src/HOL/Probability/Borel_Space.thy
author hoelzl
Fri, 28 Oct 2011 14:10:19 +0200
changeset 45288 fc3c7db5bb2f
parent 45287 97df8c08291c
child 46731 5302e932d1e5
permissions -rw-r--r--
correct import path
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(*  Title:      HOL/Probability/Borel_Space.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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40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
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theory Borel_Space
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     9
  imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Multivariate_Analysis"
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40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
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begin
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
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d5d342611edb Rewrite the Probability theory.
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section "Generic Borel spaces"
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c10485a6a7af make HOL-Probability respect set/pred distinction
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definition "borel = sigma \<lparr> space = UNIV::'a::topological_space set, sets = {S. open S}\<rparr>"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
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    15
abbreviation "borel_measurable M \<equiv> measurable M borel"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
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40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
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    17
interpretation borel: sigma_algebra borel
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    18
  by (auto simp: borel_def intro!: sigma_algebra_sigma)
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parents:
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    19
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
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lemma in_borel_measurable:
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
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   "f \<in> borel_measurable M \<longleftrightarrow>
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    22
    (\<forall>S \<in> sets (sigma \<lparr> space = UNIV, sets = {S. open S}\<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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    25
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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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40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
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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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    38
  have "A \<in> {S. open S}" unfolding mem_Collect_eq using assms .
40859
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    39
  thus ?thesis unfolding borel_def sigma_def by (auto intro!: sigma_sets.Basic)
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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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    41
40859
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lemma borel_closed[simp]:
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    43
  assumes "closed A" shows "A \<in> sets borel"
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parents:
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    44
proof -
40859
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hoelzl
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    45
  have "space borel - (- A) \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
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    46
    using assms unfolding closed_def by (blast intro: borel_open)
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  thus ?thesis by simp
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40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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    48
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
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    49
41830
719b0a517c33 log is borel measurable
hoelzl
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    50
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
hoelzl
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    52
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    53
lemma (in sigma_algebra) borel_measurable_vimage:
d5d342611edb Rewrite the Probability theory.
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    54
  fixes f :: "'a \<Rightarrow> 'x::t2_space"
d5d342611edb Rewrite the Probability theory.
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    55
  assumes borel: "f \<in> borel_measurable M"
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hoelzl
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    56
  shows "f -` {x} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
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    57
proof (cases "x \<in> f ` space M")
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    58
  case True then obtain y where "x = f y" by auto
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1cf3e4107a2a moved t2_spaces to HOL image
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    59
  from closed_singleton[of "f y"]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
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    60
  have "{f y} \<in> sets borel" by (rule borel_closed)
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d5d342611edb Rewrite the Probability theory.
hoelzl
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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.
hoelzl
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    65
  thus ?thesis by auto
33533
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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d5d342611edb Rewrite the Probability theory.
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    68
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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huffman
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    74
  fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
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    75
    using assms[of S] by simp
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
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    76
qed
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
    77
40859
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    78
lemma borel_singleton[simp, intro]:
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    79
  fixes x :: "'a::t1_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
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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.
hoelzl
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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
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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)
33533
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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diff changeset
    90
lemma (in sigma_algebra) borel_measurable_indicator[simp, intro!]:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
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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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diff changeset
    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
de0b30e6c2d2 Support product spaces on sigma finite measures.
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
parents: 39302
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
diff changeset
   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:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   115
  fixes f :: "'a \<Rightarrow> ereal" 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
parents: 39087
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
parents: 39087
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
parents: 39087
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
parents: 39302
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)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   126
    fix S :: "ereal 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"
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44666
diff changeset
   130
      using `A \<in> sets M` sets_into_space by fastforce
39092
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
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
   145
    fix S :: "ereal 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 .
44537
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
   394
  { fix S :: "'a set" assume "S \<in> {S. open S}"
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
   395
    then have "open S" unfolding mem_Collect_eq .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   396
    from open_UNION[OF this]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   397
    obtain I where *: "S =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   398
      (\<Union>(a, b)\<in>I.
