src/HOL/Probability/Lebesgue_Integration.thy
author hoelzl
Tue, 13 May 2014 11:35:47 +0200
changeset 56949 d1a937cbf858
parent 56571 f4635657d66f
permissions -rw-r--r--
clean up Lebesgue integration
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Probability/Lebesgue_Integration.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Armin Heller, TU München
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*)
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header {*Lebesgue Integration*}
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theory Lebesgue_Integration
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  imports Measure_Space Borel_Space
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begin
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lemma indicator_less_ereal[simp]:
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    13
  "indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)"
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    14
  by (simp add: indicator_def not_le)
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    15
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section "Simple function"
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text {*
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    19
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    20
Our simple functions are not restricted to positive real numbers. Instead
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    21
they are just functions with a finite range and are measurable when singleton
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    22
sets are measurable.
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    24
*}
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    25
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definition "simple_function M g \<longleftrightarrow>
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    27
    finite (g ` space M) \<and>
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    (\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
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lemma simple_functionD:
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    31
  assumes "simple_function M g"
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  shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
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    33
proof -
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
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    34
  show "finite (g ` space M)"
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
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    35
    using assms unfolding simple_function_def by auto
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    36
  have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
9a9d33f6fb46 generalized simple_functionD
hoelzl
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    37
  also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
9a9d33f6fb46 generalized simple_functionD
hoelzl
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    38
  finally show "g -` X \<inter> space M \<in> sets M" using assms
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    39
    by (auto simp del: UN_simps simp: simple_function_def)
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    40
qed
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    41
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lemma measurable_simple_function[measurable_dest]:
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  "simple_function M f \<Longrightarrow> f \<in> measurable M (count_space UNIV)"
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  unfolding simple_function_def measurable_def
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proof safe
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    46
  fix A assume "finite (f ` space M)" "\<forall>x\<in>f ` space M. f -` {x} \<inter> space M \<in> sets M"
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  then have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) \<in> sets M"
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    48
    by (intro sets.finite_UN) auto
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    49
  also have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) = f -` A \<inter> space M"
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    50
    by (auto split: split_if_asm)
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    51
  finally show "f -` A \<inter> space M \<in> sets M" .
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qed simp
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    53
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    54
lemma borel_measurable_simple_function:
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  "simple_function M f \<Longrightarrow> f \<in> borel_measurable M"
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    56
  by (auto dest!: measurable_simple_function simp: measurable_def)
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    57
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lemma simple_function_measurable2[intro]:
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  assumes "simple_function M f" "simple_function M g"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    60
  shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    61
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    62
  have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    63
    by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    64
  then show ?thesis using assms[THEN simple_functionD(2)] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    65
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    66
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    67
lemma simple_function_indicator_representation:
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    68
  fixes f ::"'a \<Rightarrow> ereal"
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    69
  assumes f: "simple_function M f" and x: "x \<in> space M"
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    70
  shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
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    71
  (is "?l = ?r")
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    72
proof -
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aaee86c0e237 moved generic lemmas in Probability to HOL
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    73
  have "?r = (\<Sum>y \<in> f ` space M.
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    74
    (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
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hoelzl
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    75
    by (auto intro!: setsum_cong2)
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hoelzl
parents: 38642
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    76
  also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
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hoelzl
parents: 38642
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    77
    using assms by (auto dest: simple_functionD simp: setsum_delta)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    78
  also have "... = f x" using x by (auto simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    79
  finally show ?thesis by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
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    80
qed
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hoelzl
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    81
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    82
lemma simple_function_notspace:
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cedb5cb948fd Rename extreal => ereal
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    83
  "simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h")
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hoelzl
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    84
proof -
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d5d342611edb Rewrite the Probability theory.
hoelzl
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    85
  have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    86
  hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    87
  have "?h -` {0} \<inter> space M = space M" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    88
  thus ?thesis unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    89
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    90
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    91
lemma simple_function_cong:
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    92
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
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hoelzl
parents: 41661
diff changeset
    93
  shows "simple_function M f \<longleftrightarrow> simple_function M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    94
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    95
  have "f ` space M = g ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    96
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    97
    using assms by (auto intro!: image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    98
  thus ?thesis unfolding simple_function_def using assms by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    99
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   100
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   101
lemma simple_function_cong_algebra:
41689
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hoelzl
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   102
  assumes "sets N = sets M" "space N = space M"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   103
  shows "simple_function M f \<longleftrightarrow> simple_function N f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   104
  unfolding simple_function_def assms ..
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   105
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
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   106
lemma simple_function_borel_measurable:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   107
  fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   108
  assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   109
  shows "simple_function M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   110
  using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   111
  by (auto intro: borel_measurable_vimage)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   112
56949
d1a937cbf858 clean up Lebesgue integration
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parents: 56571
diff changeset
   113
lemma simple_function_eq_measurable:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   114
  fixes f :: "'a \<Rightarrow> ereal"
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   115
  shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> measurable M (count_space UNIV)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   116
  using simple_function_borel_measurable[of f] measurable_simple_function[of M f]
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44666
diff changeset
   117
  by (fastforce simp: simple_function_def)
41981
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hoelzl
parents: 41831
diff changeset
   118
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05663f75964c reworked Probability theory
hoelzl
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diff changeset
   119
lemma simple_function_const[intro, simp]:
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hoelzl
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diff changeset
   120
  "simple_function M (\<lambda>x. c)"
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hoelzl
parents: 38642
diff changeset
   121
  by (auto intro: finite_subset simp: simple_function_def)
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hoelzl
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diff changeset
   122
lemma simple_function_compose[intro, simp]:
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hoelzl
parents: 41661
diff changeset
   123
  assumes "simple_function M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   124
  shows "simple_function M (g \<circ> f)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   125
  unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   126
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   127
  show "finite ((g \<circ> f) ` space M)"
56154
f0a927235162 more complete set of lemmas wrt. image and composition
haftmann
parents: 54611
diff changeset
   128
    using assms unfolding simple_function_def by (auto simp: image_comp [symmetric])
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   129
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   130
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   131
  let ?G = "g -` {g (f x)} \<inter> (f`space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   132
  have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   133
    (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   134
  show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   135
    using assms unfolding simple_function_def *
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   136
    by (rule_tac sets.finite_UN) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   137
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   138
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   139
lemma simple_function_indicator[intro, simp]:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   140
  assumes "A \<in> sets M"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   141
  shows "simple_function M (indicator A)"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   142
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   143
  have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   144
    by (auto simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   145
  hence "finite ?S" by (rule finite_subset) simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   146
  moreover have "- A \<inter> space M = space M - A" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   147
  ultimately show ?thesis unfolding simple_function_def
46905
6b1c0a80a57a prefer abs_def over def_raw;
wenzelm
parents: 46884
diff changeset
   148
    using assms by (auto simp: indicator_def [abs_def])
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   149
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   150
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   151
lemma simple_function_Pair[intro, simp]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   152
  assumes "simple_function M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   153
  assumes "simple_function M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   154
  shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   155
  unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   156
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   157
  show "finite (?p ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   158
    using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   159
    by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   160
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   161
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   162
  have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   163
      (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   164
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   165
  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
   166
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   167
qed
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   168
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   169
lemma simple_function_compose1:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   170
  assumes "simple_function M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   171
  shows "simple_function M (\<lambda>x. g (f x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   172
  using simple_function_compose[OF assms, of g]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   173
  by (simp add: comp_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   174
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   175
lemma simple_function_compose2:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   176
  assumes "simple_function M f" and "simple_function M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   177
  shows "simple_function M (\<lambda>x. h (f x) (g x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   178
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   179
  have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   180
    using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   181
  thus ?thesis by (simp_all add: comp_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   182
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   183
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   184
lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   185
  and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   186
  and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   187
  and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   188
  and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   189
  and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   190
  and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   191
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   192
lemma simple_function_setsum[intro, simp]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   193
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   194
  shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   195
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   196
  assume "finite P" from this assms show ?thesis by induct auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   197
qed auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   198
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   199
lemma simple_function_ereal[intro, simp]: 
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   200
  fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   201
  shows "simple_function M (\<lambda>x. ereal (f x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   202
  by (auto intro!: simple_function_compose1[OF sf])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   203
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   204
lemma simple_function_real_of_nat[intro, simp]: 
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   205
  fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   206
  shows "simple_function M (\<lambda>x. real (f x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   207
  by (auto intro!: simple_function_compose1[OF sf])
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   208
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   209
lemma borel_measurable_implies_simple_function_sequence:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   210
  fixes u :: "'a \<Rightarrow> ereal"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   211
  assumes u: "u \<in> borel_measurable M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   212
  shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   213
             (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   214
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   215
  def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   216
  { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   217
    proof (split split_if, intro conjI impI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   218
      assume "\<not> real j \<le> u x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   219
      then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56218
diff changeset
   220
         by (cases "u x") (auto intro!: natfloor_mono)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   221
      moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56218
diff changeset
   222
        by (intro real_natfloor_le) auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   223
      ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   224
        unfolding real_of_nat_le_iff by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   225
    qed auto }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   226
  note f_upper = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   227
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   228
  have real_f:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   229
    "\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (natfloor (real (u x) * 2 ^ i)))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   230
    unfolding f_def by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   231
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   232
  let ?g = "\<lambda>j x. real (f x j) / 2^j :: ereal"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   233
  show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   234
  proof (intro exI[of _ ?g] conjI allI ballI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   235
    fix i
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   236
    have "simple_function M (\<lambda>x. real (f x i))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   237
    proof (intro simple_function_borel_measurable)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   238
      show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
50021
d96a3f468203 add support for function application to measurability prover
hoelzl
parents: 50003
diff changeset
   239
        using u by (auto simp: real_f)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   240
      have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   241
        using f_upper[of _ i] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   242
      then show "finite ((\<lambda>x. real (f x i))`space M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   243
        by (rule finite_subset) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   244
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   245
    then show "simple_function M (?g i)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   246
      by (auto intro: simple_function_ereal simple_function_div)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   247
  next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   248
    show "incseq ?g"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   249
    proof (intro incseq_ereal incseq_SucI le_funI)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   250
      fix x and i :: nat
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   251
      have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   252
      proof ((split split_if)+, intro conjI impI)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   253
        assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   254
        then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   255
          by (cases "u x") (auto intro!: le_natfloor)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   256
      next
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   257
        assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   258
        then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   259
          by (cases "u x") auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   260
      next
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   261
        assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   262
        have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   263
          by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   264
        also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   265
        proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   266
          assume "0 \<le> u x" then show ?thesis
46671
3a40ea076230 removing unnecessary assumptions in RComplete;
bulwahn
parents: 45342
diff changeset
   267
            by (intro le_mult_natfloor) 
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   268
        next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   269
          assume "\<not> 0 \<le> u x" then show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   270
            by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   271
        qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   272
        also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   273
          by (simp add: ac_simps)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   274
        finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   275
      qed simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   276
      then show "?g i x \<le> ?g (Suc i) x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   277
        by (auto simp: field_simps)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   278
    qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   279
  next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   280
    fix x show "(SUP i. ?g i x) = max 0 (u x)"
51000
c9adb50f74ad use order topology for extended reals
hoelzl
parents: 50384
diff changeset
   281
    proof (rule SUP_eqI)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   282
      fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   283
        by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56218
diff changeset
   284
                                     mult_nonpos_nonneg)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   285
    next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   286
      fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
56571
f4635657d66f added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents: 56537
diff changeset
   287
      have "\<And>i. 0 \<le> ?g i x" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   288
      from order_trans[OF this *] have "0 \<le> y" by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   289
      show "max 0 (u x) \<le> y"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   290
      proof (cases y)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   291
        case (real r)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   292
        with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44568
diff changeset
   293
        from reals_Archimedean2[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   294
        then have "\<exists>p. max 0 (u x) = ereal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   295
        then guess p .. note ux = this
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44568
diff changeset
   296
        obtain m :: nat where m: "p < real m" using reals_Archimedean2 ..
