +0100
changeset 42067 66c8281349ec
parent 41981 cdf7693bbe08
child 42150 b0c0638c4aad
permissions -rw-r--r--
standardized headers
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Probability/Lebesgue_Integration.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Armin Heller, TU München
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*)
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header {*Borel spaces*}
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de0b30e6c2d2 Support product spaces on sigma finite measures.
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theory Borel_Space
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cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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  imports Sigma_Algebra Multivariate_Analysis
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begin
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section "Generic Borel spaces"
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40859
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definition "borel = sigma \<lparr> space = UNIV::'a::topological_space set, sets = open\<rparr>"
de0b30e6c2d2 Support product spaces on sigma finite measures.
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abbreviation "borel_measurable M \<equiv> measurable M borel"
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interpretation borel: sigma_algebra borel
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  by (auto simp: borel_def intro!: sigma_algebra_sigma)
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40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
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lemma in_borel_measurable:
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   "f \<in> borel_measurable M \<longleftrightarrow>
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    (\<forall>S \<in> sets (sigma \<lparr> space = UNIV, sets = open\<rparr>).
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      f -` S \<inter> space M \<in> sets M)"
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  by (auto simp add: measurable_def borel_def)
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lemma in_borel_measurable_borel:
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   "f \<in> borel_measurable M \<longleftrightarrow>
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    (\<forall>S \<in> sets borel.
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      f -` S \<inter> space M \<in> sets M)"
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  by (auto simp add: measurable_def borel_def)
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parents:
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lemma space_borel[simp]: "space borel = UNIV"
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  unfolding borel_def by auto
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lemma borel_open[simp]:
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  assumes "open A" shows "A \<in> sets borel"
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proof -
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  have "A \<in> open" unfolding mem_def using assms .
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    39
  thus ?thesis unfolding borel_def sigma_def by (auto intro!: sigma_sets.Basic)
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qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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40859
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lemma borel_closed[simp]:
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  assumes "closed A" shows "A \<in> sets borel"
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parents:
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proof -
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    45
  have "space borel - (- A) \<in> sets borel"
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    using assms unfolding closed_def by (blast intro: borel_open)
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  thus ?thesis by simp
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parents:
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qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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41830
719b0a517c33 log is borel measurable
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lemma borel_comp[intro,simp]: "A \<in> sets borel \<Longrightarrow> - A \<in> sets borel"
719b0a517c33 log is borel measurable
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    51
  unfolding Compl_eq_Diff_UNIV by (intro borel.Diff) auto
719b0a517c33 log is borel measurable
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    52
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lemma (in sigma_algebra) borel_measurable_vimage:
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  fixes f :: "'a \<Rightarrow> 'x::t2_space"
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    55
  assumes borel: "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
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    56
  shows "f -` {x} \<inter> space M \<in> sets M"
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proof (cases "x \<in> f ` space M")
d5d342611edb Rewrite the Probability theory.
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    58
  case True then obtain y where "x = f y" by auto
41969
1cf3e4107a2a moved t2_spaces to HOL image
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    59
  from closed_singleton[of "f y"]
40859
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    60
  have "{f y} \<in> sets borel" by (rule borel_closed)
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    61
  with assms show ?thesis
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
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    62
    unfolding in_borel_measurable_borel `x = f y` by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
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    63
next
d5d342611edb Rewrite the Probability theory.
hoelzl
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    64
  case False hence "f -` {x} \<inter> space M = {}" by auto
d5d342611edb Rewrite the Probability theory.
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    65
  thus ?thesis by auto
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40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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    66
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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    67
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lemma (in sigma_algebra) borel_measurableI:
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    69
  fixes f :: "'a \<Rightarrow> 'x\<Colon>topological_space"
d5d342611edb Rewrite the Probability theory.
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    70
  assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
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    71
  shows "f \<in> borel_measurable M"
40859
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    72
  unfolding borel_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
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    73
proof (rule measurable_sigma, simp_all)
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d5d342611edb Rewrite the Probability theory.
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    74
  fix S :: "'x set" assume "S \<in> open" thus "f -` S \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
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    75
    using assms[of S] by (simp add: mem_def)
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    76
qed
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40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
    77
40859
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lemma borel_singleton[simp, intro]:
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    79
  fixes x :: "'a::t1_space"
40859
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    80
  shows "A \<in> sets borel \<Longrightarrow> insert x A \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
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    81
  proof (rule borel.insert_in_sets)
de0b30e6c2d2 Support product spaces on sigma finite measures.
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    82
    show "{x} \<in> sets borel"
41969
1cf3e4107a2a moved t2_spaces to HOL image
hoelzl
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    83
      using closed_singleton[of x] by (rule borel_closed)
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    84
  qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
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    85
d5d342611edb Rewrite the Probability theory.
hoelzl
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    86
lemma (in sigma_algebra) borel_measurable_const[simp, intro]:
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    87
  "(\<lambda>x. c) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
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    88
  by (auto intro!: measurable_const)
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40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
    89
39083
e46acc0ea1fe introduced integration on subalgebras
hoelzl
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    90
lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]:
38656
d5d342611edb Rewrite the Probability theory.
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    91
  assumes A: "A \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
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    92
  shows "indicator A \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
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diff changeset
    93
  unfolding indicator_def_raw using A
d5d342611edb Rewrite the Probability theory.