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   399
          (\<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
   400
          (\<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
   401
      unfolding greaterThanLessThan_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   402
      unfolding eucl_greaterThan_eq_halfspaces[where 'a='a]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   403
      unfolding eucl_lessThan_eq_halfspaces[where 'a='a]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   404
      by blast
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   405
    have "S \<in> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   406
      unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   407
      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
   408
  then show ?thesis unfolding borel_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   409
    by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   410
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   411
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   412
lemma halfspace_span_halfspace_le:
40859
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\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   414
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. x $$ i \<le> a})\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   415
  (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   416
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   417
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   418
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   419
  { fix a i
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   420
    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
   421
    proof (safe, simp_all)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   422
      fix x::'a assume *: "x$$i < a"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   423
      with reals_Archimedean[of "a - x$$i"]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   424
      obtain n where "x $$ i < a - 1 / (real (Suc n))"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   425
        by (auto simp: field_simps inverse_eq_divide)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   426
      then show "\<exists>n. x $$ i \<le> a - 1 / (real (Suc n))"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   427
        by (blast intro: less_imp_le)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   428
    next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   429
      fix x::'a and n
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   430
      assume "x$$i \<le> a - 1 / real (Suc n)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   431
      also have "\<dots> < a" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   432
      finally show "x$$i < a" .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   433
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   434
    have "{x. x$$i < a} \<in> sets ?SIGMA" unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   435
      by (safe intro!: countable_UN)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   436
         (auto simp: sets_sigma intro!: sigma_sets.Basic) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   437
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   438
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   439
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   440
lemma halfspace_span_halfspace_ge:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   441
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i < a})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   442
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a \<le> x $$ i})\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   443
  (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   444
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   445
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   446
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   447
  { fix a i have *: "{x::'a. x$$i < a} = space ?SIGMA - {x::'a. a \<le> x$$i}" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   448
    have "{x. x$$i < a} \<in> sets ?SIGMA" unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   449
      by (safe intro!: Diff)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   450
         (auto simp: sets_sigma intro!: sigma_sets.Basic) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   451
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   452
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   453
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   454
lemma halfspace_le_span_halfspace_gt:
40859
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\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   456
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x. a < x $$ i})\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   457
  (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   458
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   459
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   460
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   461
  { 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
   462
    have "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   463
      by (safe intro!: Diff)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   464
         (auto simp: sets_sigma intro!: sigma_sets.Basic) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   465
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   466
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   467
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   468
lemma halfspace_le_span_atMost:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   469
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   470
   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
   471
  (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   472
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   473
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   474
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   475
  have "\<And>a i. {x. x$$i \<le> a} \<in> sets ?SIGMA"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   476
  proof cases
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   477
    fix a i assume "i < DIM('a)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   478
    then have *: "{x::'a. x$$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
   479
    proof (safe, simp_all add: eucl_le[where 'a='a] split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   480
      fix x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   481
      from real_arch_simple[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"] guess k::nat ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   482
      then have "\<And>i. i < DIM('a) \<Longrightarrow> x$$i \<le> real k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   483
        by (subst (asm) Max_le_iff) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   484
      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia \<le> real k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   485
        by (auto intro!: exI[of _ k])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   486
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   487
    show "{x. x$$i \<le> a} \<in> sets ?SIGMA" unfolding *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   488
      by (safe intro!: countable_UN)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   489
         (auto simp: sets_sigma intro!: sigma_sets.Basic)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   490
  next
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   491
    fix a i assume "\<not> i < DIM('a)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   492
    then show "{x. x$$i \<le> a} \<in> sets ?SIGMA"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   493
      using top by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   494
  qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   495
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   496
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   497
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   498
lemma halfspace_le_span_greaterThan:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   499
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. x $$ i \<le> a})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   500
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {a<..})\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   501
  (is "_ \<subseteq> sets ?SIGMA")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   502
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   503
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   504
  then interpret sigma_algebra ?SIGMA .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   505
  have "\<And>a i. {x. x$$i \<le> a} \<in> sets ?SIGMA"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   506
  proof cases
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   507
    fix a i assume "i < DIM('a)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   508
    have "{x::'a. x$$i \<le> a} = space ?SIGMA - {x::'a. a < x$$i}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   509
    also have *: "{x::'a. a < x$$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
   510
    proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   511
      fix x
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44537
diff changeset
   512
      from reals_Archimedean2[of "Max ((\<lambda>i. -x$$i)`{..<DIM('a)})"]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   513
      guess k::nat .. note k = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   514
      { fix i assume "i < DIM('a)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   515
        then have "-x$$i < real k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   516
          using k by (subst (asm) Max_less_iff) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   517
        then have "- real k < x$$i" by simp }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   518
      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> -real k < x $$ ia"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   519
        by (auto intro!: exI[of _ k])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   520
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   521
    finally show "{x. x$$i \<le> a} \<in> sets ?SIGMA"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   522
      apply (simp only:)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   523
      apply (safe intro!: countable_UN Diff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   524
      by (auto simp: sets_sigma intro!: sigma_sets.Basic)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   525
  next
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   526
    fix a i assume "\<not> i < DIM('a)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   527
    then show "{x. x$$i \<le> a} \<in> sets ?SIGMA"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   528
