src/HOL/Probability/Lebesgue_Integration.thy
author hoelzl
Mon, 24 Jan 2011 22:29:50 +0100
changeset 41661 baf1964bc468
parent 41545 9c869baf1c66
child 41689 3e39b0e730d6
permissions -rw-r--r--
use pre-image measure, instead of image
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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parents: 38642
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     2
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
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parents:
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header {*Lebesgue Integration*}
b16d99a72dc9 Add Lebesgue integral and probability space.
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parents:
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d5d342611edb Rewrite the Probability theory.
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theory Lebesgue_Integration
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
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parents: 40786
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     6
imports Measure Borel_Space
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b16d99a72dc9 Add Lebesgue integral and probability space.
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parents:
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begin
b16d99a72dc9 Add Lebesgue integral and probability space.
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parents:
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     8
38656
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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     9
lemma sums_If_finite:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    10
  assumes finite: "finite {r. P r}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    11
  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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parents: 38642
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    12
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    13
  assume "{r. P r} = {}" hence "\<And>r. \<not> P r" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    14
  thus ?thesis by (simp add: sums_zero)
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    15
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    16
  assume not_empty: "{r. P r} \<noteq> {}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    17
  have "?F sums (\<Sum>r = 0..< Suc (Max {r. P r}). ?F r)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    18
    by (rule series_zero)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    19
       (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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    20
  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
diff changeset
    21
    by (subst setsum_cases)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    22
       (auto intro!: setsum_cong simp: Max_ge_iff[OF finite not_empty] less_Suc_eq_le)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    23
  finally show ?thesis .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    24
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
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    25
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d5d342611edb Rewrite the Probability theory.
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    26
lemma sums_single:
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    27
  "(\<lambda>r. if r = i then f r else 0) sums f i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    28
  using sums_If_finite[of "\<lambda>r. r = i" f] by simp
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    29
d5d342611edb Rewrite the Probability theory.
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    30
section "Simple function"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
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parents:
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    31
38656
d5d342611edb Rewrite the Probability theory.
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    32
text {*
d5d342611edb Rewrite the Probability theory.
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    33
d5d342611edb Rewrite the Probability theory.
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    34
Our simple functions are not restricted to positive real numbers. Instead
d5d342611edb Rewrite the Probability theory.
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    35
they are just functions with a finite range and are measurable when singleton
d5d342611edb Rewrite the Probability theory.
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    36
sets are measurable.
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
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parents:
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    37
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d5d342611edb Rewrite the Probability theory.
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    38
*}
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    39
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    40
definition (in sigma_algebra) "simple_function g \<longleftrightarrow>
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    41
    finite (g ` space M) \<and>
d5d342611edb Rewrite the Probability theory.
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    42
    (\<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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    43
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    44
lemma (in sigma_algebra) simple_functionD:
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    45
  assumes "simple_function g"
40875
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
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    46
  shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
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    47
proof -
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
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    48
  show "finite (g ` space M)"
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
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    49
    using assms unfolding simple_function_def by auto
40875
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
    50
  have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
    51
  also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
    52
  finally show "g -` X \<inter> space M \<in> sets M" using assms
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
    53
    by (auto intro!: finite_UN simp del: UN_simps simp: simple_function_def)
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
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    54
qed
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
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    55
38656
d5d342611edb Rewrite the Probability theory.
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parents: 38642
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    56
lemma (in sigma_algebra) simple_function_indicator_representation:
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
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    57
  fixes f ::"'a \<Rightarrow> pextreal"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    58
  assumes f: "simple_function f" and x: "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    59
  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
diff changeset
    60
  (is "?l = ?r")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    61
proof -
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
    62
  have "?r = (\<Sum>y \<in> f ` space M.
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    63
    (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
    64
    by (auto intro!: setsum_cong2)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    65
  also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    66
    using assms by (auto dest: simple_functionD simp: setsum_delta)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    67
  also have "... = f x" using x by (auto simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    68
  finally show ?thesis by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    69
qed
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
    70
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    71
lemma (in measure_space) simple_function_notspace:
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
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    72
  "simple_function (\<lambda>x. h x * indicator (- space M) x::pextreal)" (is "simple_function ?h")
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
    73
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    74
  have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    75
  hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    76
  have "?h -` {0} \<inter> space M = space M" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    77
  thus ?thesis unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    78
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    79
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    80
lemma (in sigma_algebra) simple_function_cong:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    81
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    82
  shows "simple_function f \<longleftrightarrow> simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    83
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    84
  have "f ` space M = g ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    85
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    86
    using assms by (auto intro!: image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    87
  thus ?thesis unfolding simple_function_def using assms by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    88
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    89
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    90
lemma (in sigma_algebra) borel_measurable_simple_function:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    91
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    92
  shows "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    93
proof (rule borel_measurableI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    94
  fix S
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    95
  let ?I = "f ` (f -` S \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    96
  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
    97
  have "finite ?I"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    98
    using assms unfolding simple_function_def by (auto intro: finite_subset)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    99
  hence "?U \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   100
    apply (rule finite_UN)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   101
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   102
  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
   103
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   104
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   105
lemma (in sigma_algebra) simple_function_borel_measurable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   106
  fixes f :: "'a \<Rightarrow> 'x::t2_space"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   107
  assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   108
  shows "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   109
  using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   110
  by (auto intro: borel_measurable_vimage)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   111
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   112
lemma (in sigma_algebra) simple_function_const[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   113
  "simple_function (\<lambda>x. c)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   114
  by (auto intro: finite_subset simp: simple_function_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   115
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   116
lemma (in sigma_algebra) simple_function_compose[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   117
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   118
  shows "simple_function (g \<circ> f)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   119
  unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   120
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   121
  show "finite ((g \<circ> f) ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   122
    using assms unfolding simple_function_def by (auto simp: image_compose)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   123
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   124
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   125
  let ?G = "g -` {g (f x)} \<inter> (f`space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   126
  have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   127
    (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   128
  show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   129
    using assms unfolding simple_function_def *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   130
    by (rule_tac finite_UN) (auto intro!: finite_UN)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   131
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   132
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   133
lemma (in sigma_algebra) simple_function_indicator[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   134
  assumes "A \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   135
  shows "simple_function (indicator A)"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   136
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   137
  have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   138
    by (auto simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   139
  hence "finite ?S" by (rule finite_subset) simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   140
  moreover have "- A \<inter> space M = space M - A" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   141
  ultimately show ?thesis unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   142
    using assms by (auto simp: indicator_def_raw)
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   143
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   144
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   145
lemma (in sigma_algebra) simple_function_Pair[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   146
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   147
  assumes "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   148
  shows "simple_function (\<lambda>x. (f x, g x))" (is "simple_function ?p")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   149
  unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   150
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   151
  show "finite (?p ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   152
    using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   153
    by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   154
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   155
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   156
  have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   157
      (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   158
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   159
  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
   160
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   161
qed
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   162
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   163
lemma (in sigma_algebra) simple_function_compose1:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   164
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   165
  shows "simple_function (\<lambda>x. g (f x))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   166
  using simple_function_compose[OF assms, of g]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   167
  by (simp add: comp_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   168
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   169
lemma (in sigma_algebra) simple_function_compose2:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   170
  assumes "simple_function f" and "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   171
  shows "simple_function (\<lambda>x. h (f x) (g x))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   172
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   173
  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
   174
    using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   175
  thus ?thesis by (simp_all add: comp_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   176
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   177
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   178
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
   179
  and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   180
  and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   181
  and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   182
  and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   183
  and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   184
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   185
lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   186
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   187
  shows "simple_function (\<lambda>x. \<Sum>i\<in>P. f i x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   188
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   189
  assume "finite P" from this assms show ?thesis by induct auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   190
qed auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   191
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   192
lemma (in sigma_algebra) simple_function_le_measurable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   193
  assumes "simple_function f" "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   194
  shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   195
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   196
  have *: "{x \<in> space M. f x \<le> g x} =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   197
    (\<Union>(F, G)\<in>f`space M \<times> g`space M.