hoelzl
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    94
  by (auto intro!: measurable_If_set borel_measurable_const)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
    95
40859
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hoelzl
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    96
lemma (in sigma_algebra) borel_measurable_indicator_iff:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
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    97
  "(indicator A :: 'a \<Rightarrow> 'x::{t1_space, zero_neq_one}) \<in> borel_measurable M \<longleftrightarrow> A \<inter> space M \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
    98
    (is "?I \<in> borel_measurable M \<longleftrightarrow> _")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
    99
proof
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
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diff changeset
   100
  assume "?I \<in> borel_measurable M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   101
  then have "?I -` {1} \<inter> space M \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
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diff changeset
   102
    unfolding measurable_def by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   103
  also have "?I -` {1} \<inter> space M = A \<inter> space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   104
    unfolding indicator_def_raw by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   105
  finally show "A \<inter> space M \<in> sets M" .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
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diff changeset
   106
next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   107
  assume "A \<inter> space M \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   108
  moreover have "?I \<in> borel_measurable M \<longleftrightarrow>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
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   109
    (indicator (A \<inter> space M) :: 'a \<Rightarrow> 'x) \<in> borel_measurable M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   110
    by (intro measurable_cong) (auto simp: indicator_def)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   111
  ultimately show "?I \<in> borel_measurable M" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   112
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   113
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   114
lemma (in sigma_algebra) borel_measurable_restricted:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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diff changeset
   115
  fixes f :: "'a \<Rightarrow> extreal" assumes "A \<in> sets M"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   116
  shows "f \<in> borel_measurable (restricted_space A) \<longleftrightarrow>
98de40859858 move lemmas to correct theory files
hoelzl
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diff changeset
   117
    (\<lambda>x. f x * indicator A x) \<in> borel_measurable M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   118
    (is "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable M")
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   119
proof -
98de40859858 move lemmas to correct theory files
hoelzl
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diff changeset
   120
  interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
98de40859858 move lemmas to correct theory files
hoelzl
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diff changeset
   121
  have *: "f \<in> borel_measurable ?R \<longleftrightarrow> ?f \<in> borel_measurable ?R"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   122
    by (auto intro!: measurable_cong)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   123
  show ?thesis unfolding *
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
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diff changeset
   124
    unfolding in_borel_measurable_borel
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   125
  proof (simp, safe)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   126
    fix S :: "extreal set" assume "S \<in> sets borel"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   127
      "\<forall>S\<in>sets borel. ?f -` S \<inter> A \<in> op \<inter> A ` sets M"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   128
    then have "?f -` S \<inter> A \<in> op \<inter> A ` sets M" by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   129
    then have f: "?f -` S \<inter> A \<in> sets M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   130
      using `A \<in> sets M` sets_into_space by fastsimp
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   131
    show "?f -` S \<inter> space M \<in> sets M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   132
    proof cases
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   133
      assume "0 \<in> S"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   134
      then have "?f -` S \<inter> space M = ?f -` S \<inter> A \<union> (space M - A)"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   135
        using `A \<in> sets M` sets_into_space by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   136
      then show ?thesis using f `A \<in> sets M` by (auto intro!: Un Diff)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   137
    next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   138
      assume "0 \<notin> S"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   139
      then have "?f -` S \<inter> space M = ?f -` S \<inter> A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   140
        using `A \<in> sets M` sets_into_space
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   141
        by (auto simp: indicator_def split: split_if_asm)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   142
      then show ?thesis using f by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   143
    qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   144
  next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
   145
    fix S :: "extreal set" assume "S \<in> sets borel"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   146
      "\<forall>S\<in>sets borel. ?f -` S \<inter> space M \<in> sets M"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   147
    then have f: "?f -` S \<inter> space M \<in> sets M" by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   148
    then show "?f -` S \<inter> A \<in> op \<inter> A ` sets M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   149
      using `A \<in> sets M` sets_into_space
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   150
      apply (simp add: image_iff)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   151
      apply (rule bexI[OF _ f])
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   152
      by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   153
  qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   154
qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   155
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   156
lemma (in sigma_algebra) borel_measurable_subalgebra:
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41097
diff changeset
   157
  assumes "sets N \<subseteq> sets M" "space N = space M" "f \<in> borel_measurable N"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   158
  shows "f \<in> borel_measurable M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   159
  using assms unfolding measurable_def by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39087
diff changeset
   160
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   161
section "Borel spaces on euclidean spaces"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   162
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   163
lemma lessThan_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   164
  fixes a :: "'a\<Colon>ordered_euclidean_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   165
  shows "{..< a} \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   166
  by (blast intro: borel_open)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   167
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   168
lemma greaterThan_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   169
  fixes a :: "'a\<Colon>ordered_euclidean_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   170
  shows "{a <..} \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   171
  by (blast intro: borel_open)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   172
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   173
lemma greaterThanLessThan_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   174
  fixes a b :: "'a\<Colon>ordered_euclidean_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   175
  shows "{a<..<b} \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   176
  by (blast intro: borel_open)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   177
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   178
lemma atMost_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   179
  fixes a :: "'a\<Colon>ordered_euclidean_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   180
  shows "{..a} \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   181
  by (blast intro: borel_closed)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   182
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   183
lemma atLeast_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   184
  fixes a :: "'a\<Colon>ordered_euclidean_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   185
  shows "{a..} \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   186
  by (blast intro: borel_closed)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   187
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   188
lemma atLeastAtMost_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   189
  fixes a b :: "'a\<Colon>ordered_euclidean_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   190
  shows "{a..b} \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   191
  by (blast intro: borel_closed)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   192
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   193
lemma greaterThanAtMost_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   194
  fixes a b :: "'a\<Colon>ordered_euclidean_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   195
  shows "{a<..b} \<in> sets borel"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   196
  unfolding greaterThanAtMost_def by blast
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   197
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   198
lemma atLeastLessThan_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   199
  fixes a b :: "'a\<Colon>ordered_euclidean_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   200
  shows "{a..<b} \<in> sets borel"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   201
  unfolding atLeastLessThan_def by blast
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   202
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   203
lemma hafspace_less_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   204
  fixes a :: real
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   205
  shows "{x::'a::euclidean_space. a < x $$ i} \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   206
  by (auto intro!: borel_open open_halfspace_component_gt)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   207
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   208
lemma hafspace_greater_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   209
  fixes a :: real
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   210
  shows "{x::'a::euclidean_space. x $$ i < a} \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   211
  by (auto intro!: borel_open open_halfspace_component_lt)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   212
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   213
lemma hafspace_less_eq_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   214
  fixes a :: real
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   215
  shows "{x::'a::euclidean_space. a \<le> x $$ i} \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   216
  by (auto intro!: borel_closed closed_halfspace_component_ge)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   217
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   218
lemma hafspace_greater_eq_borel[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   219
  fixes a :: real
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   220
  shows "{x::'a::euclidean_space. x $$ i \<le> a} \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   221
  by (auto intro!: borel_closed closed_halfspace_component_le)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   222
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   223
lemma (in sigma_algebra) borel_measurable_less[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   224
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   225
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   226
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   227
  shows "{w \<in> space M. f w < g w} \<in> sets M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   228
proof -
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   229
  have "{w \<in> space M. f w < g w} =
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   230
        (\<Union>r. (f -` {..< of_rat r} \<inter> space M) \<inter> (g -` {of_rat r <..} \<inter> space M))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   231
    using Rats_dense_in_real by (auto simp add: Rats_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   232
  then show ?thesis using f g
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   233
    by simp (blast intro: measurable_sets)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   234
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   235
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   236
lemma (in sigma_algebra) borel_measurable_le[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   237
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   238
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   239
