src/HOL/Probability/Lebesgue_Integration.thy
author hoelzl
Tue, 24 Aug 2010 14:41:37 +0200
changeset 38705 aaee86c0e237
parent 38656 d5d342611edb
child 39092 98de40859858
permissions -rw-r--r--
moved generic lemmas in Probability to HOL
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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d5d342611edb Rewrite the Probability theory.
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parents: 38642
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     1
(* Author: Armin Heller, Johannes Hoelzl, TU Muenchen *)
d5d342611edb Rewrite the Probability theory.
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b16d99a72dc9 Add Lebesgue integral and probability space.
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header {*Lebesgue Integration*}
b16d99a72dc9 Add Lebesgue integral and probability space.
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theory Lebesgue_Integration
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
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imports Measure Borel
b16d99a72dc9 Add Lebesgue integral and probability space.
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begin
b16d99a72dc9 Add Lebesgue integral and probability space.
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     9
section "@{text \<mu>}-null sets"
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    10
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    11
abbreviation (in measure_space) "null_sets \<equiv> {N\<in>sets M. \<mu> N = 0}"
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    12
38656
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    13
lemma sums_If_finite:
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    14
  assumes finite: "finite {r. P r}"
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    15
  shows "(\<lambda>r. if P r then f r else 0) sums (\<Sum>r\<in>{r. P r}. f r)" (is "?F sums _")
d5d342611edb Rewrite the Probability theory.
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    16
proof cases
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    17
  assume "{r. P r} = {}" hence "\<And>r. \<not> P r" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    18
  thus ?thesis by (simp add: sums_zero)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    19
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    20
  assume not_empty: "{r. P r} \<noteq> {}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    21
  have "?F sums (\<Sum>r = 0..< Suc (Max {r. P r}). ?F r)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    22
    by (rule series_zero)
d5d342611edb Rewrite the Probability theory.
hoelzl
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    23
       (auto simp add: Max_less_iff[OF finite not_empty] less_eq_Suc_le[symmetric])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    24
  also have "(\<Sum>r = 0..< Suc (Max {r. P r}). ?F r) = (\<Sum>r\<in>{r. P r}. f r)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    25
    by (subst setsum_cases)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    26
       (auto intro!: setsum_cong simp: Max_ge_iff[OF finite not_empty] less_Suc_eq_le)
d5d342611edb Rewrite the Probability theory.
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    27
  finally show ?thesis .
d5d342611edb Rewrite the Probability theory.
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    28
qed
35582
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parents:
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    29
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lemma sums_single:
d5d342611edb Rewrite the Probability theory.
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    31
  "(\<lambda>r. if r = i then f r else 0) sums f i"
d5d342611edb Rewrite the Probability theory.
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    32
  using sums_If_finite[of "\<lambda>r. r = i" f] by simp
d5d342611edb Rewrite the Probability theory.
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    33
d5d342611edb Rewrite the Probability theory.
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    34
section "Simple function"
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text {*
d5d342611edb Rewrite the Probability theory.
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d5d342611edb Rewrite the Probability theory.
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    38
Our simple functions are not restricted to positive real numbers. Instead
d5d342611edb Rewrite the Probability theory.
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they are just functions with a finite range and are measurable when singleton
d5d342611edb Rewrite the Probability theory.
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sets are measurable.
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d5d342611edb Rewrite the Probability theory.
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*}
d5d342611edb Rewrite the Probability theory.
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    43
d5d342611edb Rewrite the Probability theory.
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    44
definition (in sigma_algebra) "simple_function g \<longleftrightarrow>
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    45
    finite (g ` space M) \<and>
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parents: 38642
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    46
    (\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
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38656
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    48
lemma (in sigma_algebra) simple_functionD:
d5d342611edb Rewrite the Probability theory.
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    49
  assumes "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    50
  shows "finite (g ` space M)"
d5d342611edb Rewrite the Probability theory.
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    51
  "x \<in> g ` space M \<Longrightarrow> g -` {x} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    52
  using assms unfolding simple_function_def by auto
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
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    53
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    54
lemma (in sigma_algebra) simple_function_indicator_representation:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    55
  fixes f ::"'a \<Rightarrow> pinfreal"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    56
  assumes f: "simple_function f" and x: "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    57
  shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    58
  (is "?l = ?r")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    59
proof -
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
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    60
  have "?r = (\<Sum>y \<in> f ` space M.
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    61
    (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    62
    by (auto intro!: setsum_cong2)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    63
  also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    64
    using assms by (auto dest: simple_functionD simp: setsum_delta)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    65
  also have "... = f x" using x by (auto simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    66
  finally show ?thesis by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    67
qed
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
    68
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    69
lemma (in measure_space) simple_function_notspace:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    70
  "simple_function (\<lambda>x. h x * indicator (- space M) x::pinfreal)" (is "simple_function ?h")
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
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    71
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    72
  have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    73
  hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    74
  have "?h -` {0} \<inter> space M = space M" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    75
  thus ?thesis unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    76
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    77
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    78
lemma (in sigma_algebra) simple_function_cong:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    79
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    80
  shows "simple_function f \<longleftrightarrow> simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    81
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    82
  have "f ` space M = g ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    83
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    84
    using assms by (auto intro!: image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    85
  thus ?thesis unfolding simple_function_def using assms by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    86
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    87
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    88
lemma (in sigma_algebra) borel_measurable_simple_function:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    89
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    90
  shows "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    91
proof (rule borel_measurableI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    92
  fix S
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    93
  let ?I = "f ` (f -` S \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    94
  have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    95
  have "finite ?I"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    96
    using assms unfolding simple_function_def by (auto intro: finite_subset)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    97
  hence "?U \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    98
    apply (rule finite_UN)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    99
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   100
  thus "f -` S \<inter> space M \<in> sets M" unfolding * .
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   101
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   102
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   103
lemma (in sigma_algebra) simple_function_borel_measurable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   104
  fixes f :: "'a \<Rightarrow> 'x::t2_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   105
  assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   106
  shows "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   107
  using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   108
  by (auto intro: borel_measurable_vimage)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   109
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   110
lemma (in sigma_algebra) simple_function_const[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   111
  "simple_function (\<lambda>x. c)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   112
  by (auto intro: finite_subset simp: simple_function_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   113
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   114
lemma (in sigma_algebra) simple_function_compose[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   115
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   116
  shows "simple_function (g \<circ> f)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   117
  unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   118
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   119
  show "finite ((g \<circ> f) ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   120
    using assms unfolding simple_function_def by (auto simp: image_compose)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   121
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   122
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   123
  let ?G = "g -` {g (f x)} \<inter> (f`space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   124
  have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   125
    (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   126
  show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   127
    using assms unfolding simple_function_def *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   128
    by (rule_tac finite_UN) (auto intro!: finite_UN)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   129
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   130
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   131
lemma (in sigma_algebra) simple_function_indicator[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   132
  assumes "A \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   133
  shows "simple_function (indicator A)"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   134
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   135
  have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   136
    by (auto simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   137
  hence "finite ?S" by (rule finite_subset) simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   138
  moreover have "- A \<inter> space M = space M - A" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   139
  ultimately show ?thesis unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   140
    using assms by (auto simp: indicator_def_raw)
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   141
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   142
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   143
lemma (in sigma_algebra) simple_function_Pair[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   144
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   145
  assumes "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   146
  shows "simple_function (\<lambda>x. (f x, g x))" (is "simple_function ?p")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   147
  unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   148
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   149
  show "finite (?p ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   150
    using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   151
    by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   152
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   153
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   154
  have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   155
      (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   156
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   157
  with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   158
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   159
qed
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   160
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   161
lemma (in sigma_algebra) simple_function_compose1:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   162
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   163
  shows "simple_function (\<lambda>x. g (f x))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   164
  using simple_function_compose[OF assms, of g]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   165
  by (simp add: comp_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   166
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   167
lemma (in sigma_algebra) simple_function_compose2:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   168
  assumes "simple_function f" and "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   169
  shows "simple_function (\<lambda>x. h (f x) (g x))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   170
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   171
  have "simple_function ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   172
    using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   173
  thus ?thesis by (simp_all add: comp_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   174
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   175
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   176
lemmas (in sigma_algebra) simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   177
  and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   178
  and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   179
  and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   180
  and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   181
  and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   182
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   183
lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   184
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   185
  shows "simple_function (\<lambda>x. \<Sum>i\<in>P. f i x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   186
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   187
  assume "finite P" from this assms show ?thesis by induct auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   188
qed auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   189
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   190
lemma (in sigma_algebra) simple_function_le_measurable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   191
  assumes "simple_function f" "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   192
  shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   193
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   194
  have *: "{x \<in> space M. f x \<le> g x} =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   195
    (\<Union>(F, G)\<in>f`space M \<times> g`space M.