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   198
      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
   199
    apply (auto split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   200
    apply (rule_tac x=x in bexI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   201
    apply (rule_tac x=x in bexI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   202
    by simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   203
  have **: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   204
    (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
   205
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   206
  have "finite (f`space M \<times> g`space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   207
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   208
  thus ?thesis unfolding *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   209
    apply (rule finite_UN)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   210
    using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   211
    by (auto intro!: **)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   212
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   213
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   214
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   215
  fixes u :: "'a \<Rightarrow> pextreal"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   216
  assumes u: "u \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   217
  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
   218
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   219
  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
   220
    (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
   221
    (is "\<exists>f. \<forall>x j. ?P x j (f x j)")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   222
  proof(rule choice, rule, rule choice, rule)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   223
    fix x j show "\<exists>n. ?P x j n"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   224
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   225
      assume *: "u x < of_nat j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   226
      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
   227
      from reals_Archimedean6a[of "r * 2^j"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   228
      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
   229
        using `0 \<le> r` by (auto simp: zero_le_power zero_le_mult_iff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   230
      thus ?thesis using r * by (auto intro!: exI[of _ n])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   231
    qed auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   232
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   233
  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
   234
    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
   235
    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
   236
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   237
  { fix j x P
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   238
    assume 1: "of_nat j \<le> u x \<Longrightarrow> P (j * 2^j)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   239
    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
   240
    have "P (f x j)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   241
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   242
      assume "of_nat j \<le> u x" thus "P (f x j)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   243
        using top[of j x] 1 by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   244
    next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   245
      assume "\<not> of_nat j \<le> u x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   246
      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
   247
        using upper lower by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   248
      from 2[OF this] show "P (f x j)" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   249
    qed }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   250
  note fI = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   251
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   252
  { fix j x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   253
    have "f x j = j * 2 ^ j \<longleftrightarrow> of_nat j \<le> u x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   254
      by (rule fI, simp, cases "u x") (auto split: split_if_asm) }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   255
  note f_eq = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   256
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   257
  { fix j x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   258
    have "f x j \<le> j * 2 ^ j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   259
    proof (rule fI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   260
      fix k assume *: "u x < of_nat j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   261
      assume "of_nat k \<le> u x * 2 ^ j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   262
      also have "\<dots> \<le> of_nat (j * 2^j)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   263
        using * by (cases "u x") (auto simp: zero_le_mult_iff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   264
      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
   265
    qed simp }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   266
  note f_upper = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   267
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   268
  let "?g j x" = "of_nat (f x j) / 2^j :: pextreal"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   269
  show ?thesis unfolding simple_function_def isoton_fun_expand unfolding isoton_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   270
  proof (safe intro!: exI[of _ ?g])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   271
    fix j
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   272
    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
   273
      using f_upper by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   274
    thus "finite (?g j ` space M)" by (rule finite_subset) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   275
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   276
    fix j t assume "t \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   277
    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
   278
      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
   279
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   280
    show "?g j -` {?g j t} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   281
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   282
      assume "of_nat j \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   283
      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
   284
        unfolding ** f_eq[symmetric] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   285
      thus "?g j -` {?g j t} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   286
        using u by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   287
    next
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   288
      assume not_t: "\<not> of_nat j \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   289
      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
   290
      have split_vimage: "?g j -` {?g j t} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   291
          {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
   292
        unfolding **
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   293
      proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   294
        fix x assume [simp]: "f t j = f x j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   295
        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
   296
        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
   297
          using upper lower by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   298
        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
   299
          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
   300
        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
   301
      next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   302
        fix x
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   303
        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
   304
        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
   305
          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
   306
        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
   307
        note 2
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   308
        also have "\<dots> \<le> of_nat (j*2^j)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   309
          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
   310
        finally have bound_ux: "u x < of_nat j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   311
          by (cases "u x") (auto simp: zero_le_mult_iff power_le_zero_eq)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   312
        show "f t j = f x j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   313
        proof (rule antisym)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   314
          from 1 lower[OF bound_ux]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   315
          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
   316
          from upper[OF bound_ux] 2
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   317
          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
   318
        qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   319
      qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   320
      show ?thesis unfolding split_vimage using u by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   321
    qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   322
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   323
    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
   324
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   325
    fix t
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   326
    { fix i
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   327
      have "f t i * 2 \<le> f t (Suc i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   328
      proof (rule fI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   329
        assume "of_nat (Suc i) \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   330
        hence "of_nat i \<le> u t" by (cases "u t") auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   331
        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
   332
      next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   333
        fix k
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   334
        assume *: "u t * 2 ^ Suc i < of_nat (Suc k)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   335
        show "f t i * 2 \<le> k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   336
        proof (rule fI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   337
          assume "of_nat i \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   338
          hence "of_nat (i * 2^Suc i) \<le> u t * 2^Suc i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   339
            by (cases "u t") (auto simp: zero_le_mult_iff power_le_zero_eq)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   340
          also have "\<dots> < of_nat (Suc k)" using * by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   341
          finally show "i * 2 ^ i * 2 \<le> k"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   342
            by (auto simp del: real_of_nat_mult)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   343
        next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   344
          fix j assume "of_nat j \<le> u t * 2 ^ i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   345
          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
   346
        qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   347
      qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   348
      thus "?g i t \<le> ?g (Suc i) t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   349
        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
   350
    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
   351
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   352
    show "(SUP j. of_nat (f t j) / 2 ^ j) = u t"
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   353
    proof (rule pextreal_SUPI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   354
      fix j show "of_nat (f t j) / 2 ^ j \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   355
      proof (rule fI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   356
        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
   357
          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
   358
      next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   359
        fix k assume "of_nat k \<le> u t * 2 ^ j"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   360
        thus "of_nat k / 2 ^ j \<le> u t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   361
          by (cases "u t")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   362
             (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
   363
      qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   364
    next
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   365
      fix y :: pextreal assume *: "\<And>j. j \<in> UNIV \<Longrightarrow> of_nat (f t j) / 2 ^ j \<le> y"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   366
      show "u t \<le> y"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   367
      proof (cases "u t")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   368
        case (preal r)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   369
        show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   370
        proof (rule ccontr)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   371