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   240
  shows "{w \<in> space M. f w \<le> g w} \<in> sets M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   241
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   242
  have "{w \<in> space M. f w \<le> g w} = space M - {w \<in> space M. g w < f w}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   243
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   244
  thus ?thesis using f g
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   245
    by simp blast
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   246
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   247
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   248
lemma (in sigma_algebra) borel_measurable_eq[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   249
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   250
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   251
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   252
  shows "{w \<in> space M. f w = g w} \<in> sets M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   253
proof -
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   254
  have "{w \<in> space M. f w = g w} =
33536
fd28b7399f2b eliminated hard tabulators;
wenzelm
parents: 33535
diff changeset
   255
        {w \<in> space M. f w \<le> g w} \<inter> {w \<in> space M. g w \<le> f w}"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   256
    by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   257
  thus ?thesis using f g by auto
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   258
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   259
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   260
lemma (in sigma_algebra) borel_measurable_neq[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   261
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   262
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   263
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   264
  shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   265
proof -
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   266
  have "{w \<in> space M. f w \<noteq> g w} = space M - {w \<in> space M. f w = g w}"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   267
    by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   268
  thus ?thesis using f g by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   269
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   270
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   271
subsection "Borel space equals sigma algebras over intervals"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   272
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   273
lemma rational_boxes:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   274
  fixes x :: "'a\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   275
  assumes "0 < e"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   276
  shows "\<exists>a b. (\<forall>i. a $$ i \<in> \<rat>) \<and> (\<forall>i. b $$ i \<in> \<rat>) \<and> x \<in> {a <..< b} \<and> {a <..< b} \<subseteq> ball x e"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   277
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   278
  def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   279
  then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   280
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x $$ i \<and> x $$ i - y < e'" (is "\<forall>i. ?th i")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   281
  proof
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   282
    fix i from Rats_dense_in_real[of "x $$ i - e'" "x $$ i"] e
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   283
    show "?th i" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   284
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   285
  from choice[OF this] guess a .. note a = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   286
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x $$ i < y \<and> y - x $$ i < e'" (is "\<forall>i. ?th i")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   287
  proof
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   288
    fix i from Rats_dense_in_real[of "x $$ i" "x $$ i + e'"] e
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   289
    show "?th i" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   290
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   291
  from choice[OF this] guess b .. note b = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   292
  { fix y :: 'a assume *: "Chi a < y" "y < Chi b"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   293
    have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x $$ i) (y $$ i))\<twosuperior>)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   294
      unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   295
    also have "\<dots> < sqrt (\<Sum>i<DIM('a). e^2 / real (DIM('a)))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   296
    proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   297
      fix i assume i: "i \<in> {..<DIM('a)}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   298
      have "a i < y$$i \<and> y$$i < b i" using * i eucl_less[where 'a='a] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   299
      moreover have "a i < x$$i" "x$$i - a i < e'" using a by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   300
      moreover have "x$$i < b i" "b i - x$$i < e'" using b by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   301
      ultimately have "\<bar>x$$i - y$$i\<bar> < 2 * e'" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   302
      then have "dist (x $$ i) (y $$ i) < e/sqrt (real (DIM('a)))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   303
        unfolding e'_def by (auto simp: dist_real_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   304
      then have "(dist (x $$ i) (y $$ i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   305
        by (rule power_strict_mono) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   306
      then show "(dist (x $$ i) (y $$ i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   307
        by (simp add: power_divide)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   308
    qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   309
    also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat DIM_positive)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   310
    finally have "dist x y < e" . }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   311
  with a b show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   312
    apply (rule_tac exI[of _ "Chi a"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   313
    apply (rule_tac exI[of _ "Chi b"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   314
    using eucl_less[where 'a='a] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   315
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   316
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   317
lemma ex_rat_list:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   318
  fixes x :: "'a\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   319
  assumes "\<And> i. x $$ i \<in> \<rat>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   320
  shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x $$ i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   321
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   322
  have "\<forall>i. \<exists>r. x $$ i = of_rat r" using assms unfolding Rats_def by blast
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   323
  from choice[OF this] guess r ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   324
  then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   325
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   326
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   327
lemma open_UNION:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   328
  fixes M :: "'a\<Colon>ordered_euclidean_space set"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   329
  assumes "open M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   330
  shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   331
                   (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   332
    (is "M = UNION ?idx ?box")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   333
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   334
  fix x assume "x \<in> M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   335
  obtain e where e: "e > 0" "ball x e \<subseteq> M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   336
    using openE[OF assms `x \<in> M`] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   337
  then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a $$ i \<in> \<rat>" "\<And>i. b $$ i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   338
    using rational_boxes[OF e(1)] by blast
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   339
  then obtain p q where pq: "length p = DIM ('a)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   340
                            "length q = DIM ('a)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   341
                            "\<forall> i < DIM ('a). of_rat (p ! i) = a $$ i \<and> of_rat (q ! i) = b $$ i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   342
    using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   343
  hence p: "Chi (of_rat \<circ> op ! p) = a"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   344
    using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   345
    unfolding o_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   346
  from pq have q: "Chi (of_rat \<circ> op ! q) = b"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   347
    using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   348
    unfolding o_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   349
  have "x \<in> ?box (p, q)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   350
    using p q ab by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   351
  thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   352
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   353
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   354
lemma halfspace_span_open:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   355
  "sigma_sets UNIV (range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a}))
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   356
    \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   357
  by (auto intro!: borel.sigma_sets_subset[simplified] borel_open
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   358
                   open_halfspace_component_lt)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   359
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   360
lemma halfspace_lt_in_halfspace:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   361
  "{x\<Colon>'a. x $$ i < a} \<in> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   362
  by (auto intro!: sigma_sets.Basic simp: sets_sigma)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   363
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   364
lemma halfspace_gt_in_halfspace:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   365
  "{x\<Colon>'a. a < x $$ i} \<in> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   366
  (is "?set \<in> sets ?SIGMA")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   367
proof -
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   368
  interpret sigma_algebra "?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   369
    by (intro sigma_algebra_sigma_sets) (simp_all add: sets_sigma)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   370
  have *: "?set = (\<Union>n. space ?SIGMA - {x\<Colon>'a. x $$ i < a + 1 / real (Suc n)})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   371
  proof (safe, simp_all add: not_less)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   372
    fix x assume "a < x $$ i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   373
    with reals_Archimedean[of "x $$ i - a"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   374
    obtain n where "a + 1 / real (Suc n) < x $$ i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   375
      by (auto simp: inverse_eq_divide field_simps)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   376
    then show "\<exists>n. a + 1 / real (Suc n) \<le> x $$ i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   377
      by (blast intro: less_imp_le)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   378
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   379
    fix x n
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   380
    have "a < a + 1 / real (Suc n)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   381
    also assume "\<dots> \<le> x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   382
    finally show "a < x" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   383
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   384
  show "?set \<in> sets ?SIGMA" unfolding *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   385
    by (safe intro!: countable_UN Diff halfspace_lt_in_halfspace)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   386
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   387
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   388
lemma open_span_halfspace:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   389
  "sets borel \<subseteq> sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x $$ i < a})\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   390
    (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   391
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   392
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   393
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   394
  { fix S :: "'a set" assume "S \<in> open" then have "open S" unfolding mem_def .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   395
    from open_UNION[OF this]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   396
    obtain I where *: "S =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   397
      (\<Union>(a, b)\<in>I.