      using top by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   529
  qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   530
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   531
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   532
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   533
lemma halfspace_le_span_lessThan:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   534
  "sets (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x\<Colon>'a\<Colon>ordered_euclidean_space. a \<le> x $$ i})\<rparr>) \<subseteq>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   535
   sets (sigma \<lparr>space=UNIV, sets=range (\<lambda>a. {..<a})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   536
  (is "_ \<subseteq> sets ?SIGMA")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   537
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   538
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   539
  then interpret sigma_algebra ?SIGMA .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   540
  have "\<And>a i. {x. a \<le> x$$i} \<in> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   541
  proof cases
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   542
    fix a i assume "i < DIM('a)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   543
    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
   544
    also have *: "{x::'a. x$$i < a} = (\<Union>k::nat. {..< (\<chi>\<chi> n. if n = i then a else real k)})" using `i <DIM('a)`
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   545
    proof (safe, simp_all add: eucl_less[where 'a='a] split: split_if_asm)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   546
      fix x
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44537
diff changeset
   547
      from reals_Archimedean2[of "Max ((\<lambda>i. x$$i)`{..<DIM('a)})"]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   548
      guess k::nat .. note k = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   549
      { fix i assume "i < DIM('a)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   550
        then have "x$$i < real k"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   551
          using k by (subst (asm) Max_less_iff) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   552
        then have "x$$i < real k" by simp }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   553
      then show "\<exists>k::nat. \<forall>ia. ia \<noteq> i \<longrightarrow> ia < DIM('a) \<longrightarrow> x $$ ia < real k"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   554
        by (auto intro!: exI[of _ k])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   555
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   556
    finally show "{x. a \<le> x$$i} \<in> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   557
      apply (simp only:)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   558
      apply (safe intro!: countable_UN Diff)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   559
      by (auto simp: sets_sigma intro!: sigma_sets.Basic)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   560
  next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   561
    fix a i assume "\<not> i < DIM('a)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   562
    then show "{x. a \<le> x$$i} \<in> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   563
      using top by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   564
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   565
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   566
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   567
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   568
lemma atMost_span_atLeastAtMost:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   569
  "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
   570
   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
   571
  (is "_ \<subseteq> sets ?SIGMA")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   572
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   573
  have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   574
  then interpret sigma_algebra ?SIGMA .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   575
  { fix a::'a
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   576
    have *: "{..a} = (\<Union>n::nat. {- real n *\<^sub>R One .. a})"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   577
    proof (safe, simp_all add: eucl_le[where 'a='a])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   578
      fix x
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   579
      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
   580
      guess k::nat .. note k = this
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   581
      { fix i assume "i < DIM('a)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   582
        with k have "- x$$i \<le> real k"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   583
          by (subst (asm) Max_le_iff) (auto simp: field_simps)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   584
        then have "- real k \<le> x$$i" by simp }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   585
      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
   586
        by (auto intro!: exI[of _ k])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   587
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   588
    have "{..a} \<in> sets ?SIGMA" unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   589
      by (safe intro!: countable_UN)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   590
         (auto simp: sets_sigma intro!: sigma_sets.Basic) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   591
  then show ?thesis by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   592
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   593
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   594
lemma borel_eq_atMost:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   595
  "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
   596
    (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   597
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   598
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   599
    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
   600
    by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   601
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   602
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   603
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   604
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   605
lemma borel_eq_atLeastAtMost:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   606
  "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
   607
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   608
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   609
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   610
    using atMost_span_atLeastAtMost halfspace_le_span_atMost
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   611
      halfspace_span_halfspace_le open_span_halfspace
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   612
    by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   613
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   614
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   615
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   616
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   617
lemma borel_eq_greaterThan:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   618
  "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
   619
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   620
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   621
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   622
    using halfspace_le_span_greaterThan
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   623
      halfspace_span_halfspace_le open_span_halfspace
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   624
    by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   625
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   626
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   627
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   628
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   629
lemma borel_eq_lessThan:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   630
  "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
   631
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   632
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   633
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   634
    using halfspace_le_span_lessThan
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   635
      halfspace_span_halfspace_ge open_span_halfspace
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   636
    by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   637
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   638
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   639
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   640
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   641
lemma borel_eq_greaterThanLessThan:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   642
  "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
   643
    (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   644
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   645
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   646
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   647
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   648
  proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   649
    have "sigma_algebra ?SIGMA" by (rule sigma_algebra_sigma) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   650
    then interpret sigma_algebra ?SIGMA .
44537
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
   651
    { fix M :: "'a set" assume "M \<in> {S. open S}"
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
   652
      then have "open M" by simp
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   653
      have "M \<in> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   654
        apply (subst open_UNION[OF `open M`])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   655
        apply (safe intro!: countable_UN)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   656
        by (auto simp add: sigma_def intro!: sigma_sets.Basic) }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   657
    then show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   658
      unfolding borel_def by (intro sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   659
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   660
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   661
42862
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   662
lemma borel_eq_atLeastLessThan:
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   663
  "borel = sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a ..< b :: real})\<rparr>" (is "_ = ?S")
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   664