          assume "\<not> u t \<le> y"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   372
          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
   373
          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
   374
          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
   375
          let ?N = "max n (natfloor r + 1)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   376
          have "u t < of_nat ?N" "n \<le> ?N"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   377
            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
   378
            using real_natfloor_add_one_gt
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
   379
            by (auto simp: max_def real_of_nat_Suc)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   380
          from lower[OF this(1)]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   381
          have "r < (real (f t ?N) + 1) / 2 ^ ?N" unfolding less_divide_eq
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   382
            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
   383
          hence "u t < of_nat (f t ?N) / 2 ^ ?N + 1 / 2 ^ ?N"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   384
            using preal by (auto simp: field_simps divide_real_def[symmetric])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   385
          with n[OF `n \<le> ?N`] p preal *[of ?N]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   386
          show False
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   387
            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
   388
        qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   389
      next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   390
        case infinite
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   391
        { 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
   392
          hence "of_nat j \<le> y" using *[of j]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   393
            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
   394
        note all_less_y = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   395
        show ?thesis unfolding infinite
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   396
        proof (rule ccontr)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   397
          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
   398
          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
   399
          with all_less_y[of n] r show False by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   400
        qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   401
      qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   402
    qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   403
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   404
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   405
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   406
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   407
  fixes u :: "'a \<Rightarrow> pextreal"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   408
  assumes "u \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   409
  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
   410
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   411
  from borel_measurable_implies_simple_function_sequence[OF assms]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   412
  obtain f where x: "\<And>i. simple_function (f i)" "f \<up> u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   413
    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
   414
  { 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
   415
  with x show thesis by (auto intro!: that[of f])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   416
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   417
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   418
lemma (in sigma_algebra) simple_function_eq_borel_measurable:
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   419
  fixes f :: "'a \<Rightarrow> pextreal"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   420
  shows "simple_function f \<longleftrightarrow>
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   421
    finite (f`space M) \<and> f \<in> borel_measurable M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   422
  using simple_function_borel_measurable[of f]
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   423
    borel_measurable_simple_function[of f]
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   424
  by (fastsimp simp: simple_function_def)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   425
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   426
lemma (in measure_space) simple_function_restricted:
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   427
  fixes f :: "'a \<Rightarrow> pextreal" assumes "A \<in> sets M"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   428
  shows "sigma_algebra.simple_function (restricted_space A) f \<longleftrightarrow> simple_function (\<lambda>x. f x * indicator A x)"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   429
    (is "sigma_algebra.simple_function ?R f \<longleftrightarrow> simple_function ?f")
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   430
proof -
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   431
  interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   432
  have "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   433
  proof cases
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   434
    assume "A = space M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   435
    then have "f`A = ?f`space M" by (fastsimp simp: image_iff)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   436
    then show ?thesis by simp
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   437
  next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   438
    assume "A \<noteq> space M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   439
    then obtain x where x: "x \<in> space M" "x \<notin> A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   440
      using sets_into_space `A \<in> sets M` by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   441
    have *: "?f`space M = f`A \<union> {0}"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   442
    proof (auto simp add: image_iff)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   443
      show "\<exists>x\<in>space M. f x = 0 \<or> indicator A x = 0"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   444
        using x by (auto intro!: bexI[of _ x])
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   445
    next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   446
      fix x assume "x \<in> A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   447
      then show "\<exists>y\<in>space M. f x = f y * indicator A y"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   448
        using `A \<in> sets M` sets_into_space by (auto intro!: bexI[of _ x])
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   449
    next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   450
      fix x
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   451
      assume "indicator A x \<noteq> (0::pextreal)"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   452
      then have "x \<in> A" by (auto simp: indicator_def split: split_if_asm)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   453
      moreover assume "x \<in> space M" "\<forall>y\<in>A. ?f x \<noteq> f y"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   454
      ultimately show "f x = 0" by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   455
    qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   456
    then show ?thesis by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   457
  qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   458
  then show ?thesis
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   459
    unfolding simple_function_eq_borel_measurable
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   460
      R.simple_function_eq_borel_measurable
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   461
    unfolding borel_measurable_restricted[OF `A \<in> sets M`]
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   462
    by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   463
qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   464
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   465
lemma (in sigma_algebra) simple_function_subalgebra:
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   466
  assumes "sigma_algebra.simple_function N f"
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   467
  and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M" "sigma_algebra N"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   468
  shows "simple_function f"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   469
  using assms
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   470
  unfolding simple_function_def
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   471
  unfolding sigma_algebra.simple_function_def[OF N_subalgebra(3)]
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   472
  by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   473
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   474
lemma (in measure_space) simple_function_vimage:
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   475
  assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   476
    and f: "sigma_algebra.simple_function M' f"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   477
  shows "simple_function (\<lambda>x. f (T x))"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   478
proof (intro simple_function_def[THEN iffD2] conjI ballI)
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   479
  interpret T: sigma_algebra M' by fact
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   480
  have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   481
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   482
  then show "finite ((\<lambda>x. f (T x)) ` space M)"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   483
    using f unfolding T.simple_function_def by (auto intro: finite_subset)
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   484
  fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   485
  then have "i \<in> f ` space M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   486
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   487
  then have "f -` {i} \<inter> space M' \<in> sets M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   488
    using f unfolding T.simple_function_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   489
  then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   490
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   491
  also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   492
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   493
  finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   494
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   495
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   496
section "Simple integral"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   497
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   498
definition (in measure_space) simple_integral (binder "\<integral>\<^isup>S " 10) where
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   499
  "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
   500
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   501
lemma (in measure_space) simple_integral_cong:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   502
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   503
  shows "simple_integral f = simple_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   504
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   505
  have "f ` space M = g ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   506
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   507
    using assms by (auto intro!: image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   508
  thus ?thesis unfolding simple_integral_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   509
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   510
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   511
lemma (in measure_space) simple_integral_cong_measure:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   512
  assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A" and "simple_function f"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   513
  shows "measure_space.simple_integral M \<nu> f = simple_integral f"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   514
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   515
  interpret v: measure_space M \<nu>
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   516
    by (rule measure_space_cong) fact
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   517
  from simple_functionD[OF `simple_function f`] assms show ?thesis
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   518
    unfolding simple_integral_def v.simple_integral_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   519
    by (auto intro!: setsum_cong)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   520
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   521
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   522
lemma (in measure_space) simple_integral_const[simp]:
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   523
  "(\<integral>\<^isup>Sx. c) = c * \<mu> (space M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   524
proof (cases "space M = {}")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   525
  case True thus ?thesis unfolding simple_integral_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   526
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   527
  case False hence "(\<lambda>x. c) ` space M = {c}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   528
  thus ?thesis unfolding simple_integral_def by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   529