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   196
      if F \<le> G then (f -` {F} \<inter> space M) \<inter> (g -` {G} \<inter> space M) else {})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   197
    apply (auto split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   198
    apply (rule_tac x=x in bexI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   199
    apply (rule_tac x=x in bexI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   200
    by simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   201
  have **: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   202
    (f -` {f x} \<inter> space M) \<inter> (g -` {g y} \<inter> space M) \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   203
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   204
  have "finite (f`space M \<times> g`space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   205
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   206
  thus ?thesis unfolding *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   207
    apply (rule finite_UN)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   208
    using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   209
    by (auto intro!: **)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   210
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   211
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   212
lemma setsum_indicator_disjoint_family:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   213
  fixes f :: "'d \<Rightarrow> 'e::semiring_1"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   214
  assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   215
  shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   216
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   217
  have "P \<inter> {i. x \<in> A i} = {j}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   218
    using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   219
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   220
  thus ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   221
    unfolding indicator_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   222
    by (simp add: if_distrib setsum_cases[OF `finite P`])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   223
qed
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   224
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   225
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   226
  fixes u :: "'a \<Rightarrow> pinfreal"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   227
  assumes u: "u \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   228
  shows "\<exists>f. (\<forall>i. simple_function (f i) \<and> (\<forall>x\<in>space M. f i x \<noteq> \<omega>)) \<and> f \<up> u"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   229
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   230
  have "\<exists>f. \<forall>x j. (of_nat j \<le> u x \<longrightarrow> f x j = j*2^j) \<and>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   231
    (u x < of_nat j \<longrightarrow> of_nat (f x j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f x j)))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   232
    (is "\<exists>f. \<forall>x j. ?P x j (f x j)")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   233
  proof(rule choice, rule, rule choice, rule)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   234
    fix x j show "\<exists>n. ?P x j n"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   235
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   236
      assume *: "u x < of_nat j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   237
      then obtain r where r: "u x = Real r" "0 \<le> r" by (cases "u x") auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   238
      from reals_Archimedean6a[of "r * 2^j"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   239
      obtain n :: nat where "real n \<le> r * 2 ^ j" "r * 2 ^ j < real (Suc n)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   240
        using `0 \<le> r` by (auto simp: zero_le_power zero_le_mult_iff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   241
      thus ?thesis using r * by (auto intro!: exI[of _ n])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   242
    qed auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   243
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   244
  then obtain f where top: "\<And>j x. of_nat j \<le> u x \<Longrightarrow> f x j = j*2^j" and
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   245
    upper: "\<And>j x. u x < of_nat j \<Longrightarrow> of_nat (f x j) \<le> u x * 2^j" and
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   246
    lower: "\<And>j x. u x < of_nat j \<Longrightarrow> u x * 2^j < of_nat (Suc (f x j))" by blast
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   247
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   248
  { fix j x P
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   249
    assume 1: "of_nat j \<le> u x \<Longrightarrow> P (j * 2^j)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   250
    assume 2: "\<And>k. \<lbrakk> u x < of_nat j ; of_nat k \<le> u x * 2^j ; u x * 2^j < of_nat (Suc k) \<rbrakk> \<Longrightarrow> P k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   251
    have "P (f x j)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   252
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   253
      assume "of_nat j \<le> u x" thus "P (f x j)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   254
        using top[of j x] 1 by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   255
    next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   256
      assume "\<not> of_nat j \<le> u x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   257
      hence "u x < of_nat j" "of_nat (f x j) \<le> u x * 2^j" "u x * 2^j < of_nat (Suc (f x j))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   258
        using upper lower by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   259
      from 2[OF this] show "P (f x j)" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   260
    qed }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   261
  note fI = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   262
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   263
  { fix j x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   264
    have "f x j = j * 2 ^ j \<longleftrightarrow> of_nat j \<le> u x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   265
      by (rule fI, simp, cases "u x") (auto split: split_if_asm) }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   266
  note f_eq = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   267
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   268
  { fix j x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   269
    have "f x j \<le> j * 2 ^ j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   270
    proof (rule fI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   271
      fix k assume *: "u x < of_nat j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   272
      assume "of_nat k \<le> u x * 2 ^ j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   273
      also have "\<dots> \<le> of_nat (j * 2^j)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   274
        using * by (cases "u x") (auto simp: zero_le_mult_iff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   275
      finally show "k \<le> j*2^j" by (auto simp del: real_of_nat_mult)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   276
    qed simp }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   277
  note f_upper = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   278
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   279
  let "?g j x" = "of_nat (f x j) / 2^j :: pinfreal"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   280
  show ?thesis unfolding simple_function_def isoton_fun_expand unfolding isoton_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   281
  proof (safe intro!: exI[of _ ?g])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   282
    fix j
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   283
    have *: "?g j ` space M \<subseteq> (\<lambda>x. of_nat x / 2^j) ` {..j * 2^j}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   284
      using f_upper by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   285
    thus "finite (?g j ` space M)" by (rule finite_subset) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   286
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   287
    fix j t assume "t \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   288
    have **: "?g j -` {?g j t} \<inter> space M = {x \<in> space M. f t j = f x j}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   289
      by (auto simp: power_le_zero_eq Real_eq_Real mult_le_0_iff)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   290
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   291
    show "?g j -` {?g j t} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   292
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   293
      assume "of_nat j \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   294
      hence "?g j -` {?g j t} \<inter> space M = {x\<in>space M. of_nat j \<le> u x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   295
        unfolding ** f_eq[symmetric] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   296
      thus "?g j -` {?g j t} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   297
        using u by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   298
    next
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   299
      assume not_t: "\<not> of_nat j \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   300
      hence less: "f t j < j*2^j" using f_eq[symmetric] `f t j \<le> j*2^j` by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   301
      have split_vimage: "?g j -` {?g j t} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   302
          {x\<in>space M. of_nat (f t j)/2^j \<le> u x} \<inter> {x\<in>space M. u x < of_nat (Suc (f t j))/2^j}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   303
        unfolding **
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   304
      proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   305
        fix x assume [simp]: "f t j = f x j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   306
        have *: "\<not> of_nat j \<le> u x" using not_t f_eq[symmetric] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   307
        hence "of_nat (f t j) \<le> u x * 2^j \<and> u x * 2^j < of_nat (Suc (f t j))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   308
          using upper lower by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   309
        hence "of_nat (f t j) / 2^j \<le> u x \<and> u x < of_nat (Suc (f t j))/2^j" using *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   310
          by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   311
        thus "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   312
      next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   313
        fix x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   314
        assume "of_nat (f t j) / 2^j \<le> u x" "u x < of_nat (Suc (f t j))/2^j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   315
        hence "of_nat (f t j) \<le> u x * 2 ^ j \<and> u x * 2 ^ j < of_nat (Suc (f t j))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   316
          by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   317
        hence 1: "of_nat (f t j) \<le> u x * 2 ^ j" and 2: "u x * 2 ^ j < of_nat (Suc (f t j))" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   318
        note 2
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   319
        also have "\<dots> \<le> of_nat (j*2^j)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   320
          using less by (auto simp: zero_le_mult_iff simp del: real_of_nat_mult)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   321
        finally have bound_ux: "u x < of_nat j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   322
          by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   323
        show "f t j = f x j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   324
        proof (rule antisym)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   325
          from 1 lower[OF bound_ux]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   326
          show "f t j \<le> f x j" by (cases "u x") (auto split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   327
          from upper[OF bound_ux] 2
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   328
          show "f x j \<le> f t j" by (cases "u x") (auto split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   329
        qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   330
      qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   331
      show ?thesis unfolding split_vimage using u by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   332
    qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   333
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   334
    fix j t assume "?g j t = \<omega>" thus False by (auto simp: power_le_zero_eq)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   335
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   336
    fix t
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   337
    { fix i
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   338
      have "f t i * 2 \<le> f t (Suc i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   339
      proof (rule fI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   340
        assume "of_nat (Suc i) \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   341
        hence "of_nat i \<le> u t" by (cases "u t") auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   342
        thus "f t i * 2 \<le> Suc i * 2 ^ Suc i" unfolding f_eq[symmetric] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   343
      next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   344
        fix k
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   345
        assume *: "u t * 2 ^ Suc i < of_nat (Suc k)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   346
        show "f t i * 2 \<le> k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   347