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   297
        have "p \<le> r"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   298
        proof (rule ccontr)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   299
          assume "\<not> p \<le> r"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   300
          with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56218
diff changeset
   301
          obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: field_simps)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   302
          then have "r * 2^max N m < p * 2^max N m - 1" by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   303
          moreover
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   304
          have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   305
            using *[of "max N m"] m unfolding real_f using ux
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56218
diff changeset
   306
            by (cases "0 \<le> u x") (simp_all add: max_def split: split_if_asm)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   307
          then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   308
            by (metis real_natfloor_gt_diff_one less_le_trans)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   309
          ultimately show False by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   310
        qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   311
        then show "max 0 (u x) \<le> y" using real ux by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   312
      qed (insert `0 \<le> y`, auto)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   313
    qed
56571
f4635657d66f added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents: 56537
diff changeset
   314
  qed auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   315
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   316
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   317
lemma borel_measurable_implies_simple_function_sequence':
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   318
  fixes u :: "'a \<Rightarrow> ereal"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   319
  assumes u: "u \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   320
  obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   321
    "\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   322
  using borel_measurable_implies_simple_function_sequence[OF u] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   323
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   324
lemma simple_function_induct[consumes 1, case_names cong set mult add, induct set: simple_function]:
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   325
  fixes u :: "'a \<Rightarrow> ereal"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   326
  assumes u: "simple_function M u"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   327
  assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   328
  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   329
  assumes mult: "\<And>u c. P u \<Longrightarrow> P (\<lambda>x. c * u x)"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   330
  assumes add: "\<And>u v. P u \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   331
  shows "P u"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   332
proof (rule cong)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   333
  from AE_space show "AE x in M. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   334
  proof eventually_elim
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   335
    fix x assume x: "x \<in> space M"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   336
    from simple_function_indicator_representation[OF u x]
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   337
    show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   338
  qed
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   339
next
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   340
  from u have "finite (u ` space M)"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   341
    unfolding simple_function_def by auto
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   342
  then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   343
  proof induct
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   344
    case empty show ?case
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   345
      using set[of "{}"] by (simp add: indicator_def[abs_def])
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   346
  qed (auto intro!: add mult set simple_functionD u)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   347
next
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   348
  show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   349
    apply (subst simple_function_cong)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   350
    apply (rule simple_function_indicator_representation[symmetric])
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   351
    apply (auto intro: u)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   352
    done
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   353
qed fact
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   354
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   355
lemma simple_function_induct_nn[consumes 2, case_names cong set mult add]:
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   356
  fixes u :: "'a \<Rightarrow> ereal"
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   357
  assumes u: "simple_function M u" and nn: "\<And>x. 0 \<le> u x"
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   358
  assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   359
  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
49797
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   360
  assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   361
  assumes add: "\<And>u v. simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> simple_function M v \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   362
  shows "P u"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   363
proof -
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   364
  show ?thesis
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   365
  proof (rule cong)
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   366
    fix x assume x: "x \<in> space M"
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   367
    from simple_function_indicator_representation[OF u x]
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   368
    show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   369
  next
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   370
    show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   371
      apply (subst simple_function_cong)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   372
      apply (rule simple_function_indicator_representation[symmetric])
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   373
      apply (auto intro: u)
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   374
      done
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   375
  next
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   376
    from u nn have "finite (u ` space M)" "\<And>x. x \<in> u ` space M \<Longrightarrow> 0 \<le> x"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   377
      unfolding simple_function_def by auto
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   378
    then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   379
    proof induct
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   380
      case empty show ?case
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   381
        using set[of "{}"] by (simp add: indicator_def[abs_def])
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   382
    qed (auto intro!: add mult set simple_functionD u setsum_nonneg
49797
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   383
       simple_function_setsum)
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   384
  qed fact
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   385
qed
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   386
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   387
lemma borel_measurable_induct[consumes 2, case_names cong set mult add seq, induct set: borel_measurable]:
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   388
  fixes u :: "'a \<Rightarrow> ereal"
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   389
  assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x"
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   390
  assumes cong: "\<And>f g. f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P g \<Longrightarrow> P f"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   391
  assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
49797
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   392
  assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   393
  assumes add: "\<And>u v. u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   394
  assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow>  (\<And>i x. 0 \<le> U i x) \<Longrightarrow>  (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> P (SUP i. U i)"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   395
  shows "P u"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   396
  using u
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   397
proof (induct rule: borel_measurable_implies_simple_function_sequence')
49797
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   398
  fix U assume U: "\<And>i. simple_function M (U i)" "incseq U" "\<And>i. \<infinity> \<notin> range (U i)" and
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   399
    sup: "\<And>x. (SUP i. U i x) = max 0 (u x)" and nn: "\<And>i x. 0 \<le> U i x"
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   400
  have u_eq: "u = (SUP i. U i)"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   401
    using nn u sup by (auto simp: max_def)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   402
  
49797
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   403
  from U have "\<And>i. U i \<in> borel_measurable M"
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   404
    by (simp add: borel_measurable_simple_function)
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   405
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   406
  show "P u"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   407
    unfolding u_eq
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   408
  proof (rule seq)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   409
    fix i show "P (U i)"
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   410
      using `simple_function M (U i)` nn
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   411
      by (induct rule: simple_function_induct_nn)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   412
         (auto intro: set mult add cong dest!: borel_measurable_simple_function)
49797
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   413
  qed fact+
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   414
qed
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   415
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   416
lemma simple_function_If_set:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   417
  assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   418
  shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   419
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   420
  def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   421
  show ?thesis unfolding simple_function_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   422
  proof safe
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   423
    have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   424
    from finite_subset[OF this] assms
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   425
    show "finite (?IF ` space M)" unfolding simple_function_def by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   426
  next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   427
    fix x assume "x \<in> space M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   428
    then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   429
      then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   430
      else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   431
      using sets.sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   432
    have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   433
      unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   434
    show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   435
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   436
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   437
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   438
lemma simple_function_If:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   439
  assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   440
  shows "simple_function M (\<lambda>x. if P x then f x else g x)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   441
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   442
  have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   443
  with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   444
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   445
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   446
lemma simple_function_subalgebra:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   447
  assumes "simple_function N f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   448
  and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   449
  shows "simple_function M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   450
  using assms unfolding simple_function_def by auto
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   451
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   452
lemma simple_function_comp:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   453
  assumes T: "T \<in> measurable M M'"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   454
    and f: "simple_function M' f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   455
  shows "simple_function M (\<lambda>x. f (T x))"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   456
proof (intro simple_function_def[THEN iffD2] conjI ballI)
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   457
  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
   458
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   459
  then show "finite ((\<lambda>x. f (T x)) ` space M)"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   460
    using f unfolding simple_function_def by (auto intro: finite_subset)
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   461
  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
   462
  then have "i \<in> f ` space M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   463
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   464
  then have "f -` {i} \<inter> space M' \<in> sets M'"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   465
    using f unfolding simple_function_def by auto
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   466
  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
   467
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   468
  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
   469
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   470
  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
   471
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   472
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   473
section "Simple integral"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   474
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   475
definition simple_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>S") where
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   476
  "integral\<^sup>S M f = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M))"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   477
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   478
syntax
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   479
  "_simple_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>S _. _ \<partial>_" [60,61] 110)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   480
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   481
translations
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   482
  "\<integral>\<^sup>S x. f \<partial>M" == "CONST simple_integral M (%x. f)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   483
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   484
lemma simple_integral_cong:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   485
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   486
  shows "integral\<^sup>S M f = integral\<^sup>S M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   487
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   488
  have "f ` space M = g ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   489
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   490
    using assms by (auto intro!: image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   491
  thus ?thesis unfolding simple_integral_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   492
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   493
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   494
lemma simple_integral_const[simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   495
  "(\<integral>\<^sup>Sx. c \<partial>M) = c * (emeasure M) (space M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   496
proof (cases "space M = {}")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   497
  case True thus ?thesis unfolding simple_integral_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   498
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   499
  case False hence "(\<lambda>x. c) ` space M = {c}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   500
  thus ?thesis unfolding simple_integral_def by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   501
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   502
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   503
lemma simple_function_partition:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   504
  assumes f: "simple_function M f" and g: "simple_function M g"
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   505
  assumes sub: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow> g x = g y \<Longrightarrow> f x = f y"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   506
  assumes v: "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   507
  shows "integral\<^sup>S M f = (\<Sum>y\<in>g ` space M. v y * emeasure M {x\<in>space M. g x = y})"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   508
    (is "_ = ?r")
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   509
proof -
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   510
  from f g have [simp]: "finite (f`space M)" "finite (g`space M)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   511
    by (auto simp: simple_function_def)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   512
  from f g have [measurable]: "f \<in> measurable M (count_space UNIV)" "g \<in> measurable M (count_space UNIV)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   513
    by (auto intro: measurable_simple_function)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   514
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   515
  { fix y assume "y \<in> space M"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   516
    then have "f ` space M \<inter> {i. \<exists>x\<in>space M. i = f x \<and> g y = g x} = {v (g y)}"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   517
      by (auto cong: sub simp: v[symmetric]) }
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   518
  note eq = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   519
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   520
  have "integral\<^sup>S M f =
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   521
    (\<Sum>y\<in>f`space M. y * (\<Sum>z\<in>g`space M. 
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   522
      if \<exists>x\<in>space M. y = f x \<and> z = g x then emeasure M {x\<in>space M. g x = z} else 0))"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   523
    unfolding simple_integral_def
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   524
  proof (safe intro!: setsum_cong ereal_left_mult_cong)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   525
    fix y assume y: "y \<in> space M" "f y \<noteq> 0"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   526
    have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} = 
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   527
        {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   528
      by auto
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   529
    have eq:"(\<Union>i\<in>{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}. {x \<in> space M. g x = i}) =
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   530
        f -` {f y} \<inter> space M"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   531
      by (auto simp: eq_commute cong: sub rev_conj_cong)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   532
    have "finite (g`space M)" by simp
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   533
    then have "finite {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   534
      by (rule rev_finite_subset) auto
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   535
    then show "emeasure M (f -` {f y} \<inter> space M) =
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   536
      (\<Sum>z\<in>g ` space M. if \<exists>x\<in>space M. f y = f x \<and> z = g x then emeasure M {x \<in> space M. g x = z} else 0)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   537
      apply (simp add: setsum_cases)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   538
      apply (subst setsum_emeasure)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   539
      apply (auto simp: disjoint_family_on_def eq)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   540
      done
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   541
  qed
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   542
  also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M. 
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   543
      if \<exists>x\<in>space M. y = f x \<and> z = g x then y * emeasure M {x\<in>space M. g x = z} else 0))"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   544
    by (auto intro!: setsum_cong simp: setsum_ereal_right_distrib emeasure_nonneg)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   545
  also have "\<dots> = ?r"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   546
    by (subst setsum_commute)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   547
       (auto intro!: setsum_cong simp: setsum_cases scaleR_setsum_right[symmetric] eq)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   548
  finally show "integral\<^sup>S M f = ?r" .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   549
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   550
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   551
lemma simple_integral_add[simp]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   552
  assumes f: "simple_function M f" and "\<And>x. 0 \<le> f x" and g: "simple_function M g" and "\<And>x. 0 \<le> g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   553
  shows "(\<integral>\<^sup>Sx. f x + g x \<partial>M) = integral\<^sup>S M f + integral\<^sup>S M g"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   554
proof -
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   555
  have "(\<integral>\<^sup>Sx. f x + g x \<partial>M) =
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   556
    (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. (fst y + snd y) * emeasure M {x\<in>space M. (f x, g x) = y})"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   557
    by (intro simple_function_partition) (auto intro: f g)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   558
  also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * emeasure M {x\<in>space M. (f x, g x) = y}) +
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   559
    (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y})"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   560
    using assms(2,4) by (auto intro!: setsum_cong ereal_left_distrib simp: setsum_addf[symmetric])
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   561
  also have "(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * emeasure M {x\<in>space M. (f x, g x) = y}) = (\<integral>\<^sup>Sx. f x \<partial>M)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   562
    by (intro simple_function_partition[symmetric]) (auto intro: f g)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   563
  also have "(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y}) = (\<integral>\<^sup>Sx. g x \<partial>M)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   564
    by (intro simple_function_partition[symmetric]) (auto intro: f g)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   565
  finally show ?thesis .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   566
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   567
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   568
lemma simple_integral_setsum[simp]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   569
  assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   570
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   571
  shows "(\<integral>\<^sup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>S M (f i))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   572
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   573
  assume "finite P"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   574
  from this assms show ?thesis
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   575
    by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   576
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   577
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   578
lemma simple_integral_mult[simp]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   579
  assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   580
  shows "(\<integral>\<^sup>Sx. c * f x \<partial>M) = c * integral\<^sup>S M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   581
proof -
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   582
  have "(\<integral>\<^sup>Sx. c * f x \<partial>M) = (\<Sum>y\<in>f ` space M. (c * y) * emeasure M {x\<in>space M. f x = y})"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   583
    using f by (intro simple_function_partition) auto
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   584
  also have "\<dots> = c * integral\<^sup>S M f"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   585
    using f unfolding simple_integral_def
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   586
    by (subst setsum_ereal_right_distrib) (auto simp: emeasure_nonneg mult_assoc Int_def conj_commute)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   587
  finally show ?thesis .