proof (intro algebra.equality antisym)
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   665
  interpret sigma_algebra ?S
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   666
    by (rule sigma_algebra_sigma) auto
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   667
  show "sets borel \<subseteq> sets ?S"
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   668
    unfolding borel_eq_lessThan
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   669
  proof (intro sets_sigma_subset subsetI)
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   670
    have move_uminus: "\<And>x y::real. -x \<le> y \<longleftrightarrow> -y \<le> x" by auto
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   671
    fix A :: "real set" assume "A \<in> sets \<lparr>space = UNIV, sets = range lessThan\<rparr>"
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   672
    then obtain x where "A = {..< x}" by auto
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   673
    then have "A = (\<Union>i::nat. {-real i ..< x})"
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   674
      by (auto simp: move_uminus real_arch_simple)
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   675
    then show "A \<in> sets ?S"
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   676
      by (auto simp: sets_sigma intro!: sigma_sets.intros)
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   677
  qed simp
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   678
  show "sets ?S \<subseteq> sets borel"
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   679
    by (intro borel.sets_sigma_subset) auto
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   680
qed simp_all
7d7627738e66 add borel_eq_atLeastLessThan
hoelzl
parents: 42150
diff changeset
   681
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   682
lemma borel_eq_halfspace_le:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   683
  "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
   684
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   685
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   686
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   687
    using open_span_halfspace halfspace_span_halfspace_le by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   688
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   689
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   690
qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   691
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   692
lemma borel_eq_halfspace_less:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   693
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. x$$i < a})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   694
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   695
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   696
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   697
    using open_span_halfspace .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   698
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   699
    by (rule borel.sets_sigma_subset) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   700
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   701
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   702
lemma borel_eq_halfspace_gt:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   703
  "borel = (sigma \<lparr>space=UNIV, sets=range (\<lambda> (a, i). {x::'a::ordered_euclidean_space. a < x$$i})\<rparr>)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   704
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   705
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   706
  show "sets borel \<subseteq> sets ?SIGMA"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   707
    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
   708
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   709
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   710
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   711
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   712
lemma borel_eq_halfspace_ge:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   713
  "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
   714
   (is "_ = ?SIGMA")
40869
251df82c0088 Replace algebra_eqI by algebra.equality;
hoelzl
parents: 40859
diff changeset
   715
proof (intro algebra.equality antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   716
  show "sets borel \<subseteq> sets ?SIGMA"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   717
    using halfspace_span_halfspace_ge open_span_halfspace by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   718
  show "sets ?SIGMA \<subseteq> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   719
    by (rule borel.sets_sigma_subset) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   720
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   721
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   722
lemma (in sigma_algebra) borel_measurable_halfspacesI:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   723
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   724
  assumes "borel = (sigma \<lparr>space=UNIV, sets=range F\<rparr>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   725
  and "\<And>a i. S a i = f -` F (a,i) \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   726
  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
   727
  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
   728
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   729
  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
   730
  then show "S a i \<in> sets M" unfolding assms
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   731
    by (auto intro!: measurable_sets sigma_sets.Basic simp: assms(1) sigma_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   732
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   733
  assume a: "\<forall>i<DIM('c). \<forall>a. S a i \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   734
  { fix a i have "S a i \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   735
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   736
      assume "i < DIM('c)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   737
      with a show ?thesis unfolding assms(2) by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   738
    next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   739
      assume "\<not> i < DIM('c)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   740
      from assms(3)[OF this] show ?thesis .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   741
    qed }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   742
  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
   743
    by (auto intro!: measurable_sigma simp: assms(2))
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   744
  then show "f \<in> borel_measurable M" unfolding measurable_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   745
    unfolding assms(1) by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   746
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   747
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   748
lemma (in sigma_algebra) borel_measurable_iff_halfspace_le:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   749
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   750
  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
   751
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_le]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   752
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   753
lemma (in sigma_algebra) borel_measurable_iff_halfspace_less:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   754
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   755
  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. f w $$ i < a} \<in> sets M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   756
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_less]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   757
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   758
lemma (in sigma_algebra) borel_measurable_iff_halfspace_ge:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   759
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   760
  shows "f \<in> borel_measurable M = (\<forall>i<DIM('c). \<forall>a. {w \<in> space M. a \<le> f w $$ i} \<in> sets M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   761
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_ge]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   762
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   763
lemma (in sigma_algebra) borel_measurable_iff_halfspace_greater:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   764
  fixes f :: "'a \<Rightarrow> 'c\<Colon>ordered_euclidean_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   765
  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
   766
  by (rule borel_measurable_halfspacesI[OF borel_eq_halfspace_gt]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   767
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   768
lemma (in sigma_algebra) borel_measurable_iff_le:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   769
  "(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
   770
  using borel_measurable_iff_halfspace_le[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   771
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   772
lemma (in sigma_algebra) borel_measurable_iff_less:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   773
  "(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
   774
  using borel_measurable_iff_halfspace_less[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   775
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   776
lemma (in sigma_algebra) borel_measurable_iff_ge:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   777
  "(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
   778
  using borel_measurable_iff_halfspace_ge[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   779
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   780
lemma (in sigma_algebra) borel_measurable_iff_greater:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   781
  "(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
   782
  using borel_measurable_iff_halfspace_greater[where 'c=real] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   783
41025
8b2cd85ecf11 fixed spelling errors
hoelzl
parents: 41023
diff changeset
   784
lemma borel_measurable_euclidean_component:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   785
  "(\<lambda>x::'a::euclidean_space. x $$ i) \<in> borel_measurable borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   786
  unfolding borel_def[where 'a=real]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   787
proof (rule borel.measurable_sigma, simp_all)
44537
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
   788
  fix S::"real set" assume "open S"
39087
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   789
  from open_vimage_euclidean_component[OF this]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   790