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   398
          (\<Inter> i<DIM('a). {x. (Chi (real_of_rat \<circ> op ! a)::'a) $$ i < x $$ i}) \<inter>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   399
          (\<Inter> i<DIM('a). {x. x $$ i < (Chi (real_of_rat \<circ> op ! b)::'a) $$ i}))"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   400
      unfolding greaterThanLessThan_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   401
      unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   402
      unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   403
      by blast
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   404
    have "S \<in> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   405
      unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   406
      by (auto intro!: countable_UN Int countable_INT halfspace_lt_in_halfspace halfspace_gt_in_halfspace) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   407
  then show ?thesis unfolding borel_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   408
    by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   409
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   410
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   411
lemma halfspace_span_halfspace_le:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   412
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   413
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. x $$ i \<le> a})\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   414
  (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   415
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   416
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   417
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   418
  { fix a i
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   419
    have *: "{x::'a. x$$i < a} = (\<Union>n. {x. x$$i \<le> a - 1/real (Suc n)})"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   420
    proof (safe, simp_all)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   421
      fix x::'a assume *: "x$$i < a"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   422
      with reals_Archimedean[of "a - x$$i"]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   423
      obtain n where "x $$ i < a - 1 / (real (Suc n))"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   424
        by (auto simp: field_simps inverse_eq_divide)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   425
      then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   426
        by (blast intro: less_imp_le)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   427
    next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   428
      fix x::'a and n
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   429
      assume "x$$i \<le> a - 1 / real (Suc n)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   430
      also have "\<dots> < a" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   431
      finally show "x$$i < a" .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   432
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   433
    have "{x. x$$i < a} \<in> sets ?SIGMA" unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   434
      by (safe intro!: countable_UN)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   435
         (auto simp: sets_sigma intro!: sigma_sets.Basic) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   436
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   437
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   438
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   439
lemma halfspace_span_halfspace_ge:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   440
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   441
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a \<le> x $$ i})\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   442
  (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   443
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   444
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   445
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   446
  { fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   447
    have "{x. x$$i < a} \<in> sets ?SIGMA" unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   448
      by (safe intro!: Diff)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   449
         (auto simp: sets_sigma intro!: sigma_sets.Basic) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   450
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   451
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   452
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   453
lemma halfspace_le_span_halfspace_gt:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   454
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   455
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a < x $$ i})\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   456
  (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   457
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   458
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   459
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   460
  { fix a i have *: "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   461
    have "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   462
      by (safe intro!: Diff)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   463
         (auto simp: sets_sigma intro!: sigma_sets.Basic) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   464
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   465
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   466
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   467
lemma halfspace_le_span_atMost:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   468
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   469
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   470
  (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   471
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   472
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   473
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   474
  have "\<And>a i. {x. x$$i \<le> a} \<in> sets ?SIGMA"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   475
  proof cases
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   476
    fix a i assume "i < DIM('a)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   477
    then have *: "{x::'a. x$$i \<le> a} = (\<Union>k::nat. {.. (\<chi>\<chi> n. if n = i then a else real k)})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   478
    proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   479
      fix x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   480
      from real_arch_simple[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] guess k::nat ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   481
      then have "\<And>i. i < DIM('a) \<Longrightarrow> x$$i \<le> real k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   482
        by (subst (asm) Max_le_iff) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   483
      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia \<le> real k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   484
        by (auto intro!: exI[of _ k])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   485
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   486
    show "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   487
      by (safe intro!: countable_UN)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   488
         (auto simp: sets_sigma intro!: sigma_sets.Basic)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   489
  next
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   490
    fix a i assume "\<not> i < DIM('a)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   491
    then show "{x. x$$i \<le> a} \<in> sets ?SIGMA"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   492
      using top by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   493
  qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   494
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   495
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   496
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   497
lemma halfspace_le_span_greaterThan:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   498
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   499
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {a<..})\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   500
  (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   501
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   502
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   503
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   504
  have "\<And>a i. {x. x$$i \<le> a} \<in> sets ?SIGMA"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   505
  proof cases
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   506
    fix a i assume "i < DIM('a)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   507
    have "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   508
    also have *: "{x::'a. a < x$$i} = (\<Union>k::nat. {(\<chi>\<chi> n. if n = i then a else -real k) <..})" using `i <DIM('a)`
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   509
    proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   510
      fix x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   511
      from real_arch_lt[of "Max ((\<lambda>i. -x$$i)`{..<DIM('a)})"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   512
      guess k::nat .. note k = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   513
      { fix i assume "i < DIM('a)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   514
        then have "-x$$i < real k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   515
          using k by (subst (asm) Max_less_iff) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   516
        then have "- real k < x$$i" by simp }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   517
      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x $$ ia"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   518
        by (auto intro!: exI[of _ k])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   519
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   520
    finally show "{x. x$$i \<le> a} \<in> sets ?SIGMA"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   521
      apply (simp only:)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   522
      apply (safe intro!: countable_UN Diff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   523
      by (auto simp: sets_sigma intro!: sigma_sets.Basic)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   524
  next
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   525
    fix a i assume "\<not> i < DIM('a)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   526
    then show "{x. x$$i \<le> a} \<in> sets ?SIGMA"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   527
      using top by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   528
  qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   529
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   530
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   531
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   532
lemma halfspace_le_span_lessThan:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   533
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x $$ i})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   534
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..<a})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   535
  (is "_ \<subseteq> sets ?SIGMA")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   536
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   537
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   538
  then interpret sigma_algebra ?SIGMA .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   539
  have "\<And>a i. {x. a \<le> x$$i} \<in> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   540
  proof cases
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   541
    fix a i assume "i < DIM('a)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   542
    have "{x::'a. a \<le> x$$i} = space ?SIGMA - {x::'a. x$$i < a}" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   543
    also have *: "{x::'a. x$$i < a} = (\<Union>k::nat. {..< (\<chi>\<chi> n. if n = i then a else real k)})" using `i <DIM('a)`
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   544
    proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   545
      fix x
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   546
      from real_arch_lt[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   547
      guess k::nat .. note k = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   548
      { fix i assume "i < DIM('a)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   549
        then have "x$$i < real k"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   550
          using k by (subst (asm) Max_less_iff) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   551
        then have "x$$i < real k" by simp }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   552
      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia < real k"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   553
        by (auto intro!: exI[of _ k])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   554
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   555
    finally show "{x. a \<le> x$$i} \<in> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   556
      apply (simp only:)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   557
      apply (safe intro!: countable_UN Diff)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   558
      by (auto simp: sets_sigma intro!: sigma_sets.Basic)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   559
  next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   560
    fix a i assume "\<not> i < DIM('a)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   561
    then show "{x. a \<le> x$$i} \<in> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   562
      using top by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   563
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   564
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   565
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   566
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   567
lemma atMost_span_atLeastAtMost:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   568
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..a\<Colon>'a\<Colon>ordered_euclidean_space})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   569
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a,b). {a..b})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   570
  (is "_ \<subseteq> sets ?SIGMA")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   571
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   572
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   573
  then interpret sigma_algebra ?SIGMA .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   574
  { fix a::'a
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   575
    have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   576
    proof (safe, simp_all add: eucl_le[where 'a='a])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   577
      fix x
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   578
      from real_arch_simple[of "Max ((\<lambda>i. - x$$i)`{..<DIM('a)})"]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   579
      guess k::nat .. note k = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   580
      { fix i assume "i < DIM('a)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   581
        with k have "- x$$i \<le> real k"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   582
          by (subst (asm) Max_le_iff) (auto simp: field_simps)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   583
        then have "- real k \<le> x$$i" by simp }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   584
      then show "\<exists>n::nat. \<forall>i<DIM('a). - real n \<le> x $$ i"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   585
        by (auto intro!: exI[of _ k])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   586
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   587
    have "{..a} \<in> sets ?SIGMA" unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   588
      by (safe intro!: countable_UN)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   589
         (auto simp: sets_sigma intro!: sigma_sets.Basic) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   590
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   591
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   592
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   593
lemma borel_eq_atMost:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   594
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> a. {.. a::'a\<Colon>ordered_euclidean_space})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   595
    (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   596
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   597
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   598
    using halfspace_le_span_atMost halfspace_span_halfspace_le open_span_halfspace
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   599
    by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   600
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   601
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   602
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   603
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   604
lemma borel_eq_atLeastAtMost:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   605
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space, b). {a .. b})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   606
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   607
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   608
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   609
    using atMost_span_atLeastAtMost halfspace_le_span_atMost
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   610
      halfspace_span_halfspace_le open_span_halfspace
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   611
    by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   612
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   613
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   614
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   615
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   616
lemma borel_eq_greaterThan:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   617
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {a <..})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   618
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   619
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   620
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   621
    using halfspace_le_span_greaterThan
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   622
      halfspace_span_halfspace_le open_span_halfspace
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   623
    by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   624
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   625
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   626