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   530
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   531
lemma (in measure_space) simple_function_partition:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   532
  assumes "simple_function f" and "simple_function g"
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   533
  shows "simple_integral f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * \<mu> A)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   534
    (is "_ = setsum _ (?p ` space M)")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   535
proof-
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   536
  let "?sub x" = "?p ` (f -` {x} \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   537
  let ?SIGMA = "Sigma (f`space M) ?sub"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   538
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   539
  have [intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   540
    "finite (f ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   541
    "finite (g ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   542
    using assms unfolding simple_function_def by simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   543
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   544
  { fix A
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   545
    have "?p ` (A \<inter> space M) \<subseteq>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   546
      (\<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
   547
      by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   548
    hence "finite (?p ` (A \<inter> space M))"
40786
0a54cfc9add3 gave more standard finite set rules simp and intro attribute
nipkow
parents: 39910
diff changeset
   549
      by (rule finite_subset) auto }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   550
  note this[intro, simp]
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   551
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   552
  { fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   553
    have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   554
    moreover {
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   555
      fix x y
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   556
      have *: "f -` {f x} \<inter> g -` {g x} \<inter> space M
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   557
          = (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
   558
      assume "x \<in> space M" "y \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   559
      hence "f -` {f x} \<inter> g -` {g x} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   560
        using assms unfolding simple_function_def * by auto }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   561
    ultimately
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   562
    have "\<mu> (f -` {f x} \<inter> space M) = setsum (\<mu>) (?sub (f x))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   563
      by (subst measure_finitely_additive) auto }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   564
  hence "simple_integral f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   565
    unfolding simple_integral_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   566
    by (subst setsum_Sigma[symmetric],
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   567
       auto intro!: setsum_cong simp: setsum_right_distrib[symmetric])
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   568
  also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * \<mu> A)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   569
  proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   570
    have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   571
    have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   572
      = (\<lambda>x. (f x, ?p x)) ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   573
    proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   574
      fix x assume "x \<in> space M"
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   575
      thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   576
        by (auto intro!: image_eqI[of _ _ "?p x"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   577
    qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   578
    thus ?thesis
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   579
      apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   580
      apply (rule_tac x="xa" in image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   581
      by simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   582
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   583
  finally show ?thesis .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   584
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   585
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   586
lemma (in measure_space) simple_integral_add[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   587
  assumes "simple_function f" and "simple_function g"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   588
  shows "(\<integral>\<^isup>Sx. f x + g x) = simple_integral f + simple_integral g"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   589
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   590
  { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   591
    assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   592
    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
   593
        "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   594
      by auto }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   595
  thus ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   596
    unfolding
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   597
      simple_function_partition[OF simple_function_add[OF assms] simple_function_Pair[OF assms]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   598
      simple_function_partition[OF `simple_function f` `simple_function g`]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   599
      simple_function_partition[OF `simple_function g` `simple_function f`]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   600
    apply (subst (3) Int_commute)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   601
    by (auto simp add: field_simps setsum_addf[symmetric] intro!: setsum_cong)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   602
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   603
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   604
lemma (in measure_space) simple_integral_setsum[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   605
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function (f i)"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   606
  shows "(\<integral>\<^isup>Sx. \<Sum>i\<in>P. f i x) = (\<Sum>i\<in>P. simple_integral (f i))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   607
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   608
  assume "finite P"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   609
  from this assms show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   610
    by induct (auto simp: simple_function_setsum simple_integral_add)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   611
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   612
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   613
lemma (in measure_space) simple_integral_mult[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   614
  assumes "simple_function f"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   615
  shows "(\<integral>\<^isup>Sx. c * f x) = c * simple_integral f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   616
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   617
  note mult = simple_function_mult[OF simple_function_const[of c] assms]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   618
  { 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
   619
    assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   620
    hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   621
      by auto }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   622
  thus ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   623
    unfolding simple_function_partition[OF mult assms]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   624
      simple_function_partition[OF assms mult]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   625
    by (auto simp: setsum_right_distrib field_simps intro!: setsum_cong)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   626
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   627
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   628
lemma (in sigma_algebra) simple_function_If:
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   629
  assumes sf: "simple_function f" "simple_function g" and A: "A \<in> sets M"
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   630
  shows "simple_function (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function ?IF")
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   631
proof -
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   632
  def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   633
  show ?thesis unfolding simple_function_def
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   634
  proof safe
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   635
    have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   636
    from finite_subset[OF this] assms
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   637
    show "finite (?IF ` space M)" unfolding simple_function_def by auto
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   638
  next
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   639
    fix x assume "x \<in> space M"
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   640
    then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   641
      then ((F (f x) \<inter> A) \<union> (G (f x) - (G (f x) \<inter> A)))
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   642
      else ((F (g x) \<inter> A) \<union> (G (g x) - (G (g x) \<inter> A))))"
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   643
      using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   644
    have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   645
      unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   646
    show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   647
  qed
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   648
qed
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   649
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   650
lemma (in measure_space) simple_integral_mono_AE:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   651
  assumes "simple_function f" and "simple_function g"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   652
  and mono: "AE x. f x \<le> g x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   653
  shows "simple_integral f \<le> simple_integral g"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   654
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   655
  let "?S x" = "(g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   656
  have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   657
    "\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   658
  show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   659
    unfolding *
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   660
      simple_function_partition[OF `simple_function f` `simple_function g`]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   661
      simple_function_partition[OF `simple_function g` `simple_function f`]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   662
  proof (safe intro!: setsum_mono)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   663
    fix x assume "x \<in> space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   664
    then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   665
    show "the_elem (f`?S x) * \<mu> (?S x) \<le> the_elem (g`?S x) * \<mu> (?S x)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   666
    proof (cases "f x \<le> g x")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   667
      case True then show ?thesis using * by (auto intro!: mult_right_mono)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   668
    next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   669
      case False
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   670
      obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   671
        using mono by (auto elim!: AE_E)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   672
      have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   673
      moreover have "?S x \<in> sets M" using assms
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   674
        by (rule_tac Int) (auto intro!: simple_functionD)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   675
      ultimately have "\<mu> (?S x) \<le> \<mu> N"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   676
        using `N \<in> sets M` by (auto intro!: measure_mono)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   677
      then show ?thesis using `\<mu> N = 0` by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   678
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   679
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   680
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   681
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   682
lemma (in measure_space) simple_integral_mono:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   683
  assumes "simple_function f" and "simple_function g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   684
  and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   685
  shows "simple_integral f \<le> simple_integral g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   686
proof (rule simple_integral_mono_AE[OF assms(1, 2)])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   687
  show "AE x. f x \<le> g x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   688
    using mono by (rule AE_cong) auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   689