  show "(\<lambda>x. x $$ i) -` S \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   791
    by (auto intro: borel_open)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   792
qed
39087
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   793
41025
8b2cd85ecf11 fixed spelling errors
hoelzl
parents: 41023
diff changeset
   794
lemma (in sigma_algebra) borel_measurable_euclidean_space:
39087
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   795
  fixes f :: "'a \<Rightarrow> 'c::ordered_euclidean_space"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   796
  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>i<DIM('c). (\<lambda>x. f x $$ i) \<in> borel_measurable M)"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   797
proof safe
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   798
  fix i assume "f \<in> borel_measurable M"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   799
  then show "(\<lambda>x. f x $$ i) \<in> borel_measurable M"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   800
    using measurable_comp[of f _ _ "\<lambda>x. x $$ i", unfolded comp_def]
41025
8b2cd85ecf11 fixed spelling errors
hoelzl
parents: 41023
diff changeset
   801
    by (auto intro: borel_measurable_euclidean_component)
39087
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   802
next
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   803
  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
   804
  then show "f \<in> borel_measurable M"
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   805
    unfolding borel_measurable_iff_halfspace_le by auto
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   806
qed
96984bf6fa5b Measurable on euclidean space is equiv. to measurable components
hoelzl
parents: 39083
diff changeset
   807
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   808
subsection "Borel measurable operators"
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_vector:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   811
  fixes f :: "'a \<Rightarrow> 'x::real_normed_vector"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   812
  assumes "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   813
  shows "(\<lambda>x. a + b *\<^sub>R f x) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   814
proof (rule borel_measurableI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   815
  fix S :: "'x set" assume "open S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   816
  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
   817
  proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   818
    assume "b \<noteq> 0"
44537
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
   819
    with `open S` have "open ((\<lambda>x. (- a + x) /\<^sub>R b) ` S)" (is "open ?S")
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
   820
      by (auto intro!: open_affinity simp: scaleR_add_right)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   821
    hence "?S \<in> sets borel"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   822
      unfolding borel_def by (auto simp: sigma_def intro!: sigma_sets.Basic)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   823
    moreover
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   824
    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
   825
      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
   826
    ultimately show ?thesis using assms unfolding in_borel_measurable_borel
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   827
      by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   828
  qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   829
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   830
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   831
lemma (in sigma_algebra) affine_borel_measurable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   832
  fixes g :: "'a \<Rightarrow> real"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   833
  assumes g: "g \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   834
  shows "(\<lambda>x. a + (g x) * b) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   835
  using affine_borel_measurable_vector[OF assms] by (simp add: mult_commute)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   836
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   837
lemma (in sigma_algebra) borel_measurable_add[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   838
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   839
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   840
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   841
  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
   842
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   843
  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
   844
    by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   845
  have "\<And>a. (\<lambda>w. a + (g w) * -1) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   846
    by (rule affine_borel_measurable [OF g])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   847
  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
   848
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   849
  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
   850
    by (simp add: 1)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   851
  then show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   852
    by (simp add: borel_measurable_iff_ge)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   853
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   854
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   855
lemma (in sigma_algebra) borel_measurable_setsum[simp, intro]:
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   856
  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   857
  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
   858
  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
   859
proof cases
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   860
  assume "finite S"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   861
  thus ?thesis using assms by induct auto
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   862
qed simp
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   863
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   864
lemma (in sigma_algebra) borel_measurable_square:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   865
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   866
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   867
  shows "(\<lambda>x. (f x)^2) \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   868
proof -
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   869
  {
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   870
    fix a
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   871
    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
   872
    proof (cases rule: linorder_cases [of a 0])
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   873
      case less
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   874
      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
   875
        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
   876
      also have "... \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   877
        by (rule empty_sets)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   878
      finally show ?thesis .
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   879
    next
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   880
      case equal
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   881
      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
   882
             {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
   883
        by auto
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   884
      also have "... \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   885
        apply (insert f)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   886
        apply (rule Int)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   887
        apply (simp add: borel_measurable_iff_le)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   888
        apply (simp add: borel_measurable_iff_ge)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   889
        done
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   890
      finally show ?thesis .
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   891
    next
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   892
      case greater
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   893
      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
   894
        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
   895
                  real_sqrt_le_iff real_sqrt_power)
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   896
      hence "{w \<in> space M. (f w)\<twosuperior> \<le> a} =
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   897
             {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
   898
        using greater by auto
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   899
      also have "... \<in> sets M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   900
        apply (insert f)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   901
        apply (rule Int)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   902
        apply (simp add: borel_measurable_iff_ge)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   903
        apply (simp add: borel_measurable_iff_le)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   904
        done
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   905
      finally show ?thesis .
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   906
    qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   907
  }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   908
  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
   909
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   910
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   911
lemma times_eq_sum_squares:
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   912
   fixes x::real
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   913
   shows"x*y = ((x+y)^2)/4 - ((x-y)^ 2)/4"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   914
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
   915
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   916
lemma (in sigma_algebra) borel_measurable_uminus[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   917
  fixes g :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   918
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   919
  shows "(\<lambda>x. - g x) \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   920
proof -
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   921
  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
   922
    by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   923
  also have "... \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   924
    by (fast intro: affine_borel_measurable g)
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   925
  finally show ?thesis .