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   627
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   628
lemma borel_eq_lessThan:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   629
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a :: 'a\<Colon>ordered_euclidean_space). {..< a})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   630
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   631
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   632
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   633
    using halfspace_le_span_lessThan
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   634
      halfspace_span_halfspace_ge open_span_halfspace
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   635
    by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   636
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   637
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   638
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   639
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   640
lemma borel_eq_greaterThanLessThan:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   641
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, b). {a <..< (b :: 'a \<Colon> ordered_euclidean_space)})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   642
    (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   643
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   644
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   645
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   646
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   647
  proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   648
    have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   649
    then interpret sigma_algebra ?SIGMA .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   650
    { fix M :: "'a set" assume "M \<in> open"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   651
      then have "open M" by (simp add: mem_def)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   652
      have "M \<in> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   653
        apply (subst open_UNION[OF `open M`])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   654
        apply (safe intro!: countable_UN)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   655
        by (auto simp add: sigma_def intro!: sigma_sets.Basic) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   656
    then show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   657
      unfolding borel_def by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   658
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   659
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   660
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   661
lemma borel_eq_halfspace_le:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   662
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i \<le> a})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   663
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   664
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   665
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   666
    using open_span_halfspace halfspace_span_halfspace_le by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   667
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   668
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   669
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   670
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   671
lemma borel_eq_halfspace_less:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   672
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i < a})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   673
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   674
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   675
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   676
    using open_span_halfspace .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   677
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   678
    by (rule borel.sets_sigma_subset) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   679
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   680
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   681
lemma borel_eq_halfspace_gt:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   682
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x$$i})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   683
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   684
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   685
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   686
    using halfspace_le_span_halfspace_gt open_span_halfspace halfspace_span_halfspace_le by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   687
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   688
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   689
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   690
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   691
lemma borel_eq_halfspace_ge:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   692
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a \<le> x$$i})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   693
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   694
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   695
  show "sets borel \<subseteq> sets ?SIGMA"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   696
    using halfspace_span_halfspace_ge open_span_halfspace by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   697
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   698
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   699
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   700
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   701
lemma (in sigma_algebra) borel_measurable_halfspacesI:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   702
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   703
  assumes "borel = (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   704
  and "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   705
  and "\<And>a i. \<not> i < DIM('c) \<Longrightarrow> S a i \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   706
  shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a::real. S a i \<in> sets M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   707
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   708
  fix a :: real and i assume i: "i < DIM('c)" and f: "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   709
  then show "S a i \<in> sets M" unfolding assms
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   710
    by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1) sigma_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   711
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   712
  assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   713
  { fix a i have "S a i \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   714
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   715
      assume "i < DIM('c)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   716
      with a show ?thesis unfolding assms(2) by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   717
    next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   718
      assume "\<not> i < DIM('c)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   719
      from assms(3)[OF this] show ?thesis .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   720
    qed }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   721
  then have "f \<in> measurable M (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   722
    by (auto intro!: measurable_sigma simp: assms(2))
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   723
  then show "f \<in> borel_measurable M" unfolding measurable_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   724
    unfolding assms(1) by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   725
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   726
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   727
lemma (in sigma_algebra) borel_measurable_iff_halfspace_le:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   728
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   729
  shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i \<le> a} \<in> sets M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   730
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   731
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   732
lemma (in sigma_algebra) borel_measurable_iff_halfspace_less:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   733
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   734
  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i < a} \<in> sets M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   735
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   736
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   737
lemma (in sigma_algebra) borel_measurable_iff_halfspace_ge:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   738
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   739
  shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w $$ i} \<in> sets M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   740
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   741
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   742
lemma (in sigma_algebra) borel_measurable_iff_halfspace_greater:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   743
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   744
  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a < f w $$ i} \<in> sets M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   745
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_gt]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   746
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   747
lemma (in sigma_algebra) borel_measurable_iff_le:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   748
  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   749
  using borel_measurable_iff_halfspace_le[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   750
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   751
lemma (in sigma_algebra) borel_measurable_iff_less:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   752
  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   753
  using borel_measurable_iff_halfspace_less[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   754
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   755
lemma (in sigma_algebra) borel_measurable_iff_ge:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   756
  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   757
  using borel_measurable_iff_halfspace_ge[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   758
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   759
lemma (in sigma_algebra) borel_measurable_iff_greater:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   760
  "(f::'a \<Rightarrow> real) \<in> borel_measurable M = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   761
  using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   762
41025
8b2cd85ecf11 fixed spelling errors
hoelzl
parents: 41023
diff changeset
   763
lemma borel_measurable_euclidean_component:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   764
  "(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   765
  unfolding borel_def[where 'a=real]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   766
proof (rule borel.measurable_sigma, simp_all)
39087
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   767
  fix S::"real set" assume "S \<in> open" then have "open S" unfolding mem_def .
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   768
  from open_vimage_euclidean_component[OF this]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   769
  show "(\<lambda>x. x $$ i) -` S \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   770
    by (auto intro: borel_open)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   771
qed
39087
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   772
41025
8b2cd85ecf11 fixed spelling errors
hoelzl
parents: 41023
diff changeset
   773
lemma (in sigma_algebra) borel_measurable_euclidean_space:
39087
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   774
  fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   775
  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M)"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   776
proof safe
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   777
  fix i assume "f \<in> borel_measurable M"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   778
  then show "(\<lambda>x. f x $$ i) \<in> borel_measurable M"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   779
    using measurable_comp[of f _ _ "\<lambda>x. x $$ i", unfolded comp_def]
41025
8b2cd85ecf11 fixed spelling errors
hoelzl
parents: 41023
diff changeset
   780
    by (auto intro: borel_measurable_euclidean_component)
39087
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   781
next
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   782
  assume f: "\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   783
  then show "f \<in> borel_measurable M"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   784
    unfolding borel_measurable_iff_halfspace_le by auto
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   785
qed
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   786
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   787
subsection "Borel measurable operators"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   788
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   789
lemma (in sigma_algebra) affine_borel_measurable_vector:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   790
  fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   791
  assumes "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   792
  shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   793
proof (rule borel_measurableI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   794
  fix S :: "'x set" assume "open S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   795
  show "(\<lambda>x. a + b *\<^sub>R f x) -` S \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   796
  proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   797
    assume "b \<noteq> 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   798
    with `open S` have "((\<lambda>x. (- a + x) /\<^sub>R b) ` S) \<in> open" (is "?S \<in> open")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   799
      by (auto intro!: open_affinity simp: scaleR.add_right mem_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   800
    hence "?S \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   801
      unfolding borel_def by (auto simp: sigma_def intro!: sigma_sets.Basic)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   802
    moreover
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   803
    from `b \<noteq> 0` have "(\<lambda>x. a + b *\<^sub>R f x) -` S = f -` ?S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   804
      apply auto by (rule_tac x="a + b *\<^sub>R f x" in image_eqI, simp_all)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   805
    ultimately show ?thesis using assms unfolding in_borel_measurable_borel
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   806
      by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   807
  qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   808
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   809
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   810
lemma (in sigma_algebra) affine_borel_measurable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   811
  fixes g :: "'a \<Rightarrow> real"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   812
  assumes g: "g \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   813
  shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   814
  using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   815
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   816
lemma (in sigma_algebra) borel_measurable_add[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   817
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   818
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   819
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   820
  shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   821
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   822
  have 1: "\<And>a. {w\<in>space M. a \<le> f w + g w} = {w \<in> space M. a + g w * -1 \<le> f w}"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   823
    by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   824
  have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   825
    by (rule affine_borel_measurable [OF g])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   826
  then have "\<And>a. {w \<in> space M. (\<lambda>w. a + (g w) * -1)(w) \<le> f w} \<in> sets M" using f
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   827
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   828
  then have "\<And>a. {w \<in> space M. a \<le> f w + g w} \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   829
    by (simp add: 1)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   830
  then show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   831
    by (simp add: borel_measurable_iff_ge)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   832
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   833
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   834
lemma (in sigma_algebra) borel_measurable_setsum[simp, intro]:
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   835
  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   836
  assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   837
  shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   838
proof cases
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   839
  assume "finite S"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   840
  thus ?thesis using assms by induct auto
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   841
qed simp
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   842
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   843
lemma (in sigma_algebra) borel_measurable_square:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   844
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   845
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   846
  shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   847
proof -
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   848
  {
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   849
    fix a
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   850
    have "{w \<in> space M. (f w)\<twosuperior> \<le> a} \<in> sets M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   851
    proof (cases rule: linorder_cases [of a 0])
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   852
      case less
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   853
      hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} = {}"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   854
        by auto (metis less order_le_less_trans power2_less_0)
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   855
      also have "... \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   856
        by (rule empty_sets)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   857
      finally show ?thesis .