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   588
qed
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   589
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   590
lemma simple_integral_mono_AE:
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   591
  assumes f[measurable]: "simple_function M f" and g[measurable]: "simple_function M g"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   592
  and mono: "AE x in M. f x \<le> g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   593
  shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   594
proof -
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   595
  let ?\<mu> = "\<lambda>P. emeasure M {x\<in>space M. P x}"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   596
  have "integral\<^sup>S M f = (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * ?\<mu> (\<lambda>x. (f x, g x) = y))"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   597
    using f g by (intro simple_function_partition) auto
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   598
  also have "\<dots> \<le> (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * ?\<mu> (\<lambda>x. (f x, g x) = y))"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   599
  proof (clarsimp intro!: setsum_mono)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   600
    fix x assume "x \<in> space M"
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   601
    let ?M = "?\<mu> (\<lambda>y. f y = f x \<and> g y = g x)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   602
    show "f x * ?M \<le> g x * ?M"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   603
    proof cases
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   604
      assume "?M \<noteq> 0"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   605
      then have "0 < ?M"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   606
        by (simp add: less_le emeasure_nonneg)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   607
      also have "\<dots> \<le> ?\<mu> (\<lambda>y. f x \<le> g x)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   608
        using mono by (intro emeasure_mono_AE) auto
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   609
      finally have "\<not> \<not> f x \<le> g x"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   610
        by (intro notI) auto
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   611
      then show ?thesis
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   612
        by (intro ereal_mult_right_mono) auto
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   613
    qed simp
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   614
  qed
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   615
  also have "\<dots> = integral\<^sup>S M g"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   616
    using f g by (intro simple_function_partition[symmetric]) auto
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   617
  finally show ?thesis .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   618
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   619
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   620
lemma simple_integral_mono:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   621
  assumes "simple_function M f" and "simple_function M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   622
  and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   623
  shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   624
  using assms by (intro simple_integral_mono_AE) auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   625
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   626
lemma simple_integral_cong_AE:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   627
  assumes "simple_function M f" and "simple_function M g"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   628
  and "AE x in M. f x = g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   629
  shows "integral\<^sup>S M f = integral\<^sup>S M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   630
  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
   631
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   632
lemma simple_integral_cong':
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   633
  assumes sf: "simple_function M f" "simple_function M g"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   634
  and mea: "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   635
  shows "integral\<^sup>S M f = integral\<^sup>S M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   636
proof (intro simple_integral_cong_AE sf AE_I)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   637
  show "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" by fact
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   638
  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
   639
    using sf[THEN borel_measurable_simple_function] by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   640
qed simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   641
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   642
lemma simple_integral_indicator:
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   643
  assumes A: "A \<in> sets M"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   644
  assumes f: "simple_function M f"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   645
  shows "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   646
    (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   647
proof -
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   648
  have eq: "(\<lambda>x. (f x, indicator A x)) ` space M \<inter> {x. snd x = 1} = (\<lambda>x. (f x, 1::ereal))`A"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   649
    using A[THEN sets.sets_into_space] by (auto simp: indicator_def image_iff split: split_if_asm)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   650
  have eq2: "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   651
    by (auto simp: image_iff)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   652
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   653
  have "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   654
    (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x))`space M. (fst y * snd y) * emeasure M {x\<in>space M. (f x, indicator A x) = y})"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   655
    using assms by (intro simple_function_partition) auto
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   656
  also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x::ereal))`space M.
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   657
    if snd y = 1 then fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A) else 0)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   658
    by (auto simp: indicator_def split: split_if_asm intro!: arg_cong2[where f="op *"] arg_cong2[where f=emeasure] setsum_cong)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   659
  also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, 1::ereal))`A. fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A))"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   660
    using assms by (subst setsum_cases) (auto intro!: simple_functionD(1) simp: eq)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   661
  also have "\<dots> = (\<Sum>y\<in>fst`(\<lambda>x. (f x, 1::ereal))`A. y * emeasure M (f -` {y} \<inter> space M \<inter> A))"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   662
    by (subst setsum_reindex[where f=fst]) (auto simp: inj_on_def)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   663
  also have "\<dots> = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   664
    using A[THEN sets.sets_into_space]
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   665
    by (intro setsum_mono_zero_cong_left simple_functionD f) (auto simp: image_comp comp_def eq2)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   666
  finally show ?thesis .
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   667
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   668
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   669
lemma simple_integral_indicator_only[simp]:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   670
  assumes "A \<in> sets M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   671
  shows "integral\<^sup>S M (indicator A) = emeasure M A"
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   672
  using simple_integral_indicator[OF assms, of "\<lambda>x. 1"] sets.sets_into_space[OF assms]
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   673
  by (simp_all add: image_constant_conv Int_absorb1 split: split_if_asm)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   674
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   675
lemma simple_integral_null_set:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   676
  assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   677
  shows "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   678
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   679
  have "AE x in M. indicator N x = (0 :: ereal)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   680
    using `N \<in> null_sets M` by (auto simp: indicator_def intro!: AE_I[of _ _ N])
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   681
  then have "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^sup>Sx. 0 \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   682
    using assms apply (intro simple_integral_cong_AE) by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   683
  then show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   684
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   685
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   686
lemma simple_integral_cong_AE_mult_indicator:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   687
  assumes sf: "simple_function M f" and eq: "AE x in M. x \<in> S" and "S \<in> sets M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   688
  shows "integral\<^sup>S M f = (\<integral>\<^sup>Sx. f x * indicator S x \<partial>M)"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   689
  using assms by (intro simple_integral_cong_AE) auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   690
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   691
lemma simple_integral_cmult_indicator:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   692
  assumes A: "A \<in> sets M"
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   693
  shows "(\<integral>\<^sup>Sx. c * indicator A x \<partial>M) = c * emeasure M A"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   694
  using simple_integral_mult[OF simple_function_indicator[OF A]]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   695
  unfolding simple_integral_indicator_only[OF A] by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   696
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   697
lemma simple_integral_positive:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   698
  assumes f: "simple_function M f" and ae: "AE x in M. 0 \<le> f x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   699
  shows "0 \<le> integral\<^sup>S M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   700
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   701
  have "integral\<^sup>S M (\<lambda>x. 0) \<le> integral\<^sup>S M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   702
    using simple_integral_mono_AE[OF _ f ae] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   703
  then show ?thesis by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   704
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   705
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   706
section "Continuous positive integration"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   707
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   708
definition positive_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>P") where
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   709
  "integral\<^sup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^sup>S M g)"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   710
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   711
syntax
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   712
  "_positive_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>+ _. _ \<partial>_" [60,61] 110)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   713
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   714
translations
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   715
  "\<integral>\<^sup>+ x. f \<partial>M" == "CONST positive_integral M (%x. f)"
40872
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   716
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   717
lemma positive_integral_positive:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   718
  "0 \<le> integral\<^sup>P M f"
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   719
  by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   720
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   721
lemma positive_integral_not_MInfty[simp]: "integral\<^sup>P M f \<noteq> -\<infinity>"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   722
  using positive_integral_positive[of M f] by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   723
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   724
lemma positive_integral_def_finite:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   725
  "integral\<^sup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^sup>S M g)"
56218
1c3f1f2431f9 elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents: 56213
diff changeset
   726
    (is "_ = SUPREMUM ?A ?f")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   727
  unfolding positive_integral_def
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   728
proof (safe intro!: antisym SUP_least)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   729
  fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   730
  let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   731
  note gM = g(1)[THEN borel_measurable_simple_function]
50252
4aa34bd43228 eliminated slightly odd identifiers;
wenzelm
parents: 50244
diff changeset
   732
  have \<mu>_G_pos: "0 \<le> (emeasure M) ?G" using gM by auto
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   733
  let ?g = "\<lambda>y x. if g x = \<infinity> then y else max 0 (g x)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   734
  from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   735
    apply (safe intro!: simple_function_max simple_function_If)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   736
    apply (force simp: max_def le_fun_def split: split_if_asm)+
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   737
    done
56218
1c3f1f2431f9 elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents: 56213
diff changeset
   738
  show "integral\<^sup>S M g \<le> SUPREMUM ?A ?f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   739
  proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   740
    have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   741
    assume "(emeasure M) ?G = 0"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   742
    with gM have "AE x in M. x \<notin> ?G"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   743
      by (auto simp add: AE_iff_null intro!: null_setsI)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   744
    with gM g show ?thesis
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   745
      by (intro SUP_upper2[OF g0] simple_integral_mono_AE)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   746
         (auto simp: max_def intro!: simple_function_If)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   747
  next
50252
4aa34bd43228 eliminated slightly odd identifiers;
wenzelm
parents: 50244
diff changeset
   748
    assume \<mu>_G: "(emeasure M) ?G \<noteq> 0"
56218
1c3f1f2431f9 elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents: 56213
diff changeset
   749
    have "SUPREMUM ?A (integral\<^sup>S M) = \<infinity>"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   750
    proof (intro SUP_PInfty)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   751
      fix n :: nat
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   752
      let ?y = "ereal (real n) / (if (emeasure M) ?G = \<infinity> then 1 else (emeasure M) ?G)"
50252
4aa34bd43228 eliminated slightly odd identifiers;
wenzelm
parents: 50244
diff changeset
   753
      have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>_G \<mu>_G_pos by (auto simp: ereal_divide_eq)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   754
      then have "?g ?y \<in> ?A" by (rule g_in_A)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   755
      have "real n \<le> ?y * (emeasure M) ?G"
50252
4aa34bd43228 eliminated slightly odd identifiers;
wenzelm
parents: 50244
diff changeset
   756
        using \<mu>_G \<mu>_G_pos by (cases "(emeasure M) ?G") (auto simp: field_simps)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   757
      also have "\<dots> = (\<integral>\<^sup>Sx. ?y * indicator ?G x \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   758
        using `0 \<le> ?y` `?g ?y \<in> ?A` gM
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   759
        by (subst simple_integral_cmult_indicator) auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   760
      also have "\<dots> \<le> integral\<^sup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   761
        by (intro simple_integral_mono) auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   762
      finally show "\<exists>i\<in>?A. real n \<le> integral\<^sup>S M i"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   763
        using `?g ?y \<in> ?A` by blast
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   764
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   765
    then show ?thesis by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   766
  qed
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   767
qed (auto intro: SUP_upper)
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   768
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   769
lemma positive_integral_mono_AE:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   770
  assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^sup>P M u \<le> integral\<^sup>P M v"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   771
  unfolding positive_integral_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   772
proof (safe intro!: SUP_mono)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   773
  fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   774
  from ae[THEN AE_E] guess N . note N = this
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   775
  then have ae_N: "AE x in M. x \<notin> N" by (auto intro: AE_not_in)
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   776
  let ?n = "\<lambda>x. n x * indicator (space M - N) x"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   777
  have "AE x in M. n x \<le> ?n x" "simple_function M ?n"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   778
    using n N ae_N by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   779
  moreover
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   780
  { fix x have "?n x \<le> max 0 (v x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   781
    proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   782
      assume x: "x \<in> space M - N"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   783
      with N have "u x \<le> v x" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   784
      with n(2)[THEN le_funD, of x] x show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   785
        by (auto simp: max_def split: split_if_asm)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   786
    qed simp }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   787
  then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   788
  moreover have "integral\<^sup>S M n \<le> integral\<^sup>S M ?n"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   789
    using ae_N N n by (auto intro!: simple_integral_mono_AE)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   790
  ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^sup>S M n \<le> integral\<^sup>S M m"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   791
    by force
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   792
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   793
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   794
lemma positive_integral_mono:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   795
  "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^sup>P M u \<le> integral\<^sup>P M v"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   796
  by (auto intro: positive_integral_mono_AE)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   797
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   798
lemma positive_integral_cong_AE:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   799
  "AE x in M. u x = v x \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P M v"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   800
  by (auto simp: eq_iff intro!: positive_integral_mono_AE)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   801
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   802
lemma positive_integral_cong:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   803
  "(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P M v"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   804
  by (auto intro: positive_integral_cong_AE)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   805
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   806
lemma positive_integral_eq_simple_integral:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   807
  assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^sup>P M f = integral\<^sup>S M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   808
proof -
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   809
  let ?f = "\<lambda>x. f x * indicator (space M) x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   810
  have f': "simple_function M ?f" using f by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   811
  with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   812
    by (auto simp: fun_eq_iff max_def split: split_indicator)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   813
  have "integral\<^sup>P M ?f \<le> integral\<^sup>S M ?f" using f'
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   814
    by (force intro!: SUP_least simple_integral_mono simp: le_fun_def positive_integral_def)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   815
  moreover have "integral\<^sup>S M ?f \<le> integral\<^sup>P M ?f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   816
    unfolding positive_integral_def
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   817
    using f' by (auto intro!: SUP_upper)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   818
  ultimately show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   819
    by (simp cong: positive_integral_cong simple_integral_cong)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   820
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   821
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   822
lemma positive_integral_eq_simple_integral_AE:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   823
  assumes f: "simple_function M f" "AE x in M. 0 \<le> f x" shows "integral\<^sup>P M f = integral\<^sup>S M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   824
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   825
  have "AE x in M. f x = max 0 (f x)" using f by (auto split: split_max)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   826
  with f have "integral\<^sup>P M f = integral\<^sup>S M (\<lambda>x. max 0 (f x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   827
    by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   828
             add: positive_integral_eq_simple_integral)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   829