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   690
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   691
lemma (in measure_space) simple_integral_cong_AE:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   692
  assumes "simple_function f" "simple_function g" and "AE x. f x = g x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   693
  shows "simple_integral f = simple_integral g"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   694
  using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   695
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   696
lemma (in measure_space) simple_integral_cong':
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   697
  assumes sf: "simple_function f" "simple_function g"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   698
  and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 0"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   699
  shows "simple_integral f = simple_integral g"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   700
proof (intro simple_integral_cong_AE sf AE_I)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   701
  show "\<mu> {x\<in>space M. f x \<noteq> g x} = 0" by fact
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   702
  show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   703
    using sf[THEN borel_measurable_simple_function] by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   704
qed simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   705
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   706
lemma (in measure_space) simple_integral_indicator:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   707
  assumes "A \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   708
  assumes "simple_function f"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   709
  shows "(\<integral>\<^isup>Sx. f x * indicator A x) =
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   710
    (\<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
   711
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   712
  assume "A = space M"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   713
  moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x) = simple_integral f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   714
    by (auto intro!: simple_integral_cong)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   715
  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
   716
  ultimately show ?thesis by (simp add: simple_integral_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   717
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   718
  assume "A \<noteq> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   719
  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
   720
  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
   721
  proof safe
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   722
    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
   723
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   724
    fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   725
      using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   726
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   727
    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
   728
  qed
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   729
  have *: "(\<integral>\<^isup>Sx. f x * indicator A x) =
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   730
    (\<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
   731
    unfolding simple_integral_def I
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   732
  proof (rule setsum_mono_zero_cong_left)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   733
    show "finite (f ` space M \<union> {0})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   734
      using assms(2) unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   735
    show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   736
      using sets_into_space[OF assms(1)] by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   737
    have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   738
      by (auto simp: image_iff)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   739
    thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   740
      i * \<mu> (f -` {i} \<inter> space M \<inter> A) = 0" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   741
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   742
    fix x assume "x \<in> f`A \<union> {0}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   743
    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
   744
      by (auto simp: indicator_def split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   745
    thus "x * \<mu> (?I -` {x} \<inter> space M) =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   746
      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
   747
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   748
  show ?thesis unfolding *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   749
    using assms(2) unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   750
    by (auto intro!: setsum_mono_zero_cong_right)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   751
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   752
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   753
lemma (in measure_space) simple_integral_indicator_only[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   754
  assumes "A \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   755
  shows "simple_integral (indicator A) = \<mu> A"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   756
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   757
  assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   758
  thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   759
next
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   760
  assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::pextreal}" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   761
  thus ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   762
    using simple_integral_indicator[OF assms simple_function_const[of 1]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   763
    using sets_into_space[OF assms]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   764
    by (auto intro!: arg_cong[where f="\<mu>"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   765
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   766
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   767
lemma (in measure_space) simple_integral_null_set:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   768
  assumes "simple_function u" "N \<in> null_sets"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   769
  shows "(\<integral>\<^isup>Sx. u x * indicator N x) = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   770
proof -
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   771
  have "AE x. indicator N x = (0 :: pextreal)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   772
    using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   773
  then have "(\<integral>\<^isup>Sx. u x * indicator N x) = (\<integral>\<^isup>Sx. 0)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   774
    using assms by (intro simple_integral_cong_AE) (auto intro!: AE_disjI2)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   775
  then show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   776
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   777
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   778
lemma (in measure_space) simple_integral_cong_AE_mult_indicator:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   779
  assumes sf: "simple_function f" and eq: "AE x. x \<in> S" and "S \<in> sets M"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   780
  shows "simple_integral f = (\<integral>\<^isup>Sx. f x * indicator S x)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   781
proof (rule simple_integral_cong_AE)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   782
  show "simple_function f" by fact
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   783
  show "simple_function (\<lambda>x. f x * indicator S x)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   784
    using sf `S \<in> sets M` by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   785
  from eq show "AE x. f x = f x * indicator S x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   786
    by (rule AE_mp) simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   787
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   788
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   789
lemma (in measure_space) simple_integral_restricted:
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   790
  assumes "A \<in> sets M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   791
  assumes sf: "simple_function (\<lambda>x. f x * indicator A x)"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   792
  shows "measure_space.simple_integral (restricted_space A) \<mu> f = (\<integral>\<^isup>Sx. f x * indicator A x)"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   793
    (is "_ = simple_integral ?f")
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   794
  unfolding measure_space.simple_integral_def[OF restricted_measure_space[OF `A \<in> sets M`]]
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   795
  unfolding simple_integral_def
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   796
proof (simp, safe intro!: setsum_mono_zero_cong_left)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   797
  from sf show "finite (?f ` space M)"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   798
    unfolding simple_function_def by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   799
next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   800
  fix x assume "x \<in> A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   801
  then show "f x \<in> ?f ` space M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   802
    using sets_into_space `A \<in> sets M` by (auto intro!: image_eqI[of _ _ x])
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   803
next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   804
  fix x assume "x \<in> space M" "?f x \<notin> f`A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   805
  then have "x \<notin> A" by (auto simp: image_iff)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   806
  then show "?f x * \<mu> (?f -` {?f x} \<inter> space M) = 0" by simp
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   807
next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   808
  fix x assume "x \<in> A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   809
  then have "f x \<noteq> 0 \<Longrightarrow>
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   810
    f -` {f x} \<inter> A = ?f -` {f x} \<inter> space M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   811
    using `A \<in> sets M` sets_into_space
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   812
    by (auto simp: indicator_def split: split_if_asm)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   813
  then show "f x * \<mu> (f -` {f x} \<inter> A) =
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   814
    f x * \<mu> (?f -` {f x} \<inter> space M)"
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   815
    unfolding pextreal_mult_cancel_left by auto
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   816
qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   817
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   818
lemma (in measure_space) simple_integral_subalgebra:
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   819
  assumes N: "measure_space N \<mu>" and [simp]: "space N = space M"
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   820
  shows "measure_space.simple_integral N \<mu> = simple_integral"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   821
  unfolding simple_integral_def_raw
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   822
  unfolding measure_space.simple_integral_def_raw[OF N] by simp
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   823
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   824
lemma (in measure_space) simple_integral_vimage:
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   825
  assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   826
    and f: "sigma_algebra.simple_function M' f"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   827
  shows "measure_space.simple_integral M' (\<lambda>A. \<mu> (T -` A \<inter> space M)) f = (\<integral>\<^isup>S x. f (T x))"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   828
    (is "measure_space.simple_integral M' ?nu f = _")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   829
proof -
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   830
  interpret T: measure_space M' ?nu using T by (rule measure_space_vimage) auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   831
  show "T.simple_integral f = (\<integral>\<^isup>S x. f (T x))"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   832
    unfolding simple_integral_def T.simple_integral_def
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   833
  proof (intro setsum_mono_zero_cong_right ballI)
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   834
    show "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   835
      using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   836
    show "finite (f ` space M')"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   837
      using f unfolding T.simple_function_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   838
  next
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   839
    fix i assume "i \<in> f ` space M' - (\<lambda>x. f (T x)) ` space M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   840
    then have "T -` (f -` {i} \<inter> space M') \<inter> space M = {}" by (auto simp: image_iff)
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   841
    then show "i * \<mu> (T -` (f -` {i} \<inter> space M') \<inter> space M) = 0" by simp