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   858
    next
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   859
      case equal
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   860
      hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   861
             {w \<in> space M. f w \<le> 0} \<inter> {w \<in> space M. 0 \<le> f w}"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   862
        by auto
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   863
      also have "... \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   864
        apply (insert f)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   865
        apply (rule Int)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   866
        apply (simp add: borel_measurable_iff_le)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   867
        apply (simp add: borel_measurable_iff_ge)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   868
        done
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   869
      finally show ?thesis .
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   870
    next
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   871
      case greater
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   872
      have "\<forall>x. (f x ^ 2 \<le> sqrt a ^ 2) = (- sqrt a  \<le> f x & f x \<le> sqrt a)"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   873
        by (metis abs_le_interval_iff abs_of_pos greater real_sqrt_abs
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   874
                  real_sqrt_le_iff real_sqrt_power)
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   875
      hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   876
             {w \<in> space M. -(sqrt a) \<le> f w} \<inter> {w \<in> space M. f w \<le> sqrt a}"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   877
        using greater by auto
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   878
      also have "... \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   879
        apply (insert f)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   880
        apply (rule Int)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   881
        apply (simp add: borel_measurable_iff_ge)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   882
        apply (simp add: borel_measurable_iff_le)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   883
        done
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   884
      finally show ?thesis .
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   885
    qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   886
  }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   887
  thus ?thesis by (auto simp add: borel_measurable_iff_le)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   888
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   889
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   890
lemma times_eq_sum_squares:
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   891
   fixes x::real
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   892
   shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   893
by (simp add: power2_eq_square ring_distribs diff_divide_distrib [symmetric])
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   894
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   895
lemma (in sigma_algebra) borel_measurable_uminus[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   896
  fixes g :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   897
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   898
  shows "(\<lambda>x. - g x) \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   899
proof -
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   900
  have "(\<lambda>x. - g x) = (\<lambda>x. 0 + (g x) * -1)"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   901
    by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   902
  also have "... \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   903
    by (fast intro: affine_borel_measurable g)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   904
  finally show ?thesis .
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   905
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   906
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   907
lemma (in sigma_algebra) borel_measurable_times[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   908
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   909
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   910
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   911
  shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   912
proof -
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   913
  have 1: "(\<lambda>x. 0 + (f x + g x)\<twosuperior> * inverse 4) \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   914
    using assms by (fast intro: affine_borel_measurable borel_measurable_square)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   915
  have "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) =
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   916
        (\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents: 35347
diff changeset
   917
    by (simp add: minus_divide_right)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   918
  also have "... \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   919
    using f g by (fast intro: affine_borel_measurable borel_measurable_square f g)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   920
  finally have 2: "(\<lambda>x. -((f x + -g x) ^ 2 * inverse 4)) \<in> borel_measurable M" .
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   921
  show ?thesis
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   922
    apply (simp add: times_eq_sum_squares diff_minus)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   923
    using 1 2 by simp
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   924
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   925
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   926
lemma (in sigma_algebra) borel_measurable_setprod[simp, intro]:
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   927
  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   928
  assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   929
  shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   930
proof cases
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   931
  assume "finite S"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   932
  thus ?thesis using assms by induct auto
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   933
qed simp
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   934
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   935
lemma (in sigma_algebra) borel_measurable_diff[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   936
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   937
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   938
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   939
  shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   940
  unfolding diff_minus using assms by fast
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   941
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   942
lemma (in sigma_algebra) borel_measurable_inverse[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   943
  fixes f :: "'a \<Rightarrow> real"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   944
  assumes "f \<in> borel_measurable M"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   945
  shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   946
  unfolding borel_measurable_iff_ge unfolding inverse_eq_divide
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   947
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   948
  fix a :: real
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   949
  have *: "{w \<in> space M. a \<le> 1 / f w} =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   950
      ({w \<in> space M. 0 < f w} \<inter> {w \<in> space M. a * f w \<le> 1}) \<union>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   951
      ({w \<in> space M. f w < 0} \<inter> {w \<in> space M. 1 \<le> a * f w}) \<union>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   952
      ({w \<in> space M. f w = 0} \<inter> {w \<in> space M. a \<le> 0})" by (auto simp: le_divide_eq)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   953
  show "{w \<in> space M. a \<le> 1 / f w} \<in> sets M" using assms unfolding *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   954
    by (auto intro!: Int Un)
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   955
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   956
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   957
lemma (in sigma_algebra) borel_measurable_divide[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   958
  fixes f :: "'a \<Rightarrow> real"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   959
  assumes "f \<in> borel_measurable M"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   960
  and "g \<in> borel_measurable M"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   961
  shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   962
  unfolding field_divide_inverse
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   963
  by (rule borel_measurable_inverse borel_measurable_times assms)+
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   964
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   965
lemma (in sigma_algebra) borel_measurable_max[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   966
  fixes f g :: "'a \<Rightarrow> real"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   967
  assumes "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   968
  assumes "g \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   969
  shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   970
  unfolding borel_measurable_iff_le
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   971
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   972
  fix a
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   973
  have "{x \<in> space M. max (g x) (f x) \<le> a} =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   974
    {x \<in> space M. g x \<le> a} \<inter> {x \<in> space M. f x \<le> a}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   975
  thus "{x \<in> space M. max (g x) (f x) \<le> a} \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   976
    using assms unfolding borel_measurable_iff_le
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   977
    by (auto intro!: Int)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   978
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   979
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   980
lemma (in sigma_algebra) borel_measurable_min[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   981
  fixes f g :: "'a \<Rightarrow> real"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   982
  assumes "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   983
  assumes "g \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   984
  shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   985
  unfolding borel_measurable_iff_ge
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   986
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   987
  fix a
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   988
  have "{x \<in> space M. a \<le> min (g x) (f x)} =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   989
    {x \<in> space M. a \<le> g x} \<inter> {x \<in> space M. a \<le> f x}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   990
  thus "{x \<in> space M. a \<le> min (g x) (f x)} \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   991
    using assms unfolding borel_measurable_iff_ge
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   992
    by (auto intro!: Int)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   993
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   994
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   995
lemma (in sigma_algebra) borel_measurable_abs[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   996
  assumes "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   997
  shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   998