        proof (rule fI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   348
          assume "of_nat i \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   349
          hence "of_nat (i * 2^Suc i) \<le> u t * 2^Suc i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   350
            by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   351
          also have "\<dots> < of_nat (Suc k)" using * by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   352
          finally show "i * 2 ^ i * 2 \<le> k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   353
            by (auto simp del: real_of_nat_mult)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   354
        next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   355
          fix j assume "of_nat j \<le> u t * 2 ^ i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   356
          with * show "j * 2 \<le> k" by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   357
        qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   358
      qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   359
      thus "?g i t \<le> ?g (Suc i) t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   360
        by (auto simp: zero_le_mult_iff power_le_zero_eq divide_real_def[symmetric] field_simps) }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   361
    hence mono: "mono (\<lambda>i. ?g i t)" unfolding mono_iff_le_Suc by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   362
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   363
    show "(SUP j. of_nat (f t j) / 2 ^ j) = u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   364
    proof (rule pinfreal_SUPI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   365
      fix j show "of_nat (f t j) / 2 ^ j \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   366
      proof (rule fI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   367
        assume "of_nat j \<le> u t" thus "of_nat (j * 2 ^ j) / 2 ^ j \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   368
          by (cases "u t") (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   369
      next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   370
        fix k assume "of_nat k \<le> u t * 2 ^ j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   371
        thus "of_nat k / 2 ^ j \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   372
          by (cases "u t")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   373
             (auto simp: power_le_zero_eq divide_real_def[symmetric] field_simps zero_le_mult_iff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   374
      qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   375
    next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   376
      fix y :: pinfreal assume *: "\<And>j. j \<in> UNIV \<Longrightarrow> of_nat (f t j) / 2 ^ j \<le> y"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   377
      show "u t \<le> y"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   378
      proof (cases "u t")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   379
        case (preal r)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   380
        show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   381
        proof (rule ccontr)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   382
          assume "\<not> u t \<le> y"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   383
          then obtain p where p: "y = Real p" "0 \<le> p" "p < r" using preal by (cases y) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   384
          with LIMSEQ_inverse_realpow_zero[of 2, unfolded LIMSEQ_iff, rule_format, of "r - p"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   385
          obtain n where n: "\<And>N. n \<le> N \<Longrightarrow> inverse (2^N) < r - p" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   386
          let ?N = "max n (natfloor r + 1)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   387
          have "u t < of_nat ?N" "n \<le> ?N"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   388
            using ge_natfloor_plus_one_imp_gt[of r n] preal
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
   389
            using real_natfloor_add_one_gt
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
   390
            by (auto simp: max_def real_of_nat_Suc)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   391
          from lower[OF this(1)]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   392
          have "r < (real (f t ?N) + 1) / 2 ^ ?N" unfolding less_divide_eq
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   393
            using preal by (auto simp add: power_le_zero_eq zero_le_mult_iff real_of_nat_Suc split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   394
          hence "u t < of_nat (f t ?N) / 2 ^ ?N + 1 / 2 ^ ?N"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   395
            using preal by (auto simp: field_simps divide_real_def[symmetric])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   396
          with n[OF `n \<le> ?N`] p preal *[of ?N]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   397
          show False
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   398
            by (cases "f t ?N") (auto simp: power_le_zero_eq split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   399
        qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   400
      next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   401
        case infinite
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   402
        { fix j have "f t j = j*2^j" using top[of j t] infinite by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   403
          hence "of_nat j \<le> y" using *[of j]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   404
            by (cases y) (auto simp: power_le_zero_eq zero_le_mult_iff field_simps) }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   405
        note all_less_y = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   406
        show ?thesis unfolding infinite
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   407
        proof (rule ccontr)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   408
          assume "\<not> \<omega> \<le> y" then obtain r where r: "y = Real r" "0 \<le> r" by (cases y) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   409
          moreover obtain n ::nat where "r < real n" using ex_less_of_nat by (auto simp: real_eq_of_nat)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   410
          with all_less_y[of n] r show False by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   411
        qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   412
      qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   413
    qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   414
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   415
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   416
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   417
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   418
  fixes u :: "'a \<Rightarrow> pinfreal"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   419
  assumes "u \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   420
  obtains (x) f where "f \<up> u" "\<And>i. simple_function (f i)" "\<And>i. \<omega>\<notin>f i`space M"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   421
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   422
  from borel_measurable_implies_simple_function_sequence[OF assms]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   423
  obtain f where x: "\<And>i. simple_function (f i)" "f \<up> u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   424
    and fin: "\<And>i. \<And>x. x\<in>space M \<Longrightarrow> f i x \<noteq> \<omega>" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   425
  { fix i from fin[of _ i] have "\<omega> \<notin> f i`space M" by fastsimp }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   426
  with x show thesis by (auto intro!: that[of f])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   427
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   428
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   429
section "Simple integral"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   430
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   431
definition (in measure_space)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   432
  "simple_integral f = (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M))"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   433
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   434
lemma (in measure_space) simple_integral_cong:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   435
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   436
  shows "simple_integral f = simple_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   437
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   438
  have "f ` space M = g ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   439
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   440
    using assms by (auto intro!: image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   441
  thus ?thesis unfolding simple_integral_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   442
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   443
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   444
lemma (in measure_space) simple_integral_const[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   445
  "simple_integral (\<lambda>x. c) = c * \<mu> (space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   446
proof (cases "space M = {}")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   447
  case True thus ?thesis unfolding simple_integral_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   448
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   449
  case False hence "(\<lambda>x. c) ` space M = {c}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   450
  thus ?thesis unfolding simple_integral_def by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   451
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   452
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   453
lemma (in measure_space) simple_function_partition:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   454
  assumes "simple_function f" and "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   455
  shows "simple_integral f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. contents (f`A) * \<mu> A)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   456
    (is "_ = setsum _ (?p ` space M)")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   457
proof-
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   458
  let "?sub x" = "?p ` (f -` {x} \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   459
  let ?SIGMA = "Sigma (f`space M) ?sub"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   460
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   461
  have [intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   462
    "finite (f ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   463
    "finite (g ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   464
    using assms unfolding simple_function_def by simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   465
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   466
  { fix A
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   467
    have "?p ` (A \<inter> space M) \<subseteq>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   468
      (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   469
      by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   470
    hence "finite (?p ` (A \<inter> space M))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   471
      by (rule finite_subset) (auto intro: finite_SigmaI finite_imageI) }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   472
  note this[intro, simp]
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   473
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   474
  { fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   475
    have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   476
    moreover {
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   477
      fix x y
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   478
      have *: "f -` {f x} \<inter> g -` {g x} \<inter> space M
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   479
          = (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   480
      assume "x \<in> space M" "y \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   481
      hence "f -` {f x} \<inter> g -` {g x} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   482
        using assms unfolding simple_function_def * by auto }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   483
    ultimately
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   484
    have "\<mu> (f -` {f x} \<inter> space M) = setsum (\<mu>) (?sub (f x))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   485
      by (subst measure_finitely_additive) auto }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   486
  hence "simple_integral f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   487
    unfolding simple_integral_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   488
    by (subst setsum_Sigma[symmetric],
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   489
       auto intro!: setsum_cong simp: setsum_right_distrib[symmetric])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   490
  also have "\<dots> = (\<Sum>A\<in>?p ` space M. contents (f`A) * \<mu> A)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   491
  proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   492
    have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   493
    have "(\<lambda>A. (contents (f ` A), A)) ` ?p ` space M
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   494
      = (\<lambda>x. (f x, ?p x)) ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   495
    proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   496
      fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   497
      thus "(f x, ?p x) \<in> (\<lambda>A. (contents (f`A), A)) ` ?p ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   498
        by (auto intro!: image_eqI[of _ _ "?p x"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   499
    qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   500
    thus ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   501
      apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (contents (f`A), A)"] inj_onI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   502
      apply (rule_tac x="xa" in image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   503
      by simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   504
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   505
  finally show ?thesis .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   506
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   507
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   508
lemma (in measure_space) simple_integral_add[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   509
  assumes "simple_function f" and "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   510
  shows "simple_integral (\<lambda>x. f x + g x) = simple_integral f + simple_integral g"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   511
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   512
  { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   513
    assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   514
    hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   515
        "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   516
      by auto }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   517
  thus ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   518
    unfolding
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   519
      simple_function_partition[OF simple_function_add[OF assms] simple_function_Pair[OF assms]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   520
      simple_function_partition[OF `simple_function f` `simple_function g`]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   521
      simple_function_partition[OF `simple_function g` `simple_function f`]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   522
    apply (subst (3) Int_commute)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   523
    by (auto simp add: field_simps setsum_addf[symmetric] intro!: setsum_cong)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   524
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   525
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   526
lemma (in measure_space) simple_integral_setsum[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   527
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   528
  shows "simple_integral (\<lambda>x. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. simple_integral (f i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   529
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   530
  assume "finite P"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   531
  from this assms show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   532
    by induct (auto simp: simple_function_setsum simple_integral_add)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   533
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   534
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   535
lemma (in measure_space) simple_integral_mult[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   536
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   537
  shows "simple_integral (\<lambda>x. c * f x) = c * simple_integral f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   538
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   539
  note mult = simple_function_mult[OF simple_function_const[of c] assms]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   540
  { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   541
    assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   542
    hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   543
      by auto }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   544
  thus ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   545
    unfolding simple_function_partition[OF mult assms]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   546
      simple_function_partition[OF assms mult]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   547
    by (auto simp: setsum_right_distrib field_simps intro!: setsum_cong)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   548
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   549
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   550
lemma (in measure_space) simple_integral_mono:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   551
  assumes "simple_function f" and "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   552
  and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   553
  shows "simple_integral f \<le> simple_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   554
  unfolding
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   555
    simple_function_partition[OF `simple_function f` `simple_function g`]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   556
    simple_function_partition[OF `simple_function g` `simple_function f`]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   557
  apply (subst Int_commute)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   558
proof (safe intro!: setsum_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   559
  fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   560
  assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   561
  hence "f ` ?S = {f x}" "g ` ?S = {g x}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   562
  thus "contents (f`?S) * \<mu> ?S \<le> contents (g`?S) * \<mu> ?S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   563
    using mono[OF `x \<in> space M`] by (auto intro!: mult_right_mono)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   564
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   565
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   566
lemma (in measure_space) simple_integral_indicator:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   567
  assumes "A \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   568
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   569
  shows "simple_integral (\<lambda>x. f x * indicator A x) =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   570
    (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   571
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   572
  assume "A = space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   573
  moreover hence "simple_integral (\<lambda>x. f x * indicator A x) = simple_integral f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   574
    by (auto intro!: simple_integral_cong)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   575
  moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   576
  ultimately show ?thesis by (simp add: simple_integral_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   577
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   578
  assume "A \<noteq> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   579
  then obtain x where x: "x \<in> space M" "x \<notin> A" using sets_into_space[OF assms(1)] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   580
  have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   581
  proof safe
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   582
    fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   583
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   584
    fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   585
      using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   586
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   587
    show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   588
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   589
  have *: "simple_integral (\<lambda>x. f x * indicator A x) =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   590
    (\<Sum>x \<in> f ` space M \<union> {0}. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   591
    unfolding simple_integral_def I
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   592
  proof (rule setsum_mono_zero_cong_left)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   593
    show "finite (f ` space M \<union> {0})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   594
      using assms(2) unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   595
    show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   596
      using sets_into_space[OF assms(1)] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   597
    have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}" by (auto simp: image_iff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   598
    thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   599
      i * \<mu> (f -` {i} \<inter> space M \<inter> A) = 0" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   600
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   601
    fix x assume "x \<in> f`A \<union> {0}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   602
    hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   603
      by (auto simp: indicator_def split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   604
    thus "x * \<mu> (?I -` {x} \<inter> space M) =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   605
      x * \<mu> (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   606
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   607
  show ?thesis unfolding *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   608
    using assms(2) unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   609
    by (auto intro!: setsum_mono_zero_cong_right)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   610
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   611
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   612
lemma (in measure_space) simple_integral_indicator_only[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   613
  assumes "A \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   614
  shows "simple_integral (indicator A) = \<mu> A"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   615
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   616
  assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   617
  thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   618
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   619
  assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::pinfreal}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   620
  thus ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   621
    using simple_integral_indicator[OF assms simple_function_const[of 1]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   622
    using sets_into_space[OF assms]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   623
    by (auto intro!: arg_cong[where f="\<mu>"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   624
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   625
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   626
lemma (in measure_space) simple_integral_null_set:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   627
  assumes "simple_function u" "N \<in> null_sets"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   628
  shows "simple_integral (\<lambda>x. u x * indicator N x) = 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   629
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   630
  have "simple_integral (\<lambda>x. u x * indicator N x) \<le>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   631
    simple_integral (\<lambda>x. \<omega> * indicator N x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   632
    using assms
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   633
    by (safe intro!: simple_integral_mono simple_function_mult simple_function_indicator simple_function_const) simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   634
  also have "... = 0" apply(subst simple_integral_mult)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   635
    using assms(2) by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   636
  finally show ?thesis by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   637
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   638
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   639
lemma (in measure_space) simple_integral_cong':
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   640
  assumes f: "simple_function f" and g: "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   641
  and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   642
  shows "simple_integral f = simple_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   643
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   644
  let ?h = "\<lambda>h. \<lambda>x. (h x * indicator {x\<in>space M. f x = g x} x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   645
    + h x * indicator {x\<in>space M. f x \<noteq> g x} x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   646
    + h x * indicator (-space M) x::pinfreal)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   647
  have *:"\<And>h. h = ?h h" unfolding indicator_def apply rule by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   648
  have mea_neq:"{x \<in> space M. f x \<noteq> g x} \<in> sets M" using f g by (auto simp: borel_measurable_simple_function)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   649
  then have mea_nullset: "{x \<in> space M. f x \<noteq> g x} \<in> null_sets" using mea by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   650
  have h1:"\<And>h::_=>pinfreal. simple_function h \<Longrightarrow>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   651
    simple_function (\<lambda>x. h x * indicator {x\<in>space M. f x = g x} x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   652
    apply(safe intro!: simple_function_add simple_function_mult simple_function_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   653
    using f g by (auto simp: borel_measurable_simple_function)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   654
  have h2:"\<And>h::_\<Rightarrow>pinfreal. simple_function h \<Longrightarrow>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   655
    simple_function (\<lambda>x. h x * indicator {x\<in>space M. f x \<noteq> g x} x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   656