  with assms show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   830
    by (auto intro!: simple_integral_cong_AE split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   831
qed
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   832
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   833
lemma positive_integral_SUP_approx:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   834
  assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   835
  and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   836
  shows "integral\<^sup>S M u \<le> (SUP i. integral\<^sup>P M (f i))" (is "_ \<le> ?S")
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   837
proof (rule ereal_le_mult_one_interval)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   838
  have "0 \<le> (SUP i. integral\<^sup>P M (f i))"
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   839
    using f(3) by (auto intro!: SUP_upper2 positive_integral_positive)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   840
  then show "(SUP i. integral\<^sup>P M (f i)) \<noteq> -\<infinity>" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   841
  have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   842
    using u(3) by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   843
  fix a :: ereal assume "0 < a" "a < 1"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   844
  hence "a \<noteq> 0" by auto
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   845
  let ?B = "\<lambda>i. {x \<in> space M. a * u x \<le> f i x}"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   846
  have B: "\<And>i. ?B i \<in> sets M"
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
   847
    using f `simple_function M u`[THEN borel_measurable_simple_function] by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   848
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   849
  let ?uB = "\<lambda>i x. u x * indicator (?B i) x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   850
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   851
  { fix i have "?B i \<subseteq> ?B (Suc i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   852
    proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   853
      fix i x assume "a * u x \<le> f i x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   854
      also have "\<dots> \<le> f (Suc i) x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   855
        using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   856
      finally show "a * u x \<le> f (Suc i) x" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   857
    qed }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   858
  note B_mono = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   859
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   860
  note B_u = sets.Int[OF u(1)[THEN simple_functionD(2)] B]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   861
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   862
  let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   863
  have measure_conv: "\<And>i. (emeasure M) (u -` {i} \<inter> space M) = (SUP n. (emeasure M) (?B' i n))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   864
  proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   865
    fix i
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   866
    have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   867
    have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   868
    have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   869
    proof safe
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   870
      fix x i assume x: "x \<in> space M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   871
      show "x \<in> (\<Union>i. ?B' (u x) i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   872
      proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   873
        assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   874
      next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   875
        assume "u x \<noteq> 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   876
        with `a < 1` u_range[OF `x \<in> space M`]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   877
        have "a * u x < 1 * u x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   878
          by (intro ereal_mult_strict_right_mono) (auto simp: image_iff)
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46731
diff changeset
   879
        also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def)
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   880
        finally obtain i where "a * u x < f i x" unfolding SUP_def
56166
9a241bc276cd normalising simp rules for compound operators
haftmann
parents: 56154
diff changeset
   881
          by (auto simp add: less_SUP_iff)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   882
        hence "a * u x \<le> f i x" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   883
        thus ?thesis using `x \<in> space M` by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   884
      qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   885
    qed
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   886
    then show "?thesis i" using SUP_emeasure_incseq[OF 1 2] by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   887
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   888
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   889
  have "integral\<^sup>S M u = (SUP i. integral\<^sup>S M (?uB i))"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   890
    unfolding simple_integral_indicator[OF B `simple_function M u`]
56212
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
   891
  proof (subst SUP_ereal_setsum, safe)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   892
    fix x n assume "x \<in> space M"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   893
    with u_range show "incseq (\<lambda>i. u x * (emeasure M) (?B' (u x) i))" "\<And>i. 0 \<le> u x * (emeasure M) (?B' (u x) i)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   894
      using B_mono B_u by (auto intro!: emeasure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   895
  next
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   896
    show "integral\<^sup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * (emeasure M) (?B' i n))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   897
      using measure_conv u_range B_u unfolding simple_integral_def
56212
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
   898
      by (auto intro!: setsum_cong SUP_ereal_cmult [symmetric])
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   899
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   900
  moreover
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   901
  have "a * (SUP i. integral\<^sup>S M (?uB i)) \<le> ?S"
56212
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
   902
    apply (subst SUP_ereal_cmult [symmetric])
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
   903
  proof (safe intro!: SUP_mono bexI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   904
    fix i
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   905
    have "a * integral\<^sup>S M (?uB i) = (\<integral>\<^sup>Sx. a * ?uB i x \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   906
      using B `simple_function M u` u_range
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   907
      by (subst simple_integral_mult) (auto split: split_indicator)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   908
    also have "\<dots> \<le> integral\<^sup>P M (f i)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   909
    proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   910
      have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   911
      show ?thesis using f(3) * u_range `0 < a`
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   912
        by (subst positive_integral_eq_simple_integral[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   913
           (auto intro!: positive_integral_mono split: split_indicator)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   914
    qed
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   915
    finally show "a * integral\<^sup>S M (?uB i) \<le> integral\<^sup>P M (f i)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   916
      by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   917
  next
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   918
    fix i show "0 \<le> \<integral>\<^sup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   919
      by (intro simple_integral_positive) (auto split: split_indicator)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   920
  qed (insert `0 < a`, auto)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   921
  ultimately show "a * integral\<^sup>S M u \<le> ?S" by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   922
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   923
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   924
lemma incseq_positive_integral:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   925
  assumes "incseq f" shows "incseq (\<lambda>i. integral\<^sup>P M (f i))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   926
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   927
  have "\<And>i x. f i x \<le> f (Suc i) x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   928
    using assms by (auto dest!: incseq_SucD simp: le_fun_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   929
  then show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   930
    by (auto intro!: incseq_SucI positive_integral_mono)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   931
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   932
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   933
text {* Beppo-Levi monotone convergence theorem *}
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   934
lemma positive_integral_monotone_convergence_SUP:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   935
  assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   936
  shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   937
proof (rule antisym)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   938
  show "(SUP j. integral\<^sup>P M (f j)) \<le> (\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M)"
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   939
    by (auto intro!: SUP_least SUP_upper positive_integral_mono)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   940
next
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   941
  show "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^sup>P M (f j))"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   942
    unfolding positive_integral_def_finite[of _ "\<lambda>x. SUP i. f i x"]
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   943
  proof (safe intro!: SUP_least)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   944
    fix g assume g: "simple_function M g"
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
   945
      and *: "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
   946
    then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}"
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   947
      using f by (auto intro!: SUP_upper2)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
   948
    with * show "integral\<^sup>S M g \<le> (SUP j. integral\<^sup>P M (f j))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   949
      by (intro  positive_integral_SUP_approx[OF f g _ g'])
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46731
diff changeset
   950
         (auto simp: le_fun_def max_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   951
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   952
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   953
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   954
lemma positive_integral_monotone_convergence_SUP_AE:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   955
  assumes f: "\<And>i. AE x in M. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   956
  shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   957
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   958
  from f have "AE x in M. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   959
    by (simp add: AE_all_countable)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   960
  from this[THEN AE_E] guess N . note N = this
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   961
  let ?f = "\<lambda>i x. if x \<in> space M - N then f i x else 0"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   962
  have f_eq: "AE x in M. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N])
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   963
  then have "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^sup>+ x. (SUP i. ?f i x) \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   964
    by (auto intro!: positive_integral_cong_AE)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   965
  also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. ?f i x \<partial>M))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   966
  proof (rule positive_integral_monotone_convergence_SUP)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   967
    show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   968
    { fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   969
        using f N(3) by (intro measurable_If_set) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   970
      fix x show "0 \<le> ?f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   971
        using N(1) by auto }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   972
  qed
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   973
  also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. f i x \<partial>M))"
56218
1c3f1f2431f9 elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents: 56213
diff changeset
   974
    using f_eq by (force intro!: arg_cong[where f="SUPREMUM UNIV"] positive_integral_cong_AE ext)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   975
  finally show ?thesis .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   976
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   977
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   978
lemma positive_integral_monotone_convergence_SUP_AE_incseq:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   979
  assumes f: "incseq f" "\<And>i. AE x in M. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   980
  shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   981
  using f[unfolded incseq_Suc_iff le_fun_def]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   982
  by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   983
     auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   984
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   985
lemma positive_integral_monotone_convergence_simple:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   986
  assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   987
  shows "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   988
  using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   989
    f(3)[THEN borel_measurable_simple_function] f(2)]
56218
1c3f1f2431f9 elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents: 56213
diff changeset
   990
  by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPREMUM UNIV"] ext)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   991
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   992
lemma positive_integral_max_0:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   993
  "(\<integral>\<^sup>+x. max 0 (f x) \<partial>M) = integral\<^sup>P M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   994
  by (simp add: le_fun_def positive_integral_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   995
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   996
lemma positive_integral_cong_pos:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   997
  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   998
  shows "integral\<^sup>P M f = integral\<^sup>P M g"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   999
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1000
  have "integral\<^sup>P M (\<lambda>x. max 0 (f x)) = integral\<^sup>P M (\<lambda>x. max 0 (g x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1001
  proof (intro positive_integral_cong)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1002
    fix x assume "x \<in> space M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1003
    from assms[OF this] show "max 0 (f x) = max 0 (g x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1004
      by (auto split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1005
  qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1006
  then show ?thesis by (simp add: positive_integral_max_0)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1007
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1008
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1009
lemma SUP_simple_integral_sequences:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1010
  assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1011
  and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1012
  and eq: "AE x in M. (SUP i. f i x) = (SUP i. g i x)"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1013
  shows "(SUP i. integral\<^sup>S M (f i)) = (SUP i. integral\<^sup>S M (g i))"
56218
1c3f1f2431f9 elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents: 56213
diff changeset
  1014
    (is "SUPREMUM _ ?F = SUPREMUM _ ?G")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1015
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1016
  have "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1017
    using f by (rule positive_integral_monotone_convergence_simple)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1018
  also have "\<dots> = (\<integral>\<^sup>+x. (SUP i. g i x) \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1019
    unfolding eq[THEN positive_integral_cong_AE] ..
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1020
  also have "\<dots> = (SUP i. ?G i)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1021
    using g by (rule positive_integral_monotone_convergence_simple[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1022
  finally show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1023
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1024
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1025
lemma positive_integral_const[simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1026
  "0 \<le> c \<Longrightarrow> (\<integral>\<^sup>+ x. c \<partial>M) = c * (emeasure M) (space M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1027
  by (subst positive_integral_eq_simple_integral) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1028
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1029
lemma positive_integral_linear:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1030
  assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1031
  and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1032
  shows "(\<integral>\<^sup>+ x. a * f x + g x \<partial>M) = a * integral\<^sup>P M f + integral\<^sup>P M g"
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1033
    (is "integral\<^sup>P M ?L = _")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1034
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1035
  from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1036
  note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1037
  from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1038
  note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1039
  let ?L' = "\<lambda>i x. a * u i x + v i x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1040
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1041
  have "?L \<in> borel_measurable M" using assms by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1042
  from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1043
  note l = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1044
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1045
  have inc: "incseq (\<lambda>i. a * integral\<^sup>S M (u i))" "incseq (\<lambda>i. integral\<^sup>S M (v i))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1046
    using u v `0 \<le> a`
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1047
    by (auto simp: incseq_Suc_iff le_fun_def
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1048
             intro!: add_mono ereal_mult_left_mono simple_integral_mono)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1049
  have pos: "\<And>i. 0 \<le> integral\<^sup>S M (u i)" "\<And>i. 0 \<le> integral\<^sup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^sup>S M (u i)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1050
    using u v `0 \<le> a` by (auto simp: simple_integral_positive)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1051
  { fix i from pos[of i] have "a * integral\<^sup>S M (u i) \<noteq> -\<infinity>" "integral\<^sup>S M (v i) \<noteq> -\<infinity>"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1052
      by (auto split: split_if_asm) }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1053
  note not_MInf = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1054
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1055
  have l': "(SUP i. integral\<^sup>S M (l i)) = (SUP i. integral\<^sup>S M (?L' i))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1056
  proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1057
    show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1058
      using u v  `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
56537
01caba82e1d2 made ereal_add_nonneg_nonneg a simp rule
nipkow
parents: 56536
diff changeset
  1059
      by (auto intro!: add_mono ereal_mult_left_mono)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1060
    { fix x
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1061
      { fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using `0 \<le> a` u(6)[of i x] v(6)[of i x]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1062
          by auto }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1063
      then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1064
        using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
56212
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
  1065
        by (subst SUP_ereal_cmult [symmetric, OF u(6) `0 \<le> a`])
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
  1066
           (auto intro!: SUP_ereal_add
56537
01caba82e1d2 made ereal_add_nonneg_nonneg a simp rule
nipkow
parents: 56536
diff changeset
  1067
                 simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono) }
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1068
    then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1069
      unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
56537
01caba82e1d2 made ereal_add_nonneg_nonneg a simp rule
nipkow
parents: 56536
diff changeset
  1070
      by (intro AE_I2) (auto split: split_max)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1071
  qed
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1072
  also have "\<dots> = (SUP i. a * integral\<^sup>S M (u i) + integral\<^sup>S M (v i))"
56218
1c3f1f2431f9 elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents: 56213
diff changeset
  1073
    using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPREMUM UNIV"] ext)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1074
  finally have "(\<integral>\<^sup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^sup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+x. max 0 (g x) \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1075
    unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1076
    unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
56212
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
  1077
    apply (subst SUP_ereal_cmult [symmetric, OF pos(1) `0 \<le> a`])
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
  1078
    apply (subst SUP_ereal_add [symmetric, OF inc not_MInf]) .