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   926
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   927
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   928
lemma (in sigma_algebra) borel_measurable_times[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   929
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   930
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   931
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   932
  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
   933
proof -
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   934
  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
   935
    using assms by (fast intro: affine_borel_measurable borel_measurable_square)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   936
  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
   937
        (\<lambda>x. 0 + ((f x + -g x) ^ 2 * inverse -4))"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents: 35347
diff changeset
   938
    by (simp add: minus_divide_right)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   939
  also have "... \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   940
    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
   941
  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
   942
  show ?thesis
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   943
    apply (simp add: times_eq_sum_squares diff_minus)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   944
    using 1 2 by simp
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   945
qed
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   946
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   947
lemma (in sigma_algebra) borel_measurable_setprod[simp, intro]:
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   948
  fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> real"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   949
  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
   950
  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
   951
proof cases
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   952
  assume "finite S"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   953
  thus ?thesis using assms by induct auto
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   954
qed simp
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
   955
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   956
lemma (in sigma_algebra) borel_measurable_diff[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   957
  fixes f :: "'a \<Rightarrow> real"
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   958
  assumes f: "f \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   959
  assumes g: "g \<in> borel_measurable M"
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   960
  shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   961
  unfolding diff_minus using assms by fast
33533
40b44cb20c8c New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
diff changeset
   962
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   963
lemma (in sigma_algebra) borel_measurable_inverse[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   964
  fixes f :: "'a \<Rightarrow> real"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   965
  assumes "f \<in> borel_measurable M"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   966
  shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   967
  unfolding borel_measurable_iff_ge unfolding inverse_eq_divide
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   968
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   969
  fix a :: real
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   970
  have *: "{w \<in> space M. a \<le> 1 / f w} =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   971
      ({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
   972
      ({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
   973
      ({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
   974
  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
   975
    by (auto intro!: Int Un)
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   976
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   977
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   978
lemma (in sigma_algebra) borel_measurable_divide[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   979
  fixes f :: "'a \<Rightarrow> real"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   980
  assumes "f \<in> borel_measurable M"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   981
  and "g \<in> borel_measurable M"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   982
  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
   983
  unfolding field_divide_inverse
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   984
  by (rule borel_measurable_inverse borel_measurable_times assms)+
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   985
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   986
lemma (in sigma_algebra) borel_measurable_max[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   987
  fixes f g :: "'a \<Rightarrow> real"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   988
  assumes "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   989
  assumes "g \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   990
  shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   991
  unfolding borel_measurable_iff_le
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   992
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   993
  fix a
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   994
  have "{x \<in> space M. max (g x) (f x) \<le> a} =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   995
    {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
   996
  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
   997
    using assms unfolding borel_measurable_iff_le
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   998
    by (auto intro!: Int)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
   999
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1000
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1001
lemma (in sigma_algebra) borel_measurable_min[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1002
  fixes f g :: "'a \<Rightarrow> real"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1003
  assumes "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1004
  assumes "g \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1005
  shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1006
  unfolding borel_measurable_iff_ge
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1007
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1008
  fix a
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1009
  have "{x \<in> space M. a \<le> min (g x) (f x)} =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1010
    {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
  1011
  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
  1012
    using assms unfolding borel_measurable_iff_ge
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1013
    by (auto intro!: Int)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1014
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1015
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1016
lemma (in sigma_algebra) borel_measurable_abs[simp, intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1017
  assumes "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1018
  shows "(\<lambda>x. \<bar>f x :: real\<bar>) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1019
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1020
  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
  1021
  show ?thesis unfolding * using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1022
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1023
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
  1024
lemma borel_measurable_nth[simp, intro]:
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
  1025
  "(\<lambda>x::real^'n. x $ i) \<in> borel_measurable borel"
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
  1026
  using borel_measurable_euclidean_component
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
  1027
  unfolding nth_conv_component by auto
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41025
diff changeset
  1028
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1029
lemma borel_measurable_continuous_on1:
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1030
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1031
  assumes "continuous_on UNIV f"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1032
  shows "f \<in> borel_measurable borel"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1033
  apply(rule borel.borel_measurableI)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1034
  using continuous_open_preimage[OF assms] unfolding vimage_def by auto
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1035
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1036
lemma borel_measurable_continuous_on:
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1037
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t1_space"
42990
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1038
  assumes cont: "continuous_on A f" "open A"
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1039
  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
  1040
proof (rule borel.borel_measurableI)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1041
  fix S :: "'b set" assume "open S"
42990
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1042
  then have "open {x\<in>A. f x \<in> S}"
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1043
    by (intro continuous_open_preimage[OF cont]) auto
42990
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1044
  then have *: "{x\<in>A. f x \<in> S} \<in> sets borel" by auto
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1045
  have "?f -` S \<inter> space borel = 
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1046
    {x\<in>A. f x \<in> S} \<union> (if c \<in> S then space borel - A else {})"
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1047
    by (auto split: split_if_asm)
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1048
  also have "\<dots> \<in> sets borel"
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1049
    using * `open A` by (auto simp del: space_borel intro!: borel.Un)
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1050
  finally show "?f -` S \<inter> space borel \<in> sets borel" .