    apply(safe intro!: simple_function_add simple_function_mult simple_function_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   657
    by(rule mea_neq)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   658
  have **:"\<And>a b c d e f. a = b \<Longrightarrow> c = d \<Longrightarrow> e = f \<Longrightarrow> a+c+e = b+d+f" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   659
  note *** = simple_integral_add[OF simple_function_add[OF h1 h2] simple_function_notspace]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   660
    simple_integral_add[OF h1 h2]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   661
  show ?thesis apply(subst *[of g]) apply(subst *[of f])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   662
    unfolding ***[OF f f] ***[OF g g]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   663
  proof(rule **) case goal1 show ?case apply(rule arg_cong[where f=simple_integral]) apply rule 
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   664
      unfolding indicator_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   665
  next note * = simple_integral_null_set[OF _ mea_nullset]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   666
    case goal2 show ?case unfolding *[OF f] *[OF g] ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   667
  next case goal3 show ?case apply(rule simple_integral_cong) by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   668
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   669
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   670
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   671
section "Continuous posititve integration"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   672
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   673
definition (in measure_space)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   674
  "positive_integral f =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   675
    (SUP g : {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}. simple_integral g)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   676
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   677
lemma (in measure_space) positive_integral_alt1:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   678
  "positive_integral f =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   679
    (SUP g : {g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}. simple_integral g)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   680
  unfolding positive_integral_def SUPR_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   681
proof (safe intro!: arg_cong[where f=Sup])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   682
  fix g let ?g = "\<lambda>x. if x \<in> space M then g x else f x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   683
  assume "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   684
  hence "?g \<le> f" "simple_function ?g" "simple_integral g = simple_integral ?g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   685
    "\<omega> \<notin> g`space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   686
    unfolding le_fun_def by (auto cong: simple_function_cong simple_integral_cong)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   687
  thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   688
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   689
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   690
  fix g assume "simple_function g" "g \<le> f" "\<omega> \<notin> g`space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   691
  hence "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   692
    by (auto simp add: le_fun_def image_iff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   693
  thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>)}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   694
    by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   695
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   696
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   697
lemma (in measure_space) positive_integral_alt:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   698
  "positive_integral f =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   699
    (SUP g : {g. simple_function g \<and> g \<le> f}. simple_integral g)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   700
  apply(rule order_class.antisym) unfolding positive_integral_def 
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   701
  apply(rule SUPR_subset) apply blast apply(rule SUPR_mono_lim)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   702
proof safe fix u assume u:"simple_function u" and uf:"u \<le> f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   703
  let ?u = "\<lambda>n x. if u x = \<omega> then Real (real (n::nat)) else u x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   704
  have su:"\<And>n. simple_function (?u n)" using simple_function_compose1[OF u] .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   705
  show "\<exists>b. \<forall>n. b n \<in> {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g ` space M} \<and>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   706
    (\<lambda>n. simple_integral (b n)) ----> simple_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   707
    apply(rule_tac x="?u" in exI, safe) apply(rule su)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   708
  proof- fix n::nat have "?u n \<le> u" unfolding le_fun_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   709
    also note uf finally show "?u n \<le> f" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   710
    let ?s = "{x \<in> space M. u x = \<omega>}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   711
    show "(\<lambda>n. simple_integral (?u n)) ----> simple_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   712
    proof(cases "\<mu> ?s = 0")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   713
      case True have *:"\<And>n. {x\<in>space M. ?u n x \<noteq> u x} = {x\<in>space M. u x = \<omega>}" by auto 
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   714
      have *:"\<And>n. simple_integral (?u n) = simple_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   715
        apply(rule simple_integral_cong'[OF su u]) unfolding * True ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   716
      show ?thesis unfolding * by auto 
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   717
    next case False note m0=this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   718
      have s:"{x \<in> space M. u x = \<omega>} \<in> sets M" using u  by (auto simp: borel_measurable_simple_function)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   719
      have "\<omega> = simple_integral (\<lambda>x. \<omega> * indicator {x\<in>space M. u x = \<omega>} x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   720
        apply(subst simple_integral_mult) using s
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   721
        unfolding simple_integral_indicator_only[OF s] using False by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   722
      also have "... \<le> simple_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   723
        apply (rule simple_integral_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   724
        apply (rule simple_function_mult)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   725
        apply (rule simple_function_const)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   726
        apply(rule ) prefer 3 apply(subst indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   727
        using s u by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   728
      finally have *:"simple_integral u = \<omega>" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   729
      show ?thesis unfolding * Lim_omega_pos
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   730
      proof safe case goal1
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   731
        from real_arch_simple[of "B / real (\<mu> ?s)"] guess N0 .. note N=this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   732
        def N \<equiv> "Suc N0" have N:"real N \<ge> B / real (\<mu> ?s)" "N > 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   733
          unfolding N_def using N by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   734
        show ?case apply-apply(rule_tac x=N in exI,safe) 
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   735
        proof- case goal1
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   736
          have "Real B \<le> Real (real N) * \<mu> ?s"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   737
          proof(cases "\<mu> ?s = \<omega>")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   738
            case True thus ?thesis using `B>0` N by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   739
          next case False
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   740
            have *:"B \<le> real N * real (\<mu> ?s)" 
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   741
              using N(1) apply-apply(subst (asm) pos_divide_le_eq)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   742
              apply rule using m0 False by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   743
            show ?thesis apply(subst Real_real'[THEN sym,OF False])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   744
              apply(subst pinfreal_times,subst if_P) defer
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   745
              apply(subst pinfreal_less_eq,subst if_P) defer
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   746
              using * N `B>0` by(auto intro: mult_nonneg_nonneg)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   747
          qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   748
          also have "... \<le> Real (real n) * \<mu> ?s"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   749
            apply(rule mult_right_mono) using goal1 by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   750
          also have "... = simple_integral (\<lambda>x. Real (real n) * indicator ?s x)" 
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   751
            apply(subst simple_integral_mult) apply(rule simple_function_indicator[OF s])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   752
            unfolding simple_integral_indicator_only[OF s] ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   753
          also have "... \<le> simple_integral (\<lambda>x. if u x = \<omega> then Real (real n) else u x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   754
            apply(rule simple_integral_mono) apply(rule simple_function_mult)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   755
            apply(rule simple_function_const)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   756
            apply(rule simple_function_indicator) apply(rule s su)+ by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   757
          finally show ?case .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   758
        qed qed qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   759
    fix x assume x:"\<omega> = (if u x = \<omega> then Real (real n) else u x)" "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   760
    hence "u x = \<omega>" apply-apply(rule ccontr) by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   761
    hence "\<omega> = Real (real n)" using x by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   762
    thus False by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   763
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   764
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   765
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   766
lemma (in measure_space) positive_integral_cong:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   767
  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   768
  shows "positive_integral f = positive_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   769
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   770
  have "\<And>h. (\<forall>x\<in>space M. h x \<le> f x \<and> h x \<noteq> \<omega>) = (\<forall>x\<in>space M. h x \<le> g x \<and> h x \<noteq> \<omega>)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   771
    using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   772
  thus ?thesis unfolding positive_integral_alt1 by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   773
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   774
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   775
lemma (in measure_space) positive_integral_eq_simple_integral:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   776
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   777
  shows "positive_integral f = simple_integral f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   778
  unfolding positive_integral_alt
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   779
proof (safe intro!: pinfreal_SUPI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   780
  fix g assume "simple_function g" "g \<le> f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   781
  with assms show "simple_integral g \<le> simple_integral f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   782
    by (auto intro!: simple_integral_mono simp: le_fun_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   783
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   784
  fix y assume "\<forall>x. x\<in>{g. simple_function g \<and> g \<le> f} \<longrightarrow> simple_integral x \<le> y"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   785
  with assms show "simple_integral f \<le> y" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   786
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   787
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   788
lemma (in measure_space) positive_integral_mono:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   789
  assumes mono: "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   790
  shows "positive_integral u \<le> positive_integral v"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   791
  unfolding positive_integral_alt1
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   792
proof (safe intro!: SUPR_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   793
  fix a assume a: "simple_function a" and "\<forall>x\<in>space M. a x \<le> u x \<and> a x \<noteq> \<omega>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   794