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   842
  next
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   843
    fix i assume "i \<in> (\<lambda>x. f (T x)) ` space M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   844
    then have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   845
      using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   846
    then show "i * \<mu> (T -` (f -` {i} \<inter> space M') \<inter> space M) = i * \<mu> ((\<lambda>x. f (T x)) -` {i} \<inter> space M)"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   847
      by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   848
  qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   849
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   850
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   851
section "Continuous posititve integration"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   852
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
   853
definition (in measure_space) positive_integral (binder "\<integral>\<^isup>+ " 10) where
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   854
  "positive_integral f = SUPR {g. simple_function g \<and> g \<le> f} simple_integral"
40872
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   855
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   856
lemma (in measure_space) positive_integral_alt:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   857
  "positive_integral f =
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   858
    (SUPR {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M} simple_integral)" (is "_ = ?alt")
40872
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   859
proof (rule antisym SUP_leI)
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   860
  show "positive_integral f \<le> ?alt" unfolding positive_integral_def
40872
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   861
  proof (safe intro!: SUP_leI)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   862
    fix g assume g: "simple_function g" "g \<le> f"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   863
    let ?G = "g -` {\<omega>} \<inter> space M"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   864
    show "simple_integral g \<le>
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   865
      SUPR {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g ` space M} simple_integral"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   866
      (is "simple_integral g \<le> SUPR ?A simple_integral")
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   867
    proof cases
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   868
      let ?g = "\<lambda>x. indicator (space M - ?G) x * g x"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   869
      have g': "simple_function ?g"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   870
        using g by (auto intro: simple_functionD)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   871
      moreover
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   872
      assume "\<mu> ?G = 0"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   873
      then have "AE x. g x = ?g x" using g
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   874
        by (intro AE_I[where N="?G"])
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   875
           (auto intro: simple_functionD simp: indicator_def)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   876
      with g(1) g' have "simple_integral g = simple_integral ?g"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   877
        by (rule simple_integral_cong_AE)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   878
      moreover have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   879
      from this `g \<le> f` have "?g \<le> f" by (rule order_trans)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   880
      moreover have "\<omega> \<notin> ?g ` space M"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   881
        by (auto simp: indicator_def split: split_if_asm)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   882
      ultimately show ?thesis by (auto intro!: le_SUPI)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   883
    next
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   884
      assume "\<mu> ?G \<noteq> 0"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   885
      then have "?G \<noteq> {}" by auto
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   886
      then have "\<omega> \<in> g`space M" by force
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   887
      then have "space M \<noteq> {}" by auto
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   888
      have "SUPR ?A simple_integral = \<omega>"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   889
      proof (intro SUP_\<omega>[THEN iffD2] allI impI)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   890
        fix x assume "x < \<omega>"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   891
        then guess n unfolding less_\<omega>_Ex_of_nat .. note n = this
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   892
        then have "0 < n" by (intro neq0_conv[THEN iffD1] notI) simp
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   893
        let ?g = "\<lambda>x. (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * indicator ?G x"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   894
        show "\<exists>i\<in>?A. x < simple_integral i"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   895
        proof (intro bexI impI CollectI conjI)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   896
          show "simple_function ?g" using g
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   897
            by (auto intro!: simple_functionD simple_function_add)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   898
          have "?g \<le> g" by (auto simp: le_fun_def indicator_def)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   899
          from this g(2) show "?g \<le> f" by (rule order_trans)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   900
          show "\<omega> \<notin> ?g ` space M"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   901
            using `\<mu> ?G \<noteq> 0` by (auto simp: indicator_def split: split_if_asm)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   902
          have "x < (of_nat n / (if \<mu> ?G = \<omega> then 1 else \<mu> ?G)) * \<mu> ?G"
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   903
            using n `\<mu> ?G \<noteq> 0` `0 < n`
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   904
            by (auto simp: pextreal_noteq_omega_Ex field_simps)
40872
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   905
          also have "\<dots> = simple_integral ?g" using g `space M \<noteq> {}`
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   906
            by (subst simple_integral_indicator)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   907
               (auto simp: image_constant ac_simps dest: simple_functionD)
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   908
          finally show "x < simple_integral ?g" .
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   909
        qed
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   910
      qed
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   911
      then show ?thesis by simp
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   912
    qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   913
  qed
40872
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   914
qed (auto intro!: SUP_subset simp: positive_integral_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   915
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   916
lemma (in measure_space) positive_integral_cong_measure:
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   917
  assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<nu> A = \<mu> A"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   918
  shows "measure_space.positive_integral M \<nu> f = positive_integral f"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   919
proof -
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   920
  interpret v: measure_space M \<nu>
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   921
    by (rule measure_space_cong) fact
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   922
  with assms show ?thesis
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   923
    unfolding positive_integral_def v.positive_integral_def SUPR_def
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   924
    by (auto intro!: arg_cong[where f=Sup] image_cong
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   925
             simp: simple_integral_cong_measure[of \<nu>])
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   926
qed
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   927
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   928
lemma (in measure_space) positive_integral_alt1:
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   929
  "positive_integral f =
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   930
    (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)"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   931
  unfolding positive_integral_alt SUPR_def
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   932
proof (safe intro!: arg_cong[where f=Sup])
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   933
  fix g let ?g = "\<lambda>x. if x \<in> space M then g x else f x"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   934
  assume "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   935
  hence "?g \<le> f" "simple_function ?g" "simple_integral g = simple_integral ?g"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   936
    "\<omega> \<notin> g`space M"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   937
    unfolding le_fun_def by (auto cong: simple_function_cong simple_integral_cong)
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   938
  thus "simple_integral g \<in> simple_integral ` {g. simple_function g \<and> g \<le> f \<and> \<omega> \<notin> g`space M}"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   939
    by auto
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   940
next
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   941
  fix g assume "simple_function g" "g \<le> f" "\<omega> \<notin> g`space M"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   942
  hence "simple_function g" "\<forall>x\<in>space M. g x \<le> f x \<and> g x \<noteq> \<omega>"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   943
    by (auto simp add: le_fun_def image_iff)
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   944
  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>)}"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   945
    by auto
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   946
qed
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   947
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   948
lemma (in measure_space) positive_integral_cong:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   949
  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   950
  shows "positive_integral f = positive_integral g"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   951
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   952
  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
   953
    using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   954
  thus ?thesis unfolding positive_integral_alt1 by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   955
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   956
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   957
lemma (in measure_space) positive_integral_eq_simple_integral:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   958
  assumes "simple_function f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   959
  shows "positive_integral f = simple_integral f"
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   960
  unfolding positive_integral_def
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
   961
proof (safe intro!: pextreal_SUPI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   962
  fix g assume "simple_function g" "g \<le> f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   963
  with assms show "simple_integral g \<le> simple_integral f"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   964
    by (auto intro!: simple_integral_mono simp: le_fun_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   965
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   966
  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
   967
  with assms show "simple_integral f \<le> y" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   968
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   969
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   970
lemma (in measure_space) positive_integral_mono_AE:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   971
  assumes ae: "AE x. u x \<le> v x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   972
  shows "positive_integral u \<le> positive_integral v"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   973
  unfolding positive_integral_alt1
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   974
proof (safe intro!: SUPR_mono)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   975
  fix a assume a: "simple_function a" and mono: "\<forall>x\<in>space M. a x \<le> u x \<and> a x \<noteq> \<omega>"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   976
  from ae obtain N where N: "{x\<in>space M. \<not> u x \<le> v x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   977
    by (auto elim!: AE_E)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   978
  have "simple_function (\<lambda>x. a x * indicator (space M - N) x)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   979
    using `N \<in> sets M` a by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   980
  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>)}.