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1051
qed
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1052
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1053
lemma (in sigma_algebra) convex_measurable:
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1054
  fixes a b :: real
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1055
  assumes X: "X \<in> borel_measurable M" "X ` space M \<subseteq> { a <..< b}"
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1056
  assumes q: "convex_on { a <..< b} q"
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1057
  shows "q \<circ> X \<in> borel_measurable M"
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1058
proof -
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1059
  have "(\<lambda>x. if x \<in> {a <..< b} then q x else 0) \<in> borel_measurable borel"
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1060
  proof (rule borel_measurable_continuous_on)
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1061
    show "open {a<..<b}" by auto
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1062
    from this q show "continuous_on {a<..<b} q"
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1063
      by (rule convex_on_continuous)
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1064
  qed
42990
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1065
  then have "(\<lambda>x. if x \<in> {a <..< b} then q x else 0) \<circ> X \<in> borel_measurable M" (is ?qX)
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1066
    using X by (intro measurable_comp) auto
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1067
  moreover have "?qX \<longleftrightarrow> q \<circ> X \<in> borel_measurable M"
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1068
    using X by (intro measurable_cong) auto
3706951a6421 composition of convex and measurable function is measurable
hoelzl
parents: 42950
diff changeset
  1069
  ultimately show ?thesis by simp
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1070
qed
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1071
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1072
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
  1073
proof -
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1074
  { fix x :: real assume x: "x \<le> 0"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1075
    { 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
  1076
    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
  1077
      by (auto simp: ln_def log_def) }
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1078
  note log_imp = this
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1079
  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
  1080
  proof (rule borel_measurable_continuous_on)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1081
    show "continuous_on {0<..} (log b)"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1082
      by (auto intro!: continuous_at_imp_continuous_on DERIV_log DERIV_isCont
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1083
               simp: continuous_isCont[symmetric])
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1084
    show "open ({0<..}::real set)" by auto
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1085
  qed
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1086
  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
  1087
    by (simp add: fun_eq_iff not_less log_imp)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1088
  finally show ?thesis .
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1089
qed
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1090
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1091
lemma (in sigma_algebra) borel_measurable_log[simp,intro]:
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1092
  assumes f: "f \<in> borel_measurable M" and "1 < b"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1093
  shows "(\<lambda>x. log b (f x)) \<in> borel_measurable M"
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1094
  using measurable_comp[OF f borel_measurable_borel_log[OF `1 < b`]]
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1095
  by (simp add: comp_def)
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1096
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1097
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
  1098
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1099
lemma borel_measurable_ereal_borel:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1100
  "ereal \<in> borel_measurable borel"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1101
  unfolding borel_def[where 'a=ereal]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1102
proof (rule borel.measurable_sigma)
44537
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
  1103
  fix X :: "ereal set" assume "X \<in> sets \<lparr>space = UNIV, sets = {S. open S} \<rparr>"
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
  1104
  then have "open X" by simp
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1105
  then have "open (ereal -` X \<inter> space borel)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1106
    by (simp add: open_ereal_vimage)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1107
  then show "ereal -` X \<inter> space borel \<in> sets borel" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1108
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1109
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1110
lemma (in sigma_algebra) borel_measurable_ereal[simp, intro]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1111
  assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1112
  using measurable_comp[OF f borel_measurable_ereal_borel] unfolding comp_def .
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1113
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1114
lemma borel_measurable_real_of_ereal_borel:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1115
  "(real :: ereal \<Rightarrow> real) \<in> borel_measurable borel"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1116
  unfolding borel_def[where 'a=real]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1117
proof (rule borel.measurable_sigma)
44537
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
  1118
  fix B :: "real set" assume "B \<in> sets \<lparr>space = UNIV, sets = {S. open S} \<rparr>"
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
  1119
  then have "open B" by simp
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1120
  have *: "ereal -` real -` (B - {0}) = B - {0}" by auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1121
  have open_real: "open (real -` (B - {0}) :: ereal set)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1122
    unfolding open_ereal_def * using `open B` by auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1123
  show "(real -` B \<inter> space borel :: ereal set) \<in> sets borel"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1124
  proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1125
    assume "0 \<in> B"
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1126
    then have *: "real -` B = real -` (B - {0}) \<union> {-\<infinity>, \<infinity>, 0::ereal}"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1127
      by (auto simp add: real_of_ereal_eq_0)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1128
    then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1129
      using open_real by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1130
  next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1131
    assume "0 \<notin> B"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1132
    then have *: "(real -` B :: ereal set) = real -` (B - {0})"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1133
      by (auto simp add: real_of_ereal_eq_0)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1134
    then show "(real -` B :: ereal set) \<inter> space borel \<in> sets borel"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1135
      using open_real by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1136
  qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1137
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1138
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1139
lemma (in sigma_algebra) borel_measurable_real_of_ereal[simp, intro]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1140
  assumes f: "f \<in> borel_measurable M" shows "(\<lambda>x. real (f x :: ereal)) \<in> borel_measurable M"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1141
  using measurable_comp[OF f borel_measurable_real_of_ereal_borel] unfolding comp_def .