  with mono have "\<forall>x\<in>space M. a x \<le> v x \<and> a x \<noteq> \<omega>" by fastsimp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   795
  with a show "\<exists>b\<in>{g. simple_function g \<and> (\<forall>x\<in>space M. g x \<le> v x \<and> g x \<noteq> \<omega>)}. simple_integral a \<le> simple_integral b"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   796
    by (auto intro!: bexI[of _ a])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   797
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   798
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   799
lemma (in measure_space) positive_integral_SUP_approx:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   800
  assumes "f \<up> s"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   801
  and f: "\<And>i. f i \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   802
  and "simple_function u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   803
  and le: "u \<le> s" and real: "\<omega> \<notin> u`space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   804
  shows "simple_integral u \<le> (SUP i. positive_integral (f i))" (is "_ \<le> ?S")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   805
proof (rule pinfreal_le_mult_one_interval)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   806
  fix a :: pinfreal assume "0 < a" "a < 1"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   807
  hence "a \<noteq> 0" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   808
  let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   809
  have B: "\<And>i. ?B i \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   810
    using f `simple_function u` by (auto simp: borel_measurable_simple_function)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   811
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   812
  let "?uB i x" = "u x * indicator (?B i) x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   813
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   814
  { fix i have "?B i \<subseteq> ?B (Suc i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   815
    proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   816
      fix i x assume "a * u x \<le> f i x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   817
      also have "\<dots> \<le> f (Suc i) x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   818
        using `f \<up> s` unfolding isoton_def le_fun_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   819
      finally show "a * u x \<le> f (Suc i) x" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   820
    qed }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   821
  note B_mono = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   822
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   823
  have u: "\<And>x. x \<in> space M \<Longrightarrow> u -` {u x} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   824
    using `simple_function u` by (auto simp add: simple_function_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   825
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   826
  { fix i
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   827
    have "(\<lambda>n. (u -` {i} \<inter> space M) \<inter> ?B n) \<up> (u -` {i} \<inter> space M)" using B_mono unfolding isoton_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   828
    proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   829
      fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   830
      show "x \<in> (\<Union>i. (u -` {u x} \<inter> space M) \<inter> ?B i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   831
      proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   832
        assume "u x = 0" thus ?thesis using `x \<in> space M` by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   833
      next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   834
        assume "u x \<noteq> 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   835
        with `a < 1` real `x \<in> space M`
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   836
        have "a * u x < 1 * u x" by (rule_tac pinfreal_mult_strict_right_mono) (auto simp: image_iff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   837
        also have "\<dots> \<le> (SUP i. f i x)" using le `f \<up> s`
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   838
          unfolding isoton_fun_expand by (auto simp: isoton_def le_fun_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   839
        finally obtain i where "a * u x < f i x" unfolding SUPR_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   840
          by (auto simp add: less_Sup_iff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   841
        hence "a * u x \<le> f i x" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   842
        thus ?thesis using `x \<in> space M` by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   843
      qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   844
    qed auto }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   845
  note measure_conv = measure_up[OF u Int[OF u B] this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   846
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   847
  have "simple_integral u = (SUP i. simple_integral (?uB i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   848
    unfolding simple_integral_indicator[OF B `simple_function u`]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   849
  proof (subst SUPR_pinfreal_setsum, safe)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   850
    fix x n assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   851
    have "\<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f n x})
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   852
      \<le> \<mu> (u -` {u x} \<inter> space M \<inter> {x \<in> space M. a * u x \<le> f (Suc n) x})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   853
      using B_mono Int[OF u[OF `x \<in> space M`] B] by (auto intro!: measure_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   854
    thus "u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B n)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   855
            \<le> u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B (Suc n))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   856
      by (auto intro: mult_left_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   857
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   858
    show "simple_integral u =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   859
      (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (u -` {i} \<inter> space M \<inter> ?B n))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   860
      using measure_conv unfolding simple_integral_def isoton_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   861
      by (auto intro!: setsum_cong simp: pinfreal_SUP_cmult)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   862
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   863
  moreover
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   864
  have "a * (SUP i. simple_integral (?uB i)) \<le> ?S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   865
    unfolding pinfreal_SUP_cmult[symmetric]
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
   866
  proof (safe intro!: SUP_mono bexI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   867
    fix i
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   868
    have "a * simple_integral (?uB i) = simple_integral (\<lambda>x. a * ?uB i x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   869
      using B `simple_function u`
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   870
      by (subst simple_integral_mult[symmetric]) (auto simp: field_simps)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   871
    also have "\<dots> \<le> positive_integral (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   872
    proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   873
      have "\<And>x. a * ?uB i x \<le> f i x" unfolding indicator_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   874
      hence *: "simple_function (\<lambda>x. a * ?uB i x)" using B assms(3)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   875
        by (auto intro!: simple_integral_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   876
      show ?thesis unfolding positive_integral_eq_simple_integral[OF *, symmetric]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   877
        by (auto intro!: positive_integral_mono simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   878
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   879
    finally show "a * simple_integral (?uB i) \<le> positive_integral (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   880
      by auto
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
   881
  qed simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   882
  ultimately show "a * simple_integral u \<le> ?S" by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   883
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   884
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   885
text {* Beppo-Levi monotone convergence theorem *}
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   886
lemma (in measure_space) positive_integral_isoton:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   887
  assumes "f \<up> u" "\<And>i. f i \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   888
  shows "(\<lambda>i. positive_integral (f i)) \<up> positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   889
  unfolding isoton_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   890
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   891
  fix i show "positive_integral (f i) \<le> positive_integral (f (Suc i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   892
    apply (rule positive_integral_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   893
    using `f \<up> u` unfolding isoton_def le_fun_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   894
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   895
  have "u \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   896
    using borel_measurable_SUP[of UNIV f] assms by (auto simp: isoton_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   897
  have u: "u = (SUP i. f i)" using `f \<up> u` unfolding isoton_def by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   898
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   899
  show "(SUP i. positive_integral (f i)) = positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   900
  proof (rule antisym)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   901
    from `f \<up> u`[THEN isoton_Sup, unfolded le_fun_def]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   902
    show "(SUP j. positive_integral (f j)) \<le> positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   903
      by (auto intro!: SUP_leI positive_integral_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   904
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   905
    show "positive_integral u \<le> (SUP i. positive_integral (f i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   906
      unfolding positive_integral_def[of u]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   907
      by (auto intro!: SUP_leI positive_integral_SUP_approx[OF assms])
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   908
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   909
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   910
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   911
lemma (in measure_space) SUP_simple_integral_sequences:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   912
  assumes f: "f \<up> u" "\<And>i. simple_function (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   913
  and g: "g \<up> u" "\<And>i. simple_function (g i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   914
  shows "(SUP i. simple_integral (f i)) = (SUP i. simple_integral (g i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   915
    (is "SUPR _ ?F = SUPR _ ?G")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   916
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   917
  have "(SUP i. ?F i) = (SUP i. positive_integral (f i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   918
    using assms by (simp add: positive_integral_eq_simple_integral)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   919
  also have "\<dots> = positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   920
    using positive_integral_isoton[OF f(1) borel_measurable_simple_function[OF f(2)]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   921
    unfolding isoton_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   922
  also have "\<dots> = (SUP i. positive_integral (g i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   923
    using positive_integral_isoton[OF g(1) borel_measurable_simple_function[OF g(2)]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   924
    unfolding isoton_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   925
  also have "\<dots> = (SUP i. ?G i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   926
    using assms by (simp add: positive_integral_eq_simple_integral)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   927
  finally show ?thesis .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   928
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   929
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   930
lemma (in measure_space) positive_integral_const[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   931
  "positive_integral (\<lambda>x. c) = c * \<mu> (space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   932
  by (subst positive_integral_eq_simple_integral) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   933
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   934
lemma (in measure_space) positive_integral_isoton_simple:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   935
  assumes "f \<up> u" and e: "\<And>i. simple_function (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   936
  shows "(\<lambda>i. simple_integral (f i)) \<up> positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   937
  using positive_integral_isoton[OF `f \<up> u` e(1)[THEN borel_measurable_simple_function]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   938