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1079
  then show ?thesis by (simp add: positive_integral_max_0)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1080
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1081
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1082
lemma positive_integral_cmult:
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1083
  assumes f: "f \<in> borel_measurable M" "0 \<le> c"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1084
  shows "(\<integral>\<^sup>+ x. c * f x \<partial>M) = c * integral\<^sup>P M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1085
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1086
  have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1087
    by (auto split: split_max simp: ereal_zero_le_0_iff)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1088
  have "(\<integral>\<^sup>+ x. c * f x \<partial>M) = (\<integral>\<^sup>+ x. c * max 0 (f x) \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1089
    by (simp add: positive_integral_max_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1090
  then show ?thesis
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1091
    using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1092
    by (auto simp: positive_integral_max_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1093
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1094
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1095
lemma positive_integral_multc:
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1096
  assumes "f \<in> borel_measurable M" "0 \<le> c"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1097
  shows "(\<integral>\<^sup>+ x. f x * c \<partial>M) = integral\<^sup>P M f * c"
41096
843c40bbc379 integral over setprod
hoelzl
parents: 41095
diff changeset
  1098
  unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
843c40bbc379 integral over setprod
hoelzl
parents: 41095
diff changeset
  1099
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1100
lemma positive_integral_indicator[simp]:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1101
  "A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. indicator A x\<partial>M) = (emeasure M) A"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1102
  by (subst positive_integral_eq_simple_integral)
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1103
     (auto simp: simple_integral_indicator)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1104
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1105
lemma positive_integral_cmult_indicator:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1106
  "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. c * indicator A x \<partial>M) = c * (emeasure M) A"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1107
  by (subst positive_integral_eq_simple_integral)
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1108
     (auto simp: simple_function_indicator simple_integral_indicator)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1109
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1110
lemma positive_integral_indicator':
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1111
  assumes [measurable]: "A \<inter> space M \<in> sets M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1112
  shows "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = emeasure M (A \<inter> space M)"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1113
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1114
  have "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = (\<integral>\<^sup>+ x. indicator (A \<inter> space M) x \<partial>M)"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1115
    by (intro positive_integral_cong) (simp split: split_indicator)
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1116
  also have "\<dots> = emeasure M (A \<inter> space M)"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1117
    by simp
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1118
  finally show ?thesis .
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1119
qed
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1120
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1121
lemma positive_integral_add:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1122
  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1123
  and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1124
  shows "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = integral\<^sup>P M f + integral\<^sup>P M g"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1125
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1126
  have ae: "AE x in M. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
56537
01caba82e1d2 made ereal_add_nonneg_nonneg a simp rule
nipkow
parents: 56536
diff changeset
  1127
    using assms by (auto split: split_max)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1128
  have "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = (\<integral>\<^sup>+ x. max 0 (f x + g x) \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1129
    by (simp add: positive_integral_max_0)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1130
  also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1131
    unfolding ae[THEN positive_integral_cong_AE] ..
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1132
  also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+ x. max 0 (g x) \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1133
    using positive_integral_linear[of "\<lambda>x. max 0 (f x)" _ 1 "\<lambda>x. max 0 (g x)"] f g
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1134
    by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1135
  finally show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1136
    by (simp add: positive_integral_max_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1137
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1138
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1139
lemma positive_integral_setsum:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1140
  assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x in M. 0 \<le> f i x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1141
  shows "(\<integral>\<^sup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>P M (f i))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1142
proof cases
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1143
  assume f: "finite P"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1144
  from assms have "AE x in M. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1145
  from f this assms(1) show ?thesis
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1146
  proof induct
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1147
    case (insert i P)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1148
    then have "f i \<in> borel_measurable M" "AE x in M. 0 \<le> f i x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1149
      "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x in M. 0 \<le> (\<Sum>i\<in>P. f i x)"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1150
      by (auto intro!: setsum_nonneg)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1151
    from positive_integral_add[OF this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1152
    show ?case using insert by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1153
  qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1154
qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1155
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1156
lemma positive_integral_Markov_inequality:
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1157
  assumes u: "u \<in> borel_measurable M" "AE x in M. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1158
  shows "(emeasure M) ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1159
    (is "(emeasure M) ?A \<le> _ * ?PI")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1160
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1161
  have "?A \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1162
    using `A \<in> sets M` u by auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1163
  hence "(emeasure M) ?A = (\<integral>\<^sup>+ x. indicator ?A x \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1164
    using positive_integral_indicator by simp
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1165
  also have "\<dots> \<le> (\<integral>\<^sup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1166
    by (auto intro!: positive_integral_mono_AE
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1167
      simp: indicator_def ereal_zero_le_0_iff)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1168
  also have "\<dots> = c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1169
    using assms
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1170
    by (auto intro!: positive_integral_cmult simp: ereal_zero_le_0_iff)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1171
  finally show ?thesis .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1172
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1173
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1174
lemma positive_integral_noteq_infinite:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1175
  assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1176
  and "integral\<^sup>P M g \<noteq> \<infinity>"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1177
  shows "AE x in M. g x \<noteq> \<infinity>"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1178
proof (rule ccontr)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1179
  assume c: "\<not> (AE x in M. g x \<noteq> \<infinity>)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1180
  have "(emeasure M) {x\<in>space M. g x = \<infinity>} \<noteq> 0"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1181
    using c g by (auto simp add: AE_iff_null)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1182
  moreover have "0 \<le> (emeasure M) {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1183
  ultimately have "0 < (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1184
  then have "\<infinity> = \<infinity> * (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1185
  also have "\<dots> \<le> (\<integral>\<^sup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1186
    using g by (subst positive_integral_cmult_indicator) auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1187
  also have "\<dots> \<le> integral\<^sup>P M g"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1188
    using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1189
  finally show False using `integral\<^sup>P M g \<noteq> \<infinity>` by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1190
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1191
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1192
lemma positive_integral_PInf:
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1193
  assumes f: "f \<in> borel_measurable M"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1194
  and not_Inf: "integral\<^sup>P M f \<noteq> \<infinity>"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1195
  shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1196
proof -
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1197
  have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^sup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1198
    using f by (subst positive_integral_cmult_indicator) (auto simp: measurable_sets)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1199
  also have "\<dots> \<le> integral\<^sup>P M (\<lambda>x. max 0 (f x))"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1200
    by (auto intro!: positive_integral_mono simp: indicator_def max_def)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1201
  finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^sup>P M f"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1202
    by (simp add: positive_integral_max_0)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1203
  moreover have "0 \<le> (emeasure M) (f -` {\<infinity>} \<inter> space M)"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1204
    by (rule emeasure_nonneg)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1205
  ultimately show ?thesis
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1206
    using assms by (auto split: split_if_asm)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1207
qed
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1208
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1209
lemma positive_integral_PInf_AE:
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1210
  assumes "f \<in> borel_measurable M" "integral\<^sup>P M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1211
proof (rule AE_I)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1212
  show "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1213
    by (rule positive_integral_PInf[OF assms])
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1214
  show "f -` {\<infinity>} \<inter> space M \<in> sets M"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1215
    using assms by (auto intro: borel_measurable_vimage)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1216
qed auto
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1217
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1218
lemma simple_integral_PInf:
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1219
  assumes "simple_function M f" "\<And>x. 0 \<le> f x"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1220
  and "integral\<^sup>S M f \<noteq> \<infinity>"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1221
  shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1222
proof (rule positive_integral_PInf)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1223
  show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1224
  show "integral\<^sup>P M f \<noteq> \<infinity>"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1225
    using assms by (simp add: positive_integral_eq_simple_integral)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1226
qed
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  1227
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1228
lemma positive_integral_diff:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1229
  assumes f: "f \<in> borel_measurable M"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1230
  and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1231
  and fin: "integral\<^sup>P M g \<noteq> \<infinity>"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1232
  and mono: "AE x in M. g x \<le> f x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1233
  shows "(\<integral>\<^sup>+ x. f x - g x \<partial>M) = integral\<^sup>P M f - integral\<^sup>P M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1234
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1235
  have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x in M. 0 \<le> f x - g x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1236
    using assms by (auto intro: ereal_diff_positive)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1237
  have pos_f: "AE x in M. 0 \<le> f x" using mono g by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1238
  { fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1239
      by (cases rule: ereal2_cases[of a b]) auto }
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1240
  note * = this
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1241
  then have "AE x in M. f x = f x - g x + g x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1242
    using mono positive_integral_noteq_infinite[OF g fin] assms by auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1243
  then have **: "integral\<^sup>P M f = (\<integral>\<^sup>+x. f x - g x \<partial>M) + integral\<^sup>P M g"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1244
    unfolding positive_integral_add[OF diff g, symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1245
    by (rule positive_integral_cong_AE)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1246
  show ?thesis unfolding **
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1247
    using fin positive_integral_positive[of M g]
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1248
    by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. f x - g x \<partial>M" "integral\<^sup>P M g"]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1249
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1250
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1251
lemma positive_integral_suminf:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1252
  assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1253
  shows "(\<integral>\<^sup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^sup>P M (f i))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1254
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1255
  have all_pos: "AE x in M. \<forall>i. 0 \<le> f i x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1256
    using assms by (auto simp: AE_all_countable)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1257
  have "(\<Sum>i. integral\<^sup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^sup>P M (f i))"
56212
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
  1258
    using positive_integral_positive by (rule suminf_ereal_eq_SUP)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1259
  also have "\<dots> = (SUP n. \<integral>\<^sup>+x. (\<Sum>i<n. f i x) \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1260
    unfolding positive_integral_setsum[OF f] ..
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1261
  also have "\<dots> = \<integral>\<^sup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1262
    by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1263
       (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1264
  also have "\<dots> = \<integral>\<^sup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
56212
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
  1265
    by (intro positive_integral_cong_AE) (auto simp: suminf_ereal_eq_SUP)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1266
  finally show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1267
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1268
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1269
text {* Fatou's lemma: convergence theorem on limes inferior *}
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1270
lemma positive_integral_lim_INF:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1271
  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1272
  assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> u i x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1273
  shows "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>P M (u n))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1274
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1275
  have pos: "AE x in M. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1276
  have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1277
    (SUP n. \<integral>\<^sup>+ x. (INF i:{n..}. u i x) \<partial>M)"
56212
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
  1278
    unfolding liminf_SUP_INF using pos u
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1279
    by (intro positive_integral_monotone_convergence_SUP_AE)
44937
22c0857b8aab removed further legacy rules from Complete_Lattices
hoelzl
parents: 44928
diff changeset
  1280
       (elim AE_mp, auto intro!: AE_I2 intro: INF_greatest INF_superset_mono)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1281
  also have "\<dots> \<le> liminf (\<lambda>n. integral\<^sup>P M (u n))"
56212
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
  1282
    unfolding liminf_SUP_INF
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
  1283
    by (auto intro!: SUP_mono exI INF_greatest positive_integral_mono INF_lower)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1284
  finally show ?thesis .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1285
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1286
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1287
lemma positive_integral_null_set:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1288
  assumes "N \<in> null_sets M" shows "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1289
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1290
  have "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1291
  proof (intro positive_integral_cong_AE AE_I)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1292
    show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1293
      by (auto simp: indicator_def)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1294
    show "(emeasure M) N = 0" "N \<in> sets M"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1295
      using assms by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1296
  qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1297
  then show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1298
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1299
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1300
lemma positive_integral_0_iff:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1301
  assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1302
  shows "integral\<^sup>P M u = 0 \<longleftrightarrow> emeasure M {x\<in>space M. u x \<noteq> 0} = 0"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1303
    (is "_ \<longleftrightarrow> (emeasure M) ?A = 0")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1304
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1305
  have u_eq: "(\<integral>\<^sup>+ x. u x * indicator ?A x \<partial>M) = integral\<^sup>P M u"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1306
    by (auto intro!: positive_integral_cong simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1307
  show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1308
  proof
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1309
    assume "(emeasure M) ?A = 0"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1310
    with positive_integral_null_set[of ?A M u] u
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1311
    show "integral\<^sup>P M u = 0" by (simp add: u_eq null_sets_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1312
  next
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1313
    { fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1314
      then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1315
      then have "0 \<le> r" by (auto simp add: ereal_zero_less_0_iff) }
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1316
    note gt_1 = this
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1317
    assume *: "integral\<^sup>P M u = 0"
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1318
    let ?M = "\<lambda>n. {x \<in> space M. 1 \<le> real (n::nat) * u x}"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1319
    have "0 = (SUP n. (emeasure M) (?M n \<inter> ?A))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1320
    proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1321
      { fix n :: nat
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1322
        from positive_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"]
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1323
        have "(emeasure M) (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1324
        moreover have "0 \<le> (emeasure M) (?M n \<inter> ?A)" using u by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1325
        ultimately have "(emeasure M) (?M n \<inter> ?A) = 0" by auto }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1326
      thus ?thesis by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1327
    qed
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1328
    also have "\<dots> = (emeasure M) (\<Union>n. ?M n \<inter> ?A)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1329
    proof (safe intro!: SUP_emeasure_incseq)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1330
      fix n show "?M n \<inter> ?A \<in> sets M"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
  1331
        using u by (auto intro!: sets.Int)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1332
    next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1333
      show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1334
      proof (safe intro!: incseq_SucI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1335
        fix n :: nat and x
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1336
        assume *: "1 \<le> real n * u x"
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
  1337
        also from gt_1[OF *] have "real n * u x \<le> real (Suc n) * u x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1338
          using `0 \<le> u x` by (auto intro!: ereal_mult_right_mono)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1339
        finally show "1 \<le> real (Suc n) * u x" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1340
      qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1341
    qed
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1342
    also have "\<dots> = (emeasure M) {x\<in>space M. 0 < u x}"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1343
    proof (safe intro!: arg_cong[where f="(emeasure M)"] dest!: gt_1)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1344
      fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1345
      show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1346
      proof (cases "u x")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1347
        case (real r) with `0 < u x` have "0 < r" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1348
        obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1349
        hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1350
        hence "1 \<le> real j * r" using real `0 < r` by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1351
        thus ?thesis using `0 < r` real by (auto simp: one_ereal_def)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1352
      qed (insert `0 < u x`, auto)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1353
    qed auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1354
    finally have "(emeasure M) {x\<in>space M. 0 < u x} = 0" by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1355
    moreover
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1356
    from pos have "AE x in M. \<not> (u x < 0)" by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1357
    then have "(emeasure M) {x\<in>space M. u x < 0} = 0"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1358
      using AE_iff_null[of M] u by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1359
    moreover have "(emeasure M) {x\<in>space M. u x \<noteq> 0} = (emeasure M) {x\<in>space M. u x < 0} + (emeasure M) {x\<in>space M. 0 < u x}"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1360
      using u by (subst plus_emeasure) (auto intro!: arg_cong[where f="emeasure M"])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1361
    ultimately show "(emeasure M) ?A = 0" by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1362
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1363
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1364
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1365
lemma positive_integral_0_iff_AE:
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1366
  assumes u: "u \<in> borel_measurable M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1367
  shows "integral\<^sup>P M u = 0 \<longleftrightarrow> (AE x in M. u x \<le> 0)"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1368
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1369
  have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1370
    using u by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1371
  from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1372
  have "integral\<^sup>P M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1373
    unfolding positive_integral_max_0
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1374
    using AE_iff_null[OF sets] u by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1375
  also have "\<dots> \<longleftrightarrow> (AE x in M. u x \<le> 0)" by (auto split: split_max)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1376
  finally show ?thesis .