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   999
  have *: "\<And>x. \<bar>f x\<bar> = max 0 (f x) + max 0 (- f x)" by (simp add: max_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1000
  show ?thesis unfolding * using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1001
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1002
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
  1003
lemma borel_measurable_nth[simp, intro]:
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
  1004
  "(\<lambda>x::real^'n. x $ i) \<in> borel_measurable borel"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
  1005
  using borel_measurable_euclidean_component
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
  1006
  unfolding nth_conv_component by auto
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
  1007
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1008
lemma borel_measurable_continuous_on1:
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1009
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1010
  assumes "continuous_on UNIV f"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1011
  shows "f \<in> borel_measurable borel"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1012
  apply(rule borel.borel_measurableI)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1013
  using continuous_open_preimage[OF assms] unfolding vimage_def by auto
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1014
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1015
lemma borel_measurable_continuous_on:
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1016
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1017
  assumes cont: "continuous_on A f" "open A" and f: "f -` {c} \<inter> A \<in> sets borel"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1018
  shows "(\<lambda>x. if x \<in> A then f x else c) \<in> borel_measurable borel" (is "?f \<in> _")
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1019
proof (rule borel.borel_measurableI)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1020
  fix S :: "'b set" assume "open S"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1021
  then have "open {x\<in>A. f x \<in> S - {c}}"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1022
    by (intro continuous_open_preimage[OF cont]) auto
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1023
  then have *: "{x\<in>A. f x \<in> S - {c}} \<in> sets borel" by auto
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1024
  show "?f -` S \<inter> space borel \<in> sets borel"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1025
  proof cases
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1026
    assume "c \<in> S"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1027
    then have "?f -` S = {x\<in>A. f x \<in> S - {c}} \<union> (f -` {c} \<inter> A) \<union> -A"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1028
      by auto
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1029
    with * show "?f -` S \<inter> space borel \<in> sets borel"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1030
      using `open A` f by (auto intro!: borel.Un)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1031
  next
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1032
    assume "c \<notin> S"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1033
    then have "?f -` S = {x\<in>A. f x \<in> S - {c}}" by (auto split: split_if_asm)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1034
    with * show "?f -` S \<inter> space borel \<in> sets borel" by auto
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1035
  qed
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1036
qed
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1037
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1038
lemma borel_measurable_borel_log: assumes "1 < b" shows "log b \<in> borel_measurable borel"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1039
proof -
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1040
  { fix x :: real assume x: "x \<le> 0"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1041
    { fix x::real assume "x \<le> 0" then have "\<And>u. exp u = x \<longleftrightarrow> False" by auto }
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1042
    from this[of x] x this[of 0] have "log b 0 = log b x"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1043
      by (auto simp: ln_def log_def) }
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1044
  note log_imp = this
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1045
  have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) \<in> borel_measurable borel"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1046
  proof (rule borel_measurable_continuous_on)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1047
    show "continuous_on {0<..} (log b)"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1048
      by (auto intro!: continuous_at_imp_continuous_on DERIV_log DERIV_isCont
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1049
               simp: continuous_isCont[symmetric])
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1050
    show "open ({0<..}::real set)" by auto
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1051
    show "log b -` {log b 0} \<inter> {0<..} \<in> sets borel"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1052
    proof cases
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1053
      assume "log b -` {log b 0} \<inter> {0<..} = {}"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1054
      then show ?thesis by simp
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1055
    next
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1056
      assume "log b -` {log b 0} \<inter> {0<..} \<noteq> {}"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1057
      then obtain x where "0 < x" "log b x = log b 0" by auto
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1058
      with log_inj[OF `1 < b`] have "log b -` {log b 0} \<inter> {0<..} = {x}"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1059
        by (auto simp: inj_on_def)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1060
      then show ?thesis by simp
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1061
    qed
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1062
  qed
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1063
  also have "(\<lambda>x. if x \<in> {0<..} then log b x else log b 0) = log b"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1064
    by (simp add: fun_eq_iff not_less log_imp)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1065
  finally show ?thesis .
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1066
qed
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1067
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1068
lemma (in sigma_algebra) borel_measurable_log[simp,intro]:
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1069
  assumes f: "f \<in> borel_measurable M" and "1 < b"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1070
  shows "(\<lambda>x. log b (f x)) \<in> borel_measurable M"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1071
  using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1072
  by (simp add: comp_def)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1073
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1074
subsection "Borel space on the extended reals"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1075
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1076
lemma borel_measurable_extreal_borel:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1077
  "extreal \<in> borel_measurable borel"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1078
  unfolding borel_def[where 'a=extreal]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1079
proof (rule borel.measurable_sigma)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1080
  fix X :: "extreal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open \<rparr>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1081
  then have "open X" by (auto simp: mem_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1082
  then have "open (extreal -` X \<inter> space borel)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1083
    by (simp add: open_extreal_vimage)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1084
  then show "extreal -` X \<inter> space borel \<in> sets borel" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1085
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1086
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1087
lemma (in sigma_algebra) borel_measurable_extreal[simp, intro]:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1088
  assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. extreal (f x)) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1089
  using measurable_comp[OF f borel_measurable_extreal_borel] unfolding comp_def .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1090
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1091
lemma borel_measurable_real_of_extreal_borel:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1092
  "(real :: extreal \<Rightarrow> real) \<in> borel_measurable borel"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1093
  unfolding borel_def[where 'a=real]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1094
proof (rule borel.measurable_sigma)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1095
  fix B :: "real set" assume "B \<in> sets \<lparr>space = UNIV, sets = open \<rparr>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1096
  then have "open B" by (auto simp: mem_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1097
  have *: "extreal -` real -` (B - {0}) = B - {0}" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1098
  have open_real: "open (real -` (B - {0}) :: extreal set)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1099
    unfolding open_extreal_def * using `open B` by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1100
  show "(real -` B \<inter> space borel :: extreal set) \<in> sets borel"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1101
  proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1102
    assume "0 \<in> B"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1103
    then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1104
      by (auto simp add: real_of_extreal_eq_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1105
    then show "(real -` B :: extreal set) \<inter> space borel \<in> sets borel"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1106
      using open_real by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1107
  next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1108
    assume "0 \<notin> B"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1109
    then have *: "(real -` B :: extreal set) = real -` (B - {0})"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1110
      by (auto simp add: real_of_extreal_eq_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1111
    then show "(real -` B :: extreal set) \<inter> space borel \<in> sets borel"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1112
      using open_real by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1113
  qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1114
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1115
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1116
lemma (in sigma_algebra) borel_measurable_real_of_extreal[simp, intro]:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1117
  assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: extreal)) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1118
  using measurable_comp[OF f borel_measurable_real_of_extreal_borel] unfolding comp_def .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1119
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1120
lemma (in sigma_algebra) borel_measurable_extreal_iff:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1121
  shows "(\<lambda>x. extreal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1122
proof
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1123
  assume "(\<lambda>x. extreal (f x)) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1124
  from borel_measurable_real_of_extreal[OF this]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1125