  unfolding positive_integral_eq_simple_integral[OF e] .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   939
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   940
lemma (in measure_space) positive_integral_linear:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   941
  assumes f: "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   942
  and g: "g \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   943
  shows "positive_integral (\<lambda>x. a * f x + g x) =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   944
      a * positive_integral f + positive_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   945
    (is "positive_integral ?L = _")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   946
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   947
  from borel_measurable_implies_simple_function_sequence'[OF f] guess u .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   948
  note u = this positive_integral_isoton_simple[OF this(1-2)]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   949
  from borel_measurable_implies_simple_function_sequence'[OF g] guess v .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   950
  note v = this positive_integral_isoton_simple[OF this(1-2)]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   951
  let "?L' i x" = "a * u i x + v i x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   952
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   953
  have "?L \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   954
    using assms by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   955
  from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   956
  note positive_integral_isoton_simple[OF this(1-2)] and l = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   957
  moreover have
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   958
      "(SUP i. simple_integral (l i)) = (SUP i. simple_integral (?L' i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   959
  proof (rule SUP_simple_integral_sequences[OF l(1-2)])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   960
    show "?L' \<up> ?L" "\<And>i. simple_function (?L' i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   961
      using u v by (auto simp: isoton_fun_expand isoton_add isoton_cmult_right)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   962
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   963
  moreover from u v have L'_isoton:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   964
      "(\<lambda>i. simple_integral (?L' i)) \<up> a * positive_integral f + positive_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   965
    by (simp add: isoton_add isoton_cmult_right)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   966
  ultimately show ?thesis by (simp add: isoton_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   967
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   968
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   969
lemma (in measure_space) positive_integral_cmult:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   970
  assumes "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   971
  shows "positive_integral (\<lambda>x. c * f x) = c * positive_integral f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   972
  using positive_integral_linear[OF assms, of "\<lambda>x. 0" c] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   973
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   974
lemma (in measure_space) positive_integral_indicator[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   975
  "A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. indicator A x) = \<mu> A"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   976
by (subst positive_integral_eq_simple_integral) (auto simp: simple_function_indicator simple_integral_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   977
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   978
lemma (in measure_space) positive_integral_cmult_indicator:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   979
  "A \<in> sets M \<Longrightarrow> positive_integral (\<lambda>x. c * indicator A x) = c * \<mu> A"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   980
by (subst positive_integral_eq_simple_integral) (auto simp: simple_function_indicator simple_integral_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   981
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   982
lemma (in measure_space) positive_integral_add:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   983
  assumes "f \<in> borel_measurable M" "g \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   984
  shows "positive_integral (\<lambda>x. f x + g x) = positive_integral f + positive_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   985
  using positive_integral_linear[OF assms, of 1] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   986
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   987
lemma (in measure_space) positive_integral_setsum:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   988
  assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   989
  shows "positive_integral (\<lambda>x. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. positive_integral (f i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   990
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   991
  assume "finite P"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   992
  from this assms show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   993
  proof induct
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   994
    case (insert i P)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   995
    have "f i \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   996
      "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   997
      using insert by (auto intro!: borel_measurable_pinfreal_setsum)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   998
    from positive_integral_add[OF this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   999
    show ?case using insert by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1000
  qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1001
qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1002
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1003
lemma (in measure_space) positive_integral_diff:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1004
  assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1005
  and fin: "positive_integral g \<noteq> \<omega>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1006
  and mono: "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1007
  shows "positive_integral (\<lambda>x. f x - g x) = positive_integral f - positive_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1008
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1009
  have borel: "(\<lambda>x. f x - g x) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1010
    using f g by (rule borel_measurable_pinfreal_diff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1011
  have "positive_integral (\<lambda>x. f x - g x) + positive_integral g =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1012
    positive_integral f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1013
    unfolding positive_integral_add[OF borel g, symmetric]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1014
  proof (rule positive_integral_cong)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1015
    fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1016
    from mono[OF this] show "f x - g x + g x = f x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1017
      by (cases "f x", cases "g x", simp, simp, cases "g x", auto)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1018
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1019
  with mono show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1020
    by (subst minus_pinfreal_eq2[OF _ fin]) (auto intro!: positive_integral_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1021
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1022
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1023
lemma (in measure_space) positive_integral_psuminf:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1024
  assumes "\<And>i. f i \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1025
  shows "positive_integral (\<lambda>x. \<Sum>\<^isub>\<infinity> i. f i x) = (\<Sum>\<^isub>\<infinity> i. positive_integral (f i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1026
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1027
  have "(\<lambda>i. positive_integral (\<lambda>x. \<Sum>i<i. f i x)) \<up> positive_integral (\<lambda>x. \<Sum>\<^isub>\<infinity>i. f i x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1028
    by (rule positive_integral_isoton)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1029
       (auto intro!: borel_measurable_pinfreal_setsum assms positive_integral_mono
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1030
                     arg_cong[where f=Sup]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1031
             simp: isoton_def le_fun_def psuminf_def expand_fun_eq SUPR_def Sup_fun_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1032
  thus ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1033
    by (auto simp: isoton_def psuminf_def positive_integral_setsum[OF assms])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1034
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1035
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1036
text {* Fatou's lemma: convergence theorem on limes inferior *}
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1037
lemma (in measure_space) positive_integral_lim_INF:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1038
  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> pinfreal"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1039
  assumes "\<And>i. u i \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1040
  shows "positive_integral (SUP n. INF m. u (m + n)) \<le>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1041
    (SUP n. INF m. positive_integral (u (m + n)))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1042
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1043
  have "(SUP n. INF m. u (m + n)) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1044
    by (auto intro!: borel_measurable_SUP borel_measurable_INF assms)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1045
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1046
  have "(\<lambda>n. INF m. u (m + n)) \<up> (SUP n. INF m. u (m + n))"
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1047
  proof (unfold isoton_def, safe intro!: INF_mono bexI)
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1048
    fix i m show "u (Suc m + i) \<le> u (m + Suc i)" by simp
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1049
  qed simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1050
  from positive_integral_isoton[OF this] assms
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1051
  have "positive_integral (SUP n. INF m. u (m + n)) =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1052
    (SUP n. positive_integral (INF m. u (m + n)))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1053
    unfolding isoton_def by (simp add: borel_measurable_INF)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1054
  also have "\<dots> \<le> (SUP n. INF m. positive_integral (u (m + n)))"
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1055
    apply (rule SUP_mono)
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1056
    apply (rule_tac x=n in bexI)
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1057
    by (auto intro!: positive_integral_mono INFI_bound INF_leI exI simp: INFI_fun_expand)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1058
  finally show ?thesis .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1059
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1060
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1061
lemma (in measure_space) measure_space_density:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1062
  assumes borel: "u \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1063
  shows "measure_space M (\<lambda>A. positive_integral (\<lambda>x. u x * indicator A x))" (is "measure_space M ?v")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1064
proof
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1065
  show "?v {} = 0" by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1066
  show "countably_additive M ?v"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1067
    unfolding countably_additive_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1068
  proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1069
    fix A :: "nat \<Rightarrow> 'a set"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1070
    assume "range A \<subseteq> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1071
    hence "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1072
      using borel by (auto intro: borel_measurable_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1073
    moreover assume "disjoint_family A"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1074
    note psuminf_indicator[OF this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1075
    ultimately show "(\<Sum>\<^isub>\<infinity>n. ?v (A n)) = ?v (\<Union>x. A x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1076
      by (simp add: positive_integral_psuminf[symmetric])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1077
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1078
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1079
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1080
lemma (in measure_space) positive_integral_null_set:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1081
  assumes borel: "u \<in> borel_measurable M" and "N \<in> null_sets"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1082
  shows "positive_integral (\<lambda>x. u x * indicator N x) = 0" (is "?I = 0")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1083
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1084
  have "N \<in> sets M" using `N \<in> null_sets` by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1085