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1377
qed
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1378
50001
382bd3173584 add syntax and a.e.-rules for (conditional) probability on predicates
hoelzl
parents: 49800
diff changeset
  1379
lemma AE_iff_positive_integral: 
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1380
  "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> integral\<^sup>P M (indicator {x. \<not> P x}) = 0"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
  1381
  by (subst positive_integral_0_iff_AE) (auto simp: one_ereal_def zero_ereal_def
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
  1382
    sets.sets_Collect_neg indicator_def[abs_def] measurable_If)
50001
382bd3173584 add syntax and a.e.-rules for (conditional) probability on predicates
hoelzl
parents: 49800
diff changeset
  1383
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1384
lemma positive_integral_const_If:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1385
  "(\<integral>\<^sup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)"
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1386
  by (auto intro!: positive_integral_0_iff_AE[THEN iffD2])
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1387
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1388
lemma positive_integral_subalgebra:
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1389
  assumes f: "f \<in> borel_measurable N" "\<And>x. 0 \<le> f x"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1390
  and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1391
  shows "integral\<^sup>P N f = integral\<^sup>P M f"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1392
proof -
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1393
  have [simp]: "\<And>f :: 'a \<Rightarrow> ereal. f \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable M"
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1394
    using N by (auto simp: measurable_def)
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1395
  have [simp]: "\<And>P. (AE x in N. P x) \<Longrightarrow> (AE x in M. P x)"
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1396
    using N by (auto simp add: eventually_ae_filter null_sets_def)
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1397
  have [simp]: "\<And>A. A \<in> sets N \<Longrightarrow> A \<in> sets M"
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1398
    using N by auto
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1399
  from f show ?thesis
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1400
    apply induct
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1401
    apply (simp_all add: positive_integral_add positive_integral_cmult positive_integral_monotone_convergence_SUP N)
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1402
    apply (auto intro!: positive_integral_cong cong: positive_integral_cong simp: N(2)[symmetric])
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  1403
    done
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1404
qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1405
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1406
lemma positive_integral_nat_function:
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1407
  fixes f :: "'a \<Rightarrow> nat"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1408
  assumes "f \<in> measurable M (count_space UNIV)"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1409
  shows "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<Sum>t. emeasure M {x\<in>space M. t < f x})"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1410
proof -
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1411
  def F \<equiv> "\<lambda>i. {x\<in>space M. i < f x}"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1412
  with assms have [measurable]: "\<And>i. F i \<in> sets M"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1413
    by auto
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1414
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1415
  { fix x assume "x \<in> space M"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1416
    have "(\<lambda>i. if i < f x then 1 else 0) sums (of_nat (f x)::real)"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1417
      using sums_If_finite[of "\<lambda>i. i < f x" "\<lambda>_. 1::real"] by simp
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1418
    then have "(\<lambda>i. ereal(if i < f x then 1 else 0)) sums (ereal(of_nat(f x)))"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1419
      unfolding sums_ereal .
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1420
    moreover have "\<And>i. ereal (if i < f x then 1 else 0) = indicator (F i) x"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1421
      using `x \<in> space M` by (simp add: one_ereal_def F_def)
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1422
    ultimately have "ereal(of_nat(f x)) = (\<Sum>i. indicator (F i) x)"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1423
      by (simp add: sums_iff) }
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1424
  then have "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. indicator (F i) x) \<partial>M)"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1425
    by (simp cong: positive_integral_cong)
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1426
  also have "\<dots> = (\<Sum>i. emeasure M (F i))"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1427
    by (simp add: positive_integral_suminf)
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1428
  finally show ?thesis
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1429
    by (simp add: F_def)
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1430
qed
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1431
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1432
section "Lebesgue Integral"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1433
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1434
definition integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1435
  "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1436
    (\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1437
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1438
lemma borel_measurable_integrable[measurable_dest]:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1439
  "integrable M f \<Longrightarrow> f \<in> borel_measurable M"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1440
  by (auto simp: integrable_def)
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1441
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1442
lemma integrableD[dest]:
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1443
  assumes "integrable M f"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1444
  shows "f \<in> borel_measurable M" "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1445
  using assms unfolding integrable_def by auto
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1446
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1447
definition lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real" ("integral\<^sup>L") where
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1448
  "integral\<^sup>L M f = real ((\<integral>\<^sup>+ x. ereal (f x) \<partial>M)) - real ((\<integral>\<^sup>+ x. ereal (- f x) \<partial>M))"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1449
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1450
syntax
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1451
  "_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> 'a measure \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1452
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1453
translations
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1454
  "\<integral> x. f \<partial>M" == "CONST lebesgue_integral M (%x. f)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1455
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1456
lemma integrableE:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1457
  assumes "integrable M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1458
  obtains r q where
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1459
    "(\<integral>\<^sup>+x. ereal (f x)\<partial>M) = ereal r"
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1460
    "(\<integral>\<^sup>+x. ereal (-f x)\<partial>M) = ereal q"
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1461
    "f \<in> borel_measurable M" "integral\<^sup>L M f = r - q"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1462
  using assms unfolding integrable_def lebesgue_integral_def
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1463
  using positive_integral_positive[of M "\<lambda>x. ereal (f x)"]
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1464
  using positive_integral_positive[of M "\<lambda>x. ereal (-f x)"]
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1465
  by (cases rule: ereal2_cases[of "(\<integral>\<^sup>+x. ereal (-f x)\<partial>M)" "(\<integral>\<^sup>+x. ereal (f x)\<partial>M)"]) auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1466
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1467
lemma integral_cong:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1468
  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1469
  shows "integral\<^sup>L M f = integral\<^sup>L M g"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1470
  using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1471
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1472
lemma integral_cong_AE:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1473
  assumes cong: "AE x in M. f x = g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1474
  shows "integral\<^sup>L M f = integral\<^sup>L M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1475
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1476
  have *: "AE x in M. ereal (f x) = ereal (g x)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1477
    "AE x in M. ereal (- f x) = ereal (- g x)" using cong by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1478
  show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1479
    unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1480
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1481
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1482
lemma integrable_cong_AE:
43339
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  1483
  assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1484
  assumes "AE x in M. f x = g x"
43339
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  1485
  shows "integrable M f = integrable M g"
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  1486
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1487
  have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = (\<integral>\<^sup>+ x. ereal (g x) \<partial>M)"
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1488
    "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) = (\<integral>\<^sup>+ x. ereal (- g x) \<partial>M)"
43339
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  1489
    using assms by (auto intro!: positive_integral_cong_AE)
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  1490
  with assms show ?thesis
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  1491
    by (auto simp: integrable_def)
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  1492
qed
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  1493
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1494
lemma integrable_cong:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1495
  "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1496
  by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1497
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1498
lemma integral_mono_AE:
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1499
  assumes fg: "integrable M f" "integrable M g"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1500
  and mono: "AE t in M. f t \<le> g t"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1501
  shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1502
proof -
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1503
  have "AE x in M. ereal (f x) \<le> ereal (g x)"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1504
    using mono by auto
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1505
  moreover have "AE x in M. ereal (- g x) \<le> ereal (- f x)"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1506
    using mono by auto
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1507
  ultimately show ?thesis using fg
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1508
    by (auto intro!: add_mono positive_integral_mono_AE real_of_ereal_positive_mono
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53374
diff changeset
  1509
             simp: positive_integral_positive lebesgue_integral_def algebra_simps)
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1510
qed
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1511
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1512
lemma integral_mono:
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1513
  assumes "integrable M f" "integrable M g" "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1514
  shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1515
  using assms by (auto intro: integral_mono_AE)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1516
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1517
lemma positive_integral_eq_integral:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1518
  assumes f: "integrable M f"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1519
  assumes nonneg: "AE x in M. 0 \<le> f x" 
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1520
  shows "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = integral\<^sup>L M f"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1521
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1522
  have "(\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1523
    using nonneg by (intro positive_integral_cong_AE) (auto split: split_max)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1524
  with f positive_integral_positive show ?thesis
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1525
    by (cases "\<integral>\<^sup>+ x. ereal (f x) \<partial>M")
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1526
       (auto simp add: lebesgue_integral_def positive_integral_max_0 integrable_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1527
qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1528
  
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1529
lemma integral_eq_positive_integral:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1530
  assumes f: "\<And>x. 0 \<le> f x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1531
  shows "integral\<^sup>L M f = real (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1532
proof -
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1533
  { fix x have "max 0 (ereal (- f x)) = 0" using f[of x] by (simp split: split_max) }
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1534
  then have "0 = (\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M)" by simp
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1535
  also have "\<dots> = (\<integral>\<^sup>+ x. ereal (- f x) \<partial>M)" unfolding positive_integral_max_0 ..