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   981
    simple_integral a \<le> simple_integral b"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   982
  proof (safe intro!: bexI[of _ "\<lambda>x. a x * indicator (space M - N) x"]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   983
                      simple_integral_mono_AE)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   984
    show "AE x. a x \<le> a x * indicator (space M - N) x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   985
    proof (rule AE_I, rule subset_refl)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   986
      have *: "{x \<in> space M. \<not> a x \<le> a x * indicator (space M - N) x} =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   987
        N \<inter> {x \<in> space M. a x \<noteq> 0}" (is "?N = _")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   988
        using `N \<in> sets M`[THEN sets_into_space] by (auto simp: indicator_def)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   989
      then show "?N \<in> sets M" 
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   990
        using `N \<in> sets M` `simple_function a`[THEN borel_measurable_simple_function]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   991
        by (auto intro!: measure_mono Int)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   992
      then have "\<mu> ?N \<le> \<mu> N"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   993
        unfolding * using `N \<in> sets M` by (auto intro!: measure_mono)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   994
      then show "\<mu> ?N = 0" using `\<mu> N = 0` by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   995
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   996
  next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   997
    fix x assume "x \<in> space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   998
    show "a x * indicator (space M - N) x \<le> v x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   999
    proof (cases "x \<in> N")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1000
      case True then show ?thesis by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1001
    next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1002
      case False
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1003
      with N mono have "a x \<le> u x" "u x \<le> v x" using `x \<in> space M` by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1004
      with False `x \<in> space M` show "a x * indicator (space M - N) x \<le> v x" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1005
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1006
    assume "a x * indicator (space M - N) x = \<omega>"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1007
    with mono `x \<in> space M` show False
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1008
      by (simp split: split_if_asm add: indicator_def)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1009
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1010
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1011
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1012
lemma (in measure_space) positive_integral_cong_AE:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1013
  "AE x. u x = v x \<Longrightarrow> positive_integral u = positive_integral v"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1014
  by (auto simp: eq_iff intro!: positive_integral_mono_AE)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1015
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1016
lemma (in measure_space) positive_integral_mono:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1017
  assumes mono: "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1018
  shows "positive_integral u \<le> positive_integral v"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1019
  using mono by (auto intro!: AE_cong positive_integral_mono_AE)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1020
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
  1021
lemma image_set_cong:
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
  1022
  assumes A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. f x = g y"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
  1023
  assumes B: "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. g y = f x"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
  1024
  shows "f ` A = g ` B"
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
  1025
  using assms by blast
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
  1026
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1027
lemma (in measure_space) positive_integral_SUP_approx:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1028
  assumes "f \<up> s"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1029
  and f: "\<And>i. f i \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1030
  and "simple_function u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1031
  and le: "u \<le> s" and real: "\<omega> \<notin> u`space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1032
  shows "simple_integral u \<le> (SUP i. positive_integral (f i))" (is "_ \<le> ?S")
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
  1033
proof (rule pextreal_le_mult_one_interval)
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
  1034
  fix a :: pextreal assume "0 < a" "a < 1"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1035
  hence "a \<noteq> 0" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1036
  let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1037
  have B: "\<And>i. ?B i \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1038
    using f `simple_function u` by (auto simp: borel_measurable_simple_function)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1039
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1040
  let "?uB i x" = "u x * indicator (?B i) x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1041
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1042
  { fix i have "?B i \<subseteq> ?B (Suc i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1043
    proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1044
      fix i x assume "a * u x \<le> f i x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1045
      also have "\<dots> \<le> f (Suc i) x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1046
        using `f \<up> s` unfolding isoton_def le_fun_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1047
      finally show "a * u x \<le> f (Suc i) x" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1048
    qed }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1049
  note B_mono = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1050
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1051
  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
  1052
    using `simple_function u` by (auto simp add: simple_function_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1053
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1054
  have "\<And>i. (\<lambda>n. (u -` {i} \<inter> space M) \<inter> ?B n) \<up> (u -` {i} \<inter> space M)" using B_mono unfolding isoton_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1055
  proof safe
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1056
    fix x i assume "x \<in> space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1057
    show "x \<in> (\<Union>i. (u -` {u x} \<inter> space M) \<inter> ?B i)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1058
    proof cases
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1059
      assume "u x = 0" thus ?thesis using `x \<in> space M` by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1060
    next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1061
      assume "u x \<noteq> 0"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1062
      with `a < 1` real `x \<in> space M`
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
  1063
      have "a * u x < 1 * u x" by (rule_tac pextreal_mult_strict_right_mono) (auto simp: image_iff)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1064
      also have "\<dots> \<le> (SUP i. f i x)" using le `f \<up> s`
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1065
        unfolding isoton_fun_expand by (auto simp: isoton_def le_fun_def)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1066
      finally obtain i where "a * u x < f i x" unfolding SUPR_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1067
        by (auto simp add: less_Sup_iff)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1068
      hence "a * u x \<le> f i x" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1069
      thus ?thesis using `x \<in> space M` by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1070
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1071
  qed auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1072
  note measure_conv = measure_up[OF Int[OF u B] this]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1073
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1074
  have "simple_integral u = (SUP i. simple_integral (?uB i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1075
    unfolding simple_integral_indicator[OF B `simple_function u`]
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
  1076
  proof (subst SUPR_pextreal_setsum, safe)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1077
    fix x n assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1078
    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
  1079
      \<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
  1080
      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
  1081
    thus "u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B n)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1082
            \<le> u x * \<mu> (u -` {u x} \<inter> space M \<inter> ?B (Suc n))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1083
      by (auto intro: mult_left_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1084
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1085
    show "simple_integral u =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1086
      (\<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
  1087
      using measure_conv unfolding simple_integral_def isoton_def
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
  1088
      by (auto intro!: setsum_cong simp: pextreal_SUP_cmult)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1089
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1090
  moreover
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1091
  have "a * (SUP i. simple_integral (?uB i)) \<le> ?S"
41023
9118eb4eb8dc it is known as the extended reals, not the infinite reals
hoelzl
parents: 40875
diff changeset
  1092
    unfolding pextreal_SUP_cmult[symmetric]
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1093
  proof (safe intro!: SUP_mono bexI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1094
    fix i
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1095
    have "a * simple_integral (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1096
      using B `simple_function u`
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1097
      by (subst simple_integral_mult[symmetric]) (auto simp: field_simps)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1098
    also have "\<dots> \<le> positive_integral (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1099
    proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1100
      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
  1101
      hence *: "simple_function (\<lambda>x. a * ?uB i x)" using B assms(3)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1102
        by (auto intro!: simple_integral_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1103
      show ?thesis unfolding positive_integral_eq_simple_integral[OF *, symmetric]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1104
        by (auto intro!: positive_integral_mono simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1105
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1106
    finally show "a * simple_integral (?uB i) \<le> positive_integral (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1107
      by auto
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1108
  qed simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1109
  ultimately show "a * simple_integral u \<le> ?S" by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1110
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1111
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1112
text {* Beppo-Levi monotone convergence theorem *}
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1113
lemma (in measure_space) positive_integral_isoton:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1114
  assumes "f \<up> u" "\<And>i. f i \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1115
  shows "(\<lambda>i. positive_integral (f i)) \<up> positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1116
  unfolding isoton_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1117
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1118
  fix i show "positive_integral (f i) \<le> positive_integral (f (Suc i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1119
    apply (rule positive_integral_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1120
    using `f \<up> u` unfolding isoton_def le_fun_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1121
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1122
  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
  1123
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1124
  show "(SUP i. positive_integral (f i)) = positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1125
  proof (rule antisym)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1126
    from `f \<up> u`[THEN isoton_Sup, unfolded le_fun_def]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1127
    show "(SUP j. positive_integral (f j)) \<le> positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1128
      by (auto intro!: SUP_leI positive_integral_mono)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1129
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1130
    show "positive_integral u \<le> (SUP i. positive_integral (f i))"
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
  1131
      unfolding positive_integral_alt[of u]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1132
      by (auto intro!: SUP_leI positive_integral_SUP_approx[OF assms])
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1133
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1134
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1135
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1136
lemma (in measure_space) positive_integral_monotone_convergence_SUP:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1137
  assumes "\<And>i x. x \<in> space M \<Longrightarrow> f i x \<le> f (Suc i) x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1138
  assumes "\<And>i. f i \<in> borel_measurable M"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1139
  shows "(SUP i. positive_integral (f i)) = (\<integral>\<^isup>+ x. SUP i. f i x)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1140
    (is "_ = positive_integral ?u")
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1141
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1142
  show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1143
  proof (rule antisym)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1144
    show "(SUP j. positive_integral (f j)) \<le> positive_integral ?u"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1145
      by (auto intro!: SUP_leI positive_integral_mono le_SUPI)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1146
  next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1147
    def rf \<equiv> "\<lambda>i. \<lambda>x\<in>space M. f i x" and ru \<equiv> "\<lambda>x\<in>space M. ?u x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1148
    have "\<And>i. rf i \<in> borel_measurable M" unfolding rf_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1149
      using assms by (simp cong: measurable_cong)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1150
    moreover have iso: "rf \<up> ru" using assms unfolding rf_def ru_def
41097
a1abfa4e2b44 use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents: 41096
diff changeset
  1151
      unfolding isoton_def le_fun_def fun_eq_iff SUPR_apply
40872
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
  1152
      using SUP_const[OF UNIV_not_empty]
41097
a1abfa4e2b44 use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents: 41096
diff changeset
  1153
      by (auto simp: restrict_def le_fun_def fun_eq_iff)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1154
    ultimately have "positive_integral ru \<le> (SUP i. positive_integral (rf i))"
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
  1155
      unfolding positive_integral_alt[of ru]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1156
      by (auto simp: le_fun_def intro!: SUP_leI positive_integral_SUP_approx)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1157
    then show "positive_integral ?u \<le> (SUP i. positive_integral (f i))"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1158
      unfolding ru_def rf_def by (simp cong: positive_integral_cong)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1159
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1160
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1161
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1162
lemma (in measure_space) SUP_simple_integral_sequences:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1163
  assumes f: "f \<up> u" "\<And>i. simple_function (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1164
  and g: "g \<up> u" "\<And>i. simple_function (g i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1165
  shows "(SUP i. simple_integral (f i)) = (SUP i. simple_integral (g i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1166
    (is "SUPR _ ?F = SUPR _ ?G")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1167
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1168
  have "(SUP i. ?F i) = (SUP i. positive_integral (f i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1169
    using assms by (simp add: positive_integral_eq_simple_integral)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1170
  also have "\<dots> = positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1171
    using positive_integral_isoton[OF f(1) borel_measurable_simple_function[OF f(2)]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1172
    unfolding isoton_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1173
  also have "\<dots> = (SUP i. positive_integral (g i))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1174
    using positive_integral_isoton[OF g(1) borel_measurable_simple_function[OF g(2)]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1175
    unfolding isoton_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1176
  also have "\<dots> = (SUP i. ?G i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1177
    using assms by (simp add: positive_integral_eq_simple_integral)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1178
  finally show ?thesis .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1179
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1180
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1181
lemma (in measure_space) positive_integral_const[simp]:
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1182
  "(\<integral>\<^isup>+ x. c) = c * \<mu> (space M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1183
  by (subst positive_integral_eq_simple_integral) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1184
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1185
lemma (in measure_space) positive_integral_isoton_simple:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1186
  assumes "f \<up> u" and e: "\<And>i. simple_function (f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1187
  shows "(\<lambda>i. simple_integral (f i)) \<up> positive_integral u"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1188
  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
  1189
  unfolding positive_integral_eq_simple_integral[OF e] .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1190
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1191
lemma (in measure_space) positive_integral_vimage:
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1192
  assumes T: "sigma_algebra M'" "T \<in> measurable M M'" and f: "f \<in> borel_measurable M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1193
  shows "measure_space.positive_integral M' (\<lambda>A. \<mu> (T -` A \<inter> space M)) f = (\<integral>\<^isup>+ x. f (T x))"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1194
    (is "measure_space.positive_integral M' ?nu f = _")
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1195
proof -
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1196
  interpret T: measure_space M' ?nu using T by (rule measure_space_vimage) auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1197
  obtain f' where f': "f' \<up> f" "\<And>i. T.simple_function (f' i)"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1198
    using T.borel_measurable_implies_simple_function_sequence[OF f] by blast
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1199
  then have f: "(\<lambda>i x. f' i (T x)) \<up> (\<lambda>x. f (T x))" "\<And>i. simple_function (\<lambda>x. f' i (T x))"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1200
    using simple_function_vimage[OF T] unfolding isoton_fun_expand by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1201
  show "T.positive_integral f = (\<integral>\<^isup>+ x. f (T x))"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1202
    using positive_integral_isoton_simple[OF f]
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1203
    using T.positive_integral_isoton_simple[OF f']
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1204
    unfolding simple_integral_vimage[OF T f'(2)] isoton_def
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1205
    by simp
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1206
qed
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1207
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1208
lemma (in measure_space) positive_integral_linear:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1209
  assumes f: "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1210
  and g: "g \<in> borel_measurable M"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1211
  shows "(\<integral>\<^isup>+ x. a * f x + g x) =
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1212