  have "(\<lambda>i x. min (of_nat i) (u x) * indicator N x) \<up> (\<lambda>x. u x * indicator N x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1086
    unfolding isoton_fun_expand
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1087
  proof (safe intro!: isoton_cmult_left, unfold isoton_def, safe)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1088
    fix j i show "min (of_nat j) (u i) \<le> min (of_nat (Suc j)) (u i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1089
      by (rule min_max.inf_mono) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1090
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1091
    fix i show "(SUP j. min (of_nat j) (u i)) = u i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1092
    proof (cases "u i")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1093
      case infinite
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1094
      moreover hence "\<And>j. min (of_nat j) (u i) = of_nat j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1095
        by (auto simp: min_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1096
      ultimately show ?thesis by (simp add: Sup_\<omega>)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1097
    next
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1098
      case (preal r)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1099
      obtain j where "r \<le> of_nat j" using ex_le_of_nat ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1100
      hence "u i \<le> of_nat j" using preal by (auto simp: real_of_nat_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1101
      show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1102
      proof (rule pinfreal_SUPI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1103
        fix y assume "\<And>j. j \<in> UNIV \<Longrightarrow> min (of_nat j) (u i) \<le> y"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1104
        note this[of j]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1105
        moreover have "min (of_nat j) (u i) = u i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1106
          using `u i \<le> of_nat j` by (auto simp: min_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1107
        ultimately show "u i \<le> y" by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1108
      qed simp
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1109
    qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1110
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1111
  from positive_integral_isoton[OF this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1112
  have "?I = (SUP i. positive_integral (\<lambda>x. min (of_nat i) (u x) * indicator N x))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1113
    unfolding isoton_def using borel `N \<in> sets M` by (simp add: borel_measurable_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1114
  also have "\<dots> \<le> (SUP i. positive_integral (\<lambda>x. of_nat i * indicator N x))"
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1115
  proof (rule SUP_mono, rule bexI, rule positive_integral_mono)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1116
    fix x i show "min (of_nat i) (u x) * indicator N x \<le> of_nat i * indicator N x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1117
      by (cases "x \<in> N") auto
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1118
  qed simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1119
  also have "\<dots> = 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1120
    using `N \<in> null_sets` by (simp add: positive_integral_cmult_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1121
  finally show ?thesis by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1122
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1123
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1124
lemma (in measure_space) positive_integral_Markov_inequality:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1125
  assumes borel: "u \<in> borel_measurable M" and "A \<in> sets M" and c: "c \<noteq> \<omega>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1126
  shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * positive_integral (\<lambda>x. u x * indicator A x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1127
    (is "\<mu> ?A \<le> _ * ?PI")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1128
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1129
  have "?A \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1130
    using `A \<in> sets M` borel by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1131
  hence "\<mu> ?A = positive_integral (\<lambda>x. indicator ?A x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1132
    using positive_integral_indicator by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1133
  also have "\<dots> \<le> positive_integral (\<lambda>x. c * (u x * indicator A x))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1134
  proof (rule positive_integral_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1135
    fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1136
    show "indicator ?A x \<le> c * (u x * indicator A x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1137
      by (cases "x \<in> ?A") auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1138
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1139
  also have "\<dots> = c * positive_integral (\<lambda>x. u x * indicator A x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1140
    using assms
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1141
    by (auto intro!: positive_integral_cmult borel_measurable_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1142
  finally show ?thesis .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1143
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1144
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1145
lemma (in measure_space) positive_integral_0_iff:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1146
  assumes borel: "u \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1147
  shows "positive_integral u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1148
    (is "_ \<longleftrightarrow> \<mu> ?A = 0")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1149
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1150
  have A: "?A \<in> sets M" using borel by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1151
  have u: "positive_integral (\<lambda>x. u x * indicator ?A x) = positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1152
    by (auto intro!: positive_integral_cong simp: indicator_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1153
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1154
  show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1155
  proof
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1156
    assume "\<mu> ?A = 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1157
    hence "?A \<in> null_sets" using `?A \<in> sets M` by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1158
    from positive_integral_null_set[OF borel this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1159
    have "0 = positive_integral (\<lambda>x. u x * indicator ?A x)" by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1160
    thus "positive_integral u = 0" unfolding u by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1161
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1162
    assume *: "positive_integral u = 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1163
    let "?M n" = "{x \<in> space M. 1 \<le> of_nat n * u x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1164
    have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1165
    proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1166
      { fix n
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1167
        from positive_integral_Markov_inequality[OF borel `?A \<in> sets M`, of "of_nat n"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1168
        have "\<mu> (?M n \<inter> ?A) = 0" unfolding * u by simp }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1169
      thus ?thesis by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1170
    qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1171
    also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1172
    proof (safe intro!: continuity_from_below)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1173
      fix n show "?M n \<inter> ?A \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1174
        using borel by (auto intro!: Int)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1175
    next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1176
      fix n x assume "1 \<le> of_nat n * u x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1177
      also have "\<dots> \<le> of_nat (Suc n) * u x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1178
        by (subst (1 2) mult_commute) (auto intro!: pinfreal_mult_cancel)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1179
      finally show "1 \<le> of_nat (Suc n) * u x" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1180
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1181
    also have "\<dots> = \<mu> ?A"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1182
    proof (safe intro!: arg_cong[where f="\<mu>"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1183
      fix x assume "u x \<noteq> 0" and [simp, intro]: "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1184
      show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1185
      proof (cases "u x")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1186
        case (preal r)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1187
        obtain j where "1 / r \<le> of_nat j" using ex_le_of_nat ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1188
        hence "1 / r * r \<le> of_nat j * r" using preal unfolding mult_le_cancel_right by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1189
        hence "1 \<le> of_nat j * r" using preal `u x \<noteq> 0` by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1190
        thus ?thesis using `u x \<noteq> 0` preal by (auto simp: real_of_nat_def[symmetric])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1191
      qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1192
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1193
    finally show "\<mu> ?A = 0" by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1194
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1195
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1196
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1197
lemma (in measure_space) positive_integral_cong_on_null_sets:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1198
  assumes f: "f \<in> borel_measurable M" and g: "g \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1199
  and measure: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1200
  shows "positive_integral f = positive_integral g"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1201
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1202
  let ?N = "{x\<in>space M. f x \<noteq> g x}" and ?E = "{x\<in>space M. f x = g x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1203
  let "?A h x" = "h x * indicator ?E x :: pinfreal"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1204
  let "?B h x" = "h x * indicator ?N x :: pinfreal"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1205
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1206
  have A: "positive_integral (?A f) = positive_integral (?A g)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1207
    by (auto intro!: positive_integral_cong simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1208
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1209
  have [intro]: "?N \<in> sets M" "?E \<in> sets M" using f g by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1210
  hence "?N \<in> null_sets" using measure by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1211
  hence B: "positive_integral (?B f) = positive_integral (?B g)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1212
    using f g by (simp add: positive_integral_null_set)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1213
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1214
  have "positive_integral f = positive_integral (\<lambda>x. ?A f x + ?B f x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1215
    by (auto intro!: positive_integral_cong simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1216
  also have "\<dots> = positive_integral (?A f) + positive_integral (?B f)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1217
    using f g by (auto intro!: positive_integral_add borel_measurable_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1218
  also have "\<dots> = positive_integral (\<lambda>x. ?A g x + ?B g x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1219
    unfolding A B using f g by (auto intro!: positive_integral_add[symmetric] borel_measurable_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1220
  also have "\<dots> = positive_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1221
    by (auto intro!: positive_integral_cong simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1222
  finally show ?thesis by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1223
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1224
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1225
section "Lebesgue Integral"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1226
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1227
definition (in measure_space) integrable where
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1228
  "integrable f \<longleftrightarrow> f \<in> borel_measurable M \<and>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1229
    positive_integral (\<lambda>x. Real (f x)) \<noteq> \<omega> \<and>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1230
    positive_integral (\<lambda>x. Real (- f x)) \<noteq> \<omega>"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1231
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1232
lemma (in measure_space) integrableD[dest]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1233
  assumes "integrable f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1234
  shows "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1235
  "positive_integral (\<lambda>x. Real (f x)) \<noteq> \<omega>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642