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1536
  finally show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1537
    unfolding lebesgue_integral_def by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1538
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1539
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1540
lemma integral_minus[intro, simp]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1541
  assumes "integrable M f"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1542
  shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^sup>L M f"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1543
  using assms by (auto simp: integrable_def lebesgue_integral_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1544
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1545
lemma integral_minus_iff[simp]:
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1546
  "integrable M (\<lambda>x. - f x) \<longleftrightarrow> integrable M f"
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1547
proof
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1548
  assume "integrable M (\<lambda>x. - f x)"
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1549
  then have "integrable M (\<lambda>x. - (- f x))"
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1550
    by (rule integral_minus)
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1551
  then show "integrable M f" by simp
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1552
qed (rule integral_minus)
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1553
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1554
lemma integral_of_positive_diff:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1555
  assumes integrable: "integrable M u" "integrable M v"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1556
  and f_def: "\<And>x. f x = u x - v x" and pos: "\<And>x. 0 \<le> u x" "\<And>x. 0 \<le> v x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1557
  shows "integrable M f" and "integral\<^sup>L M f = integral\<^sup>L M u - integral\<^sup>L M v"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1558
proof -
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1559
  let ?f = "\<lambda>x. max 0 (ereal (f x))"
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1560
  let ?mf = "\<lambda>x. max 0 (ereal (- f x))"
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1561
  let ?u = "\<lambda>x. max 0 (ereal (u x))"
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1562
  let ?v = "\<lambda>x. max 0 (ereal (v x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1563
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1564
  from borel_measurable_diff[of u M v] integrable
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1565
  have f_borel: "?f \<in> borel_measurable M" and
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1566
    mf_borel: "?mf \<in> borel_measurable M" and
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1567
    v_borel: "?v \<in> borel_measurable M" and
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1568
    u_borel: "?u \<in> borel_measurable M" and
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1569
    "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1570
    by (auto simp: f_def[symmetric] integrable_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1571
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1572
  have "(\<integral>\<^sup>+ x. ereal (u x - v x) \<partial>M) \<le> integral\<^sup>P M ?u"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1573
    using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1574
  moreover have "(\<integral>\<^sup>+ x. ereal (v x - u x) \<partial>M) \<le> integral\<^sup>P M ?v"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1575
    using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1576
  ultimately show f: "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1577
    using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1578
    by (auto simp: integrable_def f_def positive_integral_max_0)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1579
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1580
  have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1581
    unfolding f_def using pos by (simp split: split_max)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1582
  then have "(\<integral>\<^sup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^sup>+ x. ?v x + ?f x \<partial>M)" by simp
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1583
  then have "real (integral\<^sup>P M ?u + integral\<^sup>P M ?mf) =
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1584
      real (integral\<^sup>P M ?v + integral\<^sup>P M ?f)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1585
    using positive_integral_add[OF u_borel _ mf_borel]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1586
    using positive_integral_add[OF v_borel _ f_borel]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1587
    by auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1588
  then show "integral\<^sup>L M f = integral\<^sup>L M u - integral\<^sup>L M v"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1589
    unfolding positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1590
    unfolding pos[THEN integral_eq_positive_integral]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1591
    using integrable f by (auto elim!: integrableE)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1592
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1593
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1594
lemma integral_linear:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1595
  assumes "integrable M f" "integrable M g" and "0 \<le> a"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1596
  shows "integrable M (\<lambda>t. a * f t + g t)"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1597
  and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^sup>L M f + integral\<^sup>L M g" (is ?EQ)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1598
proof -
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1599
  let ?f = "\<lambda>x. max 0 (ereal (f x))"
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1600
  let ?g = "\<lambda>x. max 0 (ereal (g x))"
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1601
  let ?mf = "\<lambda>x. max 0 (ereal (- f x))"
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1602
  let ?mg = "\<lambda>x. max 0 (ereal (- g x))"
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1603
  let ?p = "\<lambda>t. max 0 (a * f t) + max 0 (g t)"
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1604
  let ?n = "\<lambda>t. max 0 (- (a * f t)) + max 0 (- g t)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1605
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1606
  from assms have linear:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1607
    "(\<integral>\<^sup>+ x. ereal a * ?f x + ?g x \<partial>M) = ereal a * integral\<^sup>P M ?f + integral\<^sup>P M ?g"
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1608
    "(\<integral>\<^sup>+ x. ereal a * ?mf x + ?mg x \<partial>M) = ereal a * integral\<^sup>P M ?mf + integral\<^sup>P M ?mg"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1609
    by (auto intro!: positive_integral_linear simp: integrable_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1610
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1611
  have *: "(\<integral>\<^sup>+x. ereal (- ?p x) \<partial>M) = 0" "(\<integral>\<^sup>+x. ereal (- ?n x) \<partial>M) = 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1612
    using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1613
  have **: "\<And>x. ereal a * ?f x + ?g x = max 0 (ereal (?p x))"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1614
           "\<And>x. ereal a * ?mf x + ?mg x = max 0 (ereal (?n x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1615
    using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1616
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1617
  have "integrable M ?p" "integrable M ?n"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1618
      "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1619
    using linear assms unfolding integrable_def ** *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1620
    by (auto simp: positive_integral_max_0)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1621
  note diff = integral_of_positive_diff[OF this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1622
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1623
  show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1624
  from assms linear show ?EQ
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1625
    unfolding diff(2) ** positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1626
    unfolding lebesgue_integral_def *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1627
    by (auto elim!: integrableE simp: field_simps)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1628
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1629
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1630
lemma integral_add[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1631
  assumes "integrable M f" "integrable M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1632
  shows "integrable M (\<lambda>t. f t + g t)"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1633
  and "(\<integral> t. f t + g t \<partial>M) = integral\<^sup>L M f + integral\<^sup>L M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1634
  using assms integral_linear[where a=1] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1635
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1636
lemma integral_zero[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1637
  shows "integrable M (\<lambda>x. 0)" "(\<integral> x.0 \<partial>M) = 0"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1638
  unfolding integrable_def lebesgue_integral_def
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1639
  by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1640
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1641
lemma lebesgue_integral_uminus:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1642
    "(\<integral>x. - f x \<partial>M) = - integral\<^sup>L M f"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1643
  unfolding lebesgue_integral_def by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1644
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1645
lemma lebesgue_integral_cmult_nonneg:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1646
  assumes f: "f \<in> borel_measurable M" and "0 \<le> c"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1647
  shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^sup>L M f"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1648
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1649
  { have "real (ereal c * integral\<^sup>P M (\<lambda>x. max 0 (ereal (f x)))) =
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1650
      real (integral\<^sup>P M (\<lambda>x. ereal c * max 0 (ereal (f x))))"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1651
      using f `0 \<le> c` by (subst positive_integral_cmult) auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1652
    also have "\<dots> = real (integral\<^sup>P M (\<lambda>x. max 0 (ereal (c * f x))))"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1653
      using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def zero_le_mult_iff)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1654
    finally have "real (integral\<^sup>P M (\<lambda>x. ereal (c * f x))) = c * real (integral\<^sup>P M (\<lambda>x. ereal (f x)))"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1655
      by (simp add: positive_integral_max_0) }
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1656
  moreover
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1657
  { have "real (ereal c * integral\<^sup>P M (\<lambda>x. max 0 (ereal (- f x)))) =
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1658
      real (integral\<^sup>P M (\<lambda>x. ereal c * max 0 (ereal (- f x))))"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1659
      using f `0 \<le> c` by (subst positive_integral_cmult) auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1660
    also have "\<dots> = real (integral\<^sup>P M (\<lambda>x. max 0 (ereal (- c * f x))))"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1661
      using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def mult_le_0_iff)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1662
    finally have "real (integral\<^sup>P M (\<lambda>x. ereal (- c * f x))) = c * real (integral\<^sup>P M (\<lambda>x. ereal (- f x)))"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1663
      by (simp add: positive_integral_max_0) }
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1664
  ultimately show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1665
    by (simp add: lebesgue_integral_def field_simps)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1666
qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1667
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1668
lemma lebesgue_integral_cmult:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1669
  assumes f: "f \<in> borel_measurable M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1670
  shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^sup>L M f"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1671
proof (cases rule: linorder_le_cases)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1672
  assume "0 \<le> c" with f show ?thesis by (rule lebesgue_integral_cmult_nonneg)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1673
next
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1674
  assume "c \<le> 0"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1675
  with lebesgue_integral_cmult_nonneg[OF f, of "-c"]
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1676
  show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1677
    by (simp add: lebesgue_integral_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1678
qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1679
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1680
lemma lebesgue_integral_multc:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1681
  "f \<in> borel_measurable M \<Longrightarrow> (\<integral>x. f x * c \<partial>M) = integral\<^sup>L M f * c"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1682
  using lebesgue_integral_cmult[of f M c] by (simp add: ac_simps)
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1683
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1684
lemma integral_multc:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1685
  "integrable M f \<Longrightarrow> (\<integral> x. f x * c \<partial>M) = integral\<^sup>L M f * c"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1686
  by (simp add: lebesgue_integral_multc)
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1687
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1688
lemma integral_cmult[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1689
  assumes "integrable M f"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1690
  shows "integrable M (\<lambda>t. a * f t)" (is ?P)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1691
  and "(\<integral> t. a * f t \<partial>M) = a * integral\<^sup>L M f" (is ?I)
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1692
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1693
  have "integrable M (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t \<partial>M) = a * integral\<^sup>L M f"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1694
  proof (cases rule: le_cases)
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1695
    assume "0 \<le> a" show ?thesis
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1696
      using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1697
      by simp
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1698
  next
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1699
    assume "a \<le> 0" hence "0 \<le> - a" by auto
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1700
    have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1701
    show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1702
        integral_minus(1)[of M "\<lambda>t. - a * f t"]
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1703
      unfolding * integral_zero by simp
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1704
  qed
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1705
  thus ?P ?I by auto
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1706
qed
41096
843c40bbc379 integral over setprod
hoelzl
parents: 41095
diff changeset
  1707
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1708
lemma integral_diff[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1709
  assumes f: "integrable M f" and g: "integrable M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1710
  shows "integrable M (\<lambda>t. f t - g t)"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1711
  and "(\<integral> t. f t - g t \<partial>M) = integral\<^sup>L M f - integral\<^sup>L M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1712
  using integral_add[OF f integral_minus(1)[OF g]]
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53374
diff changeset
  1713
  unfolding integral_minus(2)[OF g]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1714
  by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1715
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1716
lemma integral_indicator[simp, intro]:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1717
  assumes "A \<in> sets M" and "(emeasure M) A \<noteq> \<infinity>"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1718
  shows "integral\<^sup>L M (indicator A) = real (emeasure M A)" (is ?int)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1719
  and "integrable M (indicator A)" (is ?able)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1720
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1721
  from `A \<in> sets M` have *:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1722
    "\<And>x. ereal (indicator A x) = indicator A x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1723
    "(\<integral>\<^sup>+x. ereal (- indicator A x) \<partial>M) = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1724
    by (auto split: split_indicator simp: positive_integral_0_iff_AE one_ereal_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1725
  show ?int ?able
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1726
    using assms unfolding lebesgue_integral_def integrable_def
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1727
    by (auto simp: *)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1728
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1729
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1730
lemma integral_cmul_indicator:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1731
  assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> (emeasure M) A \<noteq> \<infinity>"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1732
  shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1733
  and "(\<integral>x. c * indicator A x \<partial>M) = c * real ((emeasure M) A)" (is ?I)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1734
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1735
  show ?P
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1736
  proof (cases "c = 0")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1737
    case False with assms show ?thesis by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1738
  qed simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1739
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1740
  show ?I
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1741
  proof (cases "c = 0")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1742
    case False with assms show ?thesis by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1743
  qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1744
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1745
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1746
lemma integral_setsum[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1747
  assumes "\<And>n. n \<in> S \<Longrightarrow> integrable M (f n)"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1748
  shows "(\<integral>x. (\<Sum> i \<in> S. f i x) \<partial>M) = (\<Sum> i \<in> S. integral\<^sup>L M (f i))" (is "?int S")
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1749
    and "integrable M (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1750
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1751
  have "?int S \<and> ?I S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1752
  proof (cases "finite S")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1753
    assume "finite S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1754
    from this assms show ?thesis by (induct S) simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1755
  qed simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1756
  thus "?int S" and "?I S" by auto
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1757
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1758
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1759
lemma integrable_bound:
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1760
  assumes "integrable M f" and f: "AE x in M. \<bar>g x\<bar> \<le> f x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1761
  assumes borel: "g \<in> borel_measurable M"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1762
  shows "integrable M g"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1763
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1764
  have "(\<integral>\<^sup>+ x. ereal (g x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal \<bar>g x\<bar> \<partial>M)"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1765
    by (auto intro!: positive_integral_mono)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1766
  also have "\<dots> \<le> (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1767
    using f by (auto intro!: positive_integral_mono_AE)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1768
  also have "\<dots> < \<infinity>"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1769
    using `integrable M f` unfolding integrable_def by auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1770
  finally have pos: "(\<integral>\<^sup>+ x. ereal (g x) \<partial>M) < \<infinity>" .
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1771
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1772
  have "(\<integral>\<^sup>+ x. ereal (- g x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (\<bar>g x\<bar>) \<partial>M)"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1773
    by (auto intro!: positive_integral_mono)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1774
  also have "\<dots> \<le> (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1775
    using f by (auto intro!: positive_integral_mono_AE)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1776
  also have "\<dots> < \<infinity>"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1777
    using `integrable M f` unfolding integrable_def by auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1778
  finally have neg: "(\<integral>\<^sup>+ x. ereal (- g x) \<partial>M) < \<infinity>" .
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1779
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1780
  from neg pos borel show ?thesis
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1781
    unfolding integrable_def by auto
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1782
qed
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1783
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1784
lemma integrable_abs:
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1785
  assumes f[measurable]: "integrable M f"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1786
  shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  1787
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1788
  from assms have *: "(\<integral>\<^sup>+x. ereal (- \<bar>f x\<bar>)\<partial>M) = 0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1789
    "\<And>x. ereal \<bar>f x\<bar> = max 0 (ereal (f x)) + max 0 (ereal (- f x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1790
    by (auto simp: integrable_def positive_integral_0_iff_AE split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1791
  with assms show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1792
    by (simp add: positive_integral_add positive_integral_max_0 integrable_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1793
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1794
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1795
lemma integral_subalgebra:
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1796
  assumes borel: "f \<in> borel_measurable N"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1797
  and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1798
  shows "integrable N f \<longleftrightarrow> integrable M f" (is ?P)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1799
    and "integral\<^sup>L N f = integral\<^sup>L M f" (is ?I)
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1800
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1801
  have "(\<integral>\<^sup>+ x. max 0 (ereal (f x)) \<partial>N) = (\<integral>\<^sup>+ x. max 0 (ereal (f x)) \<partial>M)"
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1802
       "(\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>N) = (\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1803
    using borel by (auto intro!: positive_integral_subalgebra N)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1804
  moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1805
    using assms unfolding measurable_def by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1806
  ultimately show ?P ?I
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1807
    by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0)
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1808
qed
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1809
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1810
lemma lebesgue_integral_nonneg:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1811
  assumes ae: "(AE x in M. 0 \<le> f x)" shows "0 \<le> integral\<^sup>L M f"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1812
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1813
  have "(\<integral>\<^sup>+x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^sup>+x. 0 \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1814
    using ae by (intro positive_integral_cong_AE) (auto simp: max_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1815
  then show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1816
    by (auto simp: lebesgue_integral_def positive_integral_max_0
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1817
             intro!: real_of_ereal_pos positive_integral_positive)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1818
qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1819
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1820
lemma integrable_abs_iff:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1821
  "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1822
  by (auto intro!: integrable_bound[where g=f] integrable_abs)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1823
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1824
lemma integrable_max:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1825
  assumes int: "integrable M f" "integrable M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1826
  shows "integrable M (\<lambda> x. max (f x) (g x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1827
proof (rule integrable_bound)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1828
  show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1829
    using int by (simp add: integrable_abs)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1830
  show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1831
    using int unfolding integrable_def by auto
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1832
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1833
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1834
lemma integrable_min:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1835
  assumes int: "integrable M f" "integrable M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1836
  shows "integrable M (\<lambda> x. min (f x) (g x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1837
proof (rule integrable_bound)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1838
  show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1839
    using int by (simp add: integrable_abs)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1840
  show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1841
    using int unfolding integrable_def by auto
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1842
qed auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1843
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1844
lemma integral_triangle_inequality:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1845
  assumes "integrable M f"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1846
  shows "\<bar>integral\<^sup>L M f\<bar> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1847
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1848
  have "\<bar>integral\<^sup>L M f\<bar> = max (integral\<^sup>L M f) (- integral\<^sup>L M f)" by auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1849
  also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1850
      using assms integral_minus(2)[of M f, symmetric]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1851
      by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1852
  finally show ?thesis .