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1142
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1143
lemma (in sigma_algebra) borel_measurable_ereal_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1144
  shows "(\<lambda>x. ereal (f x)) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1145
proof
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1146
  assume "(\<lambda>x. ereal (f x)) \<in> borel_measurable M"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1147
  from borel_measurable_real_of_ereal[OF this]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1148
  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
  1149
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1150
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1151
lemma (in sigma_algebra) borel_measurable_ereal_iff_real:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1152
  fixes f :: "'a \<Rightarrow> ereal"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1153
  shows "f \<in> borel_measurable M \<longleftrightarrow>
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1154
    ((\<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
  1155
proof safe
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1156
  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
  1157
  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
  1158
  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
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1159
  let "?f x" = "if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1160
  have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1161
  also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1162
  finally show "f \<in> borel_measurable M" .
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1163
qed (auto intro: measurable_sets borel_measurable_real_of_ereal)
41830
719b0a517c33 log is borel measurable
hoelzl
parents: 41545
diff changeset
  1164
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1165
lemma (in sigma_algebra) less_eq_ge_measurable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1166
  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
  1167
  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
  1168
proof
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1169
  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
  1170
  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
  1171
  ultimately show "f -` {..a} \<inter> space M \<in> sets M" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1172
next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1173
  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
  1174
  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
  1175
  ultimately show "f -` {a <..} \<inter> space M \<in> sets M" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1176
qed
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1177
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1178
lemma (in sigma_algebra) greater_eq_le_measurable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1179
  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
  1180
  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
  1181
proof
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1182
  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
  1183
  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
  1184
  ultimately show "f -` {..< a} \<inter> space M \<in> sets M" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1185
next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1186
  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
  1187
  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
  1188
  ultimately show "f -` {a ..} \<inter> space M \<in> sets M" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1189
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1190
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1191
lemma (in sigma_algebra) borel_measurable_uminus_borel_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1192
  "(uminus :: ereal \<Rightarrow> ereal) \<in> borel_measurable borel"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1193
proof (subst borel_def, rule borel.measurable_sigma)
44537
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
  1194
  fix X :: "ereal set" assume "X \<in> sets \<lparr>space = UNIV, sets = {S. open S}\<rparr>"
c10485a6a7af make HOL-Probability respect set/pred distinction
huffman
parents: 44282
diff changeset
  1195
  then have "open X" by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1196
  have "uminus -` X = uminus ` X" by (force simp: image_iff)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1197
  then have "open (uminus -` X)" using `open X` ereal_open_uminus by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1198
  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
  1199
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1200
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1201
lemma (in sigma_algebra) borel_measurable_uminus_ereal[intro]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1202
  assumes "f \<in> borel_measurable M"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1203
  shows "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1204
  using measurable_comp[OF assms borel_measurable_uminus_borel_ereal] by (simp add: comp_def)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1205
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1206
lemma (in sigma_algebra) borel_measurable_uminus_eq_ereal[simp]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1207
  "(\<lambda>x. - f x :: ereal) \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable M" (is "?l = ?r")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 37887
diff changeset
  1208
proof
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1209
  assume ?l from borel_measurable_uminus_ereal[OF this] show ?r by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1210
qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1211
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1212
lemma (in sigma_algebra) borel_measurable_eq_atMost_ereal:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1213
  fixes f :: "'a \<Rightarrow> ereal"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1214
  shows "f \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {..a} \<inter> space M \<in> sets M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1215
proof (intro iffI allI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1216
  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
  1217
  show "f \<in> borel_measurable M"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1218
    unfolding borel_measurable_ereal_iff_real borel_measurable_iff_le
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1219
  proof (intro conjI allI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1220
    fix a :: real
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1221
    { fix x :: ereal assume *: "\<forall>i::nat. real i < x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1222
      have "x = \<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1223
      proof (rule ereal_top)
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44537
diff changeset
  1224
        fix B from reals_Archimedean2[of B] guess n ..
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1225
        then have "ereal B < real n" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1226
        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
  1227
      qed }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1228
    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
  1229
      by (auto simp: not_le)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1230
    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
  1231
    moreover
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1232
    have "{-\<infinity>::ereal} = {..-\<infinity>}" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1233
    then show "f -` {-\<infinity>} \<inter> space M \<in> sets M" using pos by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1234
    moreover have "{x\<in>space M. f x \<le> ereal a} \<in> sets M"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1235
      using pos[of "ereal a"] by (simp add: vimage_def Int_def conj_commute)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1236
    moreover have "{w \<in> space M. real (f w) \<le> a} =
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1237
      (if a < 0 then {w \<in> space M. f w \<le> ereal a} - f -` {-\<infinity>} \<inter> space M
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1238
      else {w \<in> space M. f w \<le> ereal a} \<union> (f -` {\<infinity>} \<inter> space M) \<union> (f -` {-\<infinity>} \<inter> space M))" (is "?l = ?r")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1239
      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
  1240
    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
  1241
  qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1242
qed (simp add: measurable_sets)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents: 35347
diff changeset
  1243
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1244
lemma (in sigma_algebra) borel_measurable_eq_atLeast_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff changeset
  1245
  "(f::'a \<Rightarrow> ereal) \<in> borel_measurable M \<longleftrightarrow> (\<forall>a. f -` {a..} \<inter> space M \<in> sets M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1246
proof
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41969
diff changeset
  1247
  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
  1248
  moreover have "\<And>a. (\<lambda>x. - f x) -` {..a} = f -` {-a ..}"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42990
diff<