  show "f \<in> borel_measurable M" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1126
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1127
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1128
lemma (in sigma_algebra) borel_measurable_extreal_iff_real:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1129
  "f \<in> borel_measurable M \<longleftrightarrow>
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1130
    ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1131
proof safe
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1132
  assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1133
  have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1134
  with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1135
  let "?f x" = "if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else extreal (real (f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1136
  have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1137
  also have "?f = f" by (auto simp: fun_eq_iff extreal_real)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1138
  finally show "f \<in> borel_measurable M" .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1139
qed (auto intro: measurable_sets borel_measurable_real_of_extreal)
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1140
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1141
lemma (in sigma_algebra) less_eq_ge_measurable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1142
  fixes f :: "'a \<Rightarrow> 'c::linorder"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1143
  shows "f -` {a <..} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {..a} \<inter> space M \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1144
proof
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1145
  assume "f -` {a <..} \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1146
  moreover have "f -` {..a} \<inter> space M = space M - f -` {a <..} \<inter> space M" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1147
  ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1148
next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1149
  assume "f -` {..a} \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1150
  moreover have "f -` {a <..} \<inter> space M = space M - f -` {..a} \<inter> space M" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1151
  ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1152
qed
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1153
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1154
lemma (in sigma_algebra) greater_eq_le_measurable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1155
  fixes f :: "'a \<Rightarrow> 'c::linorder"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1156
  shows "f -` {..< a} \<inter> space M \<in> sets M \<longleftrightarrow> f -` {a ..} \<inter> space M \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1157
proof
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1158
  assume "f -` {a ..} \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1159
  moreover have "f -` {..< a} \<inter> space M = space M - f -` {a ..} \<inter> space M" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1160
  ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1161
next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1162
  assume "f -` {..< a} \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1163
  moreover have "f -` {a ..} \<inter> space M = space M - f -` {..< a} \<inter> space M" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1164
  ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1165
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1166
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1167
lemma (in sigma_algebra) borel_measurable_uminus_borel_extreal:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1168
  "(uminus :: extreal \<Rightarrow> extreal) \<in> borel_measurable borel"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1169
proof (subst borel_def, rule borel.measurable_sigma)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1170
  fix X :: "extreal set" assume "X \<in> sets \<lparr>space = UNIV, sets = open\<rparr>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1171
  then have "open X" by (simp add: mem_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1172
  have "uminus -` X = uminus ` X" by (force simp: image_iff)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1173
  then have "open (uminus -` X)" using `open X` extreal_open_uminus by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1174
  then show "uminus -` X \<inter> space borel \<in> sets borel" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1175
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1176
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1177
lemma (in sigma_algebra) borel_measurable_uminus_extreal[intro]:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1178
  assumes "f \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1179
  shows "(\<lambda>x. - f x :: extreal) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1180
  using measurable_comp[OF assms borel_measurable_uminus_borel_extreal] by (simp add: comp_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1181
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1182
lemma (in sigma_algebra) borel_measurable_uminus_eq_extreal[simp]:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1183
  "(\<lambda>x. - f x :: extreal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1184
proof
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1185
  assume ?l from borel_measurable_uminus_extreal[OF this] show ?r by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1186
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1187
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1188
lemma (in sigma_algebra) borel_measurable_eq_atMost_extreal:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1189
  "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1190
proof (intro iffI allI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1191
  assume pos[rule_format]: "\<forall>a. f -` {..a} \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1192
  show "f \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1193
    unfolding borel_measurable_extreal_iff_real borel_measurable_iff_le
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1194
  proof (intro conjI allI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1195
    fix a :: real
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1196
    { fix x :: extreal assume *: "\<forall>i::nat. real i < x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1197
      have "x = \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1198
      proof (rule extreal_top)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1199
        fix B from real_arch_lt[of B] guess n ..
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1200
        then have "extreal B < real n" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1201
        with * show "B \<le> x" by (metis less_trans less_imp_le)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1202
      qed }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1203
    then have "f -` {\<infinity>} \<inter> space M = space M - (\<Union>i::nat. f -` {.. real i} \<inter> space M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1204
      by (auto simp: not_le)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1205
    then show "f -` {\<infinity>} \<inter> space M \<in> sets M" using pos by (auto simp del: UN_simps intro!: Diff)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1206
    moreover
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1207
    have "{-\<infinity>} = {..-\<infinity>}" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1208
    then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1209
    moreover have "{x\<in>space M. f x \<le> extreal a} \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1210
      using pos[of "extreal a"] by (simp add: vimage_def Int_def conj_commute)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1211
    moreover have "{w \<in> space M. real (f w) \<le> a} =
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1212
      (if a < 0 then {w \<in> space M. f w \<le> extreal a} - f -` {-\<infinity>} \<inter> space M
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1213
      else {w \<in> space M. f w \<le> extreal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r")
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1214
      proof (intro set_eqI) fix x show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" by (cases "f x") auto qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1215
    ultimately show "{w \<in> space M. real (f w) \<le> a} \<in> sets M" by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents: 35347
diff changeset
  1216
  qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1217
qed (simp add: measurable_sets)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents: 35347
diff changeset
  1218
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1219
lemma (in sigma_algebra) borel_measurable_eq_atLeast_extreal:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1220
  "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1221
proof
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1222
  assume pos: "\<forall>a. f -` {a..} \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1223
  moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1224
    by (auto simp: extreal_uminus_le_reorder)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1225
  ultimately have "(\<lambda>x. - f x) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1226
    unfolding borel_measurable_eq_atMost_extreal by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1227
  then show "f \<in> borel_measurable M" by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1228
qed (simp add: measurable_sets)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents: 35347
diff changeset
  1229
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1230
lemma (in sigma_algebra) borel_measurable_extreal_iff_less:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1231
  "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..< a} \<inter> space M \<in> sets M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1232
  unfolding borel_measurable_eq_atLeast_extreal greater_eq_le_measurable ..
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1233
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1234
lemma (in sigma_algebra) borel_measurable_extreal_iff_ge:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1235
  "(f::'a \<Rightarrow> extreal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a <..} \<inter> space M \<in> sets M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1236
  unfolding borel_measurable_eq_atMost_extreal less_eq_ge_measurable ..
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1237
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1238
lemma (in sigma_algebra) borel_measurable_extreal_eq_const:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1239
  fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1240
  shows "{x\<in>space M. f x = c} \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1241
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1242
  have "{x\<in>space M. f x = c} = (f -` {c} \<inter> space M)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1243
  then show ?thesis using assms by (auto intro!: measurable_sets)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1244
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1245
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1246
lemma (in sigma_algebra) borel_measurable_extreal_neq_const:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1247
  fixes f :: "'a \<Rightarrow> extreal" assumes "f \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1248
  shows "{x\<in>space M. f x \<noteq> c} \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1249
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1250
  have "{x\<in>space M. f x \<noteq> c} = space M - (f -` {c} \<inter> space M)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1251
  then show ?thesis using assms by (auto intro!: measurable_sets)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1252
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1253
41981