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  1853
qed
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  1854
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1855
lemma integrable_nonneg:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1856
  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "(\<integral>\<^sup>+ x. f x \<partial>M) \<noteq> \<infinity>"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1857
  shows "integrable M f"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1858
  unfolding integrable_def
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1859
proof (intro conjI f)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1860
  have "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) = 0"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1861
    using f by (subst positive_integral_0_iff_AE) auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1862
  then show "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>" by simp
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1863
qed
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1864
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1865
lemma integral_positive:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1866
  assumes "integrable M f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1867
  shows "0 \<le> integral\<^sup>L M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1868
proof -
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  1869
  have "0 = (\<integral>x. 0 \<partial>M)" by auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1870
  also have "\<dots> \<le> integral\<^sup>L M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1871
    using assms by (rule integral_mono[OF integral_zero(1)])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1872
  finally show ?thesis .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1873
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1874
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1875
lemma integral_monotone_convergence_pos:
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1876
  assumes i: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1877
    and pos: "\<And>i. AE x in M. 0 \<le> f i x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1878
    and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1879
    and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1880
    and u: "u \<in> borel_measurable M"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1881
  shows "integrable M u"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1882
  and "integral\<^sup>L M u = x"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1883
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1884
  have "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = (SUP n. (\<integral>\<^sup>+ x. ereal (f n x) \<partial>M))"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1885
  proof (subst positive_integral_monotone_convergence_SUP_AE[symmetric])
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1886
    fix i
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1887
    from mono pos show "AE x in M. ereal (f i x) \<le> ereal (f (Suc i) x) \<and> 0 \<le> ereal (f i x)"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1888
      by eventually_elim (auto simp: mono_def)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1889
    show "(\<lambda>x. ereal (f i x)) \<in> borel_measurable M"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  1890
      using i by auto
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1891
  next
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1892
    show "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = \<integral>\<^sup>+ x. (SUP i. ereal (f i x)) \<partial>M"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1893
      apply (rule positive_integral_cong_AE)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1894
      using lim mono
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1895
      by eventually_elim (simp add: SUP_eq_LIMSEQ[THEN iffD2])
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1896
  qed
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1897
  also have "\<dots> = ereal x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1898
    using mono i unfolding positive_integral_eq_integral[OF i pos]
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1899
    by (subst SUP_eq_LIMSEQ) (auto simp: mono_def intro!: integral_mono_AE ilim)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1900
  finally have "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = ereal x" .
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1901
  moreover have "(\<integral>\<^sup>+ x. ereal (- u x) \<partial>M) = 0"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1902
  proof (subst positive_integral_0_iff_AE)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1903
    show "(\<lambda>x. ereal (- u x)) \<in> borel_measurable M"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1904
      using u by auto
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1905
    from mono pos[of 0] lim show "AE x in M. ereal (- u x) \<le> 0"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1906
    proof eventually_elim
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1907
      fix x assume "mono (\<lambda>n. f n x)" "0 \<le> f 0 x" "(\<lambda>i. f i x) ----> u x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1908
      then show "ereal (- u x) \<le> 0"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1909
        using incseq_le[of "\<lambda>n. f n x" "u x" 0] by (simp add: mono_def incseq_def)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1910
    qed
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1911
  qed
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1912
  ultimately show "integrable M u" "integral\<^sup>L M u = x"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1913
    by (auto simp: integrable_def lebesgue_integral_def u)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1914
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1915
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1916
lemma integral_monotone_convergence:
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1917
  assumes f: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1918
  and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1919
  and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1920
  and u: "u \<in> borel_measurable M"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1921
  shows "integrable M u"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1922
  and "integral\<^sup>L M u = x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1923
proof -
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1924
  have 1: "\<And>i. integrable M (\<lambda>x. f i x - f 0 x)"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1925
    using f by auto
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1926
  have 2: "AE x in M. mono (\<lambda>n. f n x - f 0 x)"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1927
    using mono by (auto simp: mono_def le_fun_def)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1928
  have 3: "\<And>n. AE x in M. 0 \<le> f n x - f 0 x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1929
    using mono by (auto simp: field_simps mono_def le_fun_def)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1930
  have 4: "AE x in M. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 43941
diff changeset
  1931
    using lim by (auto intro!: tendsto_diff)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1932
  have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^sup>L M (f 0)"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1933
    using f ilim by (auto intro!: tendsto_diff)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1934
  have 6: "(\<lambda>x. u x - f 0 x) \<in> borel_measurable M"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1935
    using f[of 0] u by auto
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1936
  note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5 6]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1937
  have "integrable M (\<lambda>x. (u x - f 0 x) + f 0 x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1938
    using diff(1) f by (rule integral_add(1))
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1939
  with diff(2) f show "integrable M u" "integral\<^sup>L M u = x"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  1940
    by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1941
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1942
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1943
lemma integral_0_iff:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1944
  assumes "integrable M f"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1945
  shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> (emeasure M) {x\<in>space M. f x \<noteq> 0} = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1946
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1947
  have *: "(\<integral>\<^sup>+x. ereal (- \<bar>f x\<bar>) \<partial>M) = 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1948
    using assms by (auto simp: positive_integral_0_iff_AE integrable_def)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1949
  have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1950
  hence "(\<lambda>x. ereal (\<bar>f x\<bar>)) \<in> borel_measurable M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1951
    "(\<integral>\<^sup>+ x. ereal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1952
  from positive_integral_0_iff[OF this(1)] this(2)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1953
  show ?thesis unfolding lebesgue_integral_def *
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1954
    using positive_integral_positive[of M "\<lambda>x. ereal \<bar>f x\<bar>"]
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1955
    by (auto simp add: real_of_ereal_eq_0)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1956
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1957
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1958
lemma integral_real:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1959
  "AE x in M. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^sup>P M f) - real (integral\<^sup>P M (\<lambda>x. - f x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1960
  using assms unfolding lebesgue_integral_def
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1961
  by (subst (1 2) positive_integral_cong_AE) (auto simp add: ereal_real)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1962
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1963
lemma (in finite_measure) lebesgue_integral_const[simp]:
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1964
  shows "integrable M (\<lambda>x. a)"
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1965
  and  "(\<integral>x. a \<partial>M) = a * measure M (space M)"
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1966
proof -
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1967
  { fix a :: real assume "0 \<le> a"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1968
    then have "(\<integral>\<^sup>+ x. ereal a \<partial>M) = ereal a * (emeasure M) (space M)"
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1969
      by (subst positive_integral_const) auto
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1970
    moreover
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1971
    from `0 \<le> a` have "(\<integral>\<^sup>+ x. ereal (-a) \<partial>M) = 0"
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1972
      by (subst positive_integral_0_iff_AE) auto
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1973
    ultimately have "integrable M (\<lambda>x. a)" by (auto simp: integrable_def) }
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1974
  note * = this
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1975
  show "integrable M (\<lambda>x. a)"
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1976
  proof cases
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1977
    assume "0 \<le> a" with * show ?thesis .
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1978
  next
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1979
    assume "\<not> 0 \<le> a"
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1980
    then have "0 \<le> -a" by auto
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1981
    from *[OF this] show ?thesis by simp
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1982
  qed
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1983
  show "(\<integral>x. a \<partial>M) = a * measure M (space M)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1984
    by (simp add: lebesgue_integral_def positive_integral_const_If emeasure_eq_measure)
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1985
qed
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1986
50097
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1987
lemma (in finite_measure) integrable_const_bound:
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1988
  assumes "AE x in M. \<bar>f x\<bar> \<le> B" and "f \<in> borel_measurable M" shows "integrable M f"
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1989
  by (auto intro: integrable_bound[where f="\<lambda>x. B"] lebesgue_integral_const assms)
32973da2d4f7 rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents: 50027
diff changeset
  1990
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1991
lemma (in finite_measure) integral_less_AE:
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1992
  assumes int: "integrable M X" "integrable M Y"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1993
  assumes A: "(emeasure M) A \<noteq> 0" "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> X x \<noteq> Y x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1994
  assumes gt: "AE x in M. X x \<le> Y x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1995
  shows "integral\<^sup>L M X < integral\<^sup>L M Y"
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1996
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1997
  have "integral\<^sup>L M X \<le> integral\<^sup>L M Y"
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1998
    using gt int by (intro integral_mono_AE) auto
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  1999
  moreover
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2000
  have "integral\<^sup>L M X \<noteq> integral\<^sup>L M Y"
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  2001
  proof
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2002
    assume eq: "integral\<^sup>L M X = integral\<^sup>L M Y"
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2003
    have "integral\<^sup>L M (\<lambda>x. \<bar>Y x - X x\<bar>) = integral\<^sup>L M (\<lambda>x. Y x - X x)"
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  2004
      using gt by (intro integral_cong_AE) auto
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  2005
    also have "\<dots> = 0"
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  2006
      using eq int by simp
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2007
    finally have "(emeasure M) {x \<in> space M. Y x - X x \<noteq> 0} = 0"
43339
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  2008
      using int by (simp add: integral_0_iff)
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  2009
    moreover
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2010
    have "(\<integral>\<^sup>+x. indicator A x \<partial>M) \<le> (\<integral>\<^sup>+x. indicator {x \<in> space M. Y x - X x \<noteq> 0} x \<partial>M)"
43339
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  2011
      using A by (intro positive_integral_mono_AE) auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2012
    then have "(emeasure M) A \<le> (emeasure M) {x \<in> space M. Y x - X x \<noteq> 0}"
43339
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  2013
      using int A by (simp add: integrable_def)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2014
    ultimately have "emeasure M A = 0"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2015
      using emeasure_nonneg[of M A] by simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2016
    with `(emeasure M) A \<noteq> 0` show False by auto
42991
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  2017
  qed
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  2018
  ultimately show ?thesis by auto
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  2019
qed
3fa22920bf86 integral strong monotone; finite subadditivity for measure
hoelzl
parents: 42950
diff changeset
  2020
43339
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  2021
lemma (in finite_measure) integral_less_AE_space:
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  2022
  assumes int: "integrable M X" "integrable M Y"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2023
  assumes gt: "AE x in M. X x < Y x" "(emeasure M) (space M) \<noteq> 0"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2024
  shows "integral\<^sup>L M X < integral\<^sup>L M Y"
43339
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  2025
  using gt by (intro integral_less_AE[OF int, where A="space M"]) auto
9ba256ad6781 jensens inequality
hoelzl
parents: 42991
diff changeset
  2026
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2027
lemma integral_dominated_convergence:
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2028
  assumes u[measurable]: "\<And>i. integrable M (u i)" and bound: "\<And>j. AE x in M. \<bar>u j x\<bar> \<le> w x"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2029
  and w[measurable]: "integrable M w"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2030
  and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2031
  and [measurable]: "u' \<in> borel_measurable M"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2032
  shows "integrable M u'"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2033
  and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar> \<partial>M)) ----> 0" (is "?lim_diff")
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2034
  and "(\<lambda>i. integral\<^sup>L M (u i)) ----> integral\<^sup>L M u'" (is ?lim)
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  2035
proof -
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2036
  have all_bound: "AE x in M. \<forall>j. \<bar>u j x\<bar> \<le> w x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2037
    using bound by (auto simp: AE_all_countable)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2038
  with u' have u'_bound: "AE x in M. \<bar>u' x\<bar> \<le> w x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2039
    by eventually_elim (auto intro: LIMSEQ_le_const2 tendsto_rabs)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2040
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2041
  from bound[of 0] have w_pos: "AE x in M. 0 \<le> w x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2042
    by eventually_elim auto
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2043
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2044
  show "integrable M u'"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2045
    by (rule integrable_bound) fact+
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2046
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  2047
  let ?diff = "\<lambda>n x. 2 * w x - \<bar>u n x - u' x\<bar>"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2048
  have diff: "\<And>n. integrable M (\<lambda>x. \<bar>u n x - u' x\<bar>)"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2049
    using w u `integrable M u'` by (auto intro!: integrable_abs)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2050
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2051
  from u'_bound all_bound
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2052
  have diff_less_2w: "AE x in M. \<forall>j. \<bar>u j x - u' x\<bar> \<le> 2 * w x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2053
  proof (eventually_elim, intro allI)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2054
    fix x j assume *: "\<bar>u' x\<bar> \<le> w x" "\<forall>j. \<bar>u j x\<bar> \<le> w x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2055
    then have "\<bar>u j x - u' x\<bar> \<le> \<bar>u j x\<bar> + \<bar>u' x\<bar>" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2056
    also have "\<dots> \<le> w x + w x"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2057
      using * by (intro add_mono) auto
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2058
    finally show "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2059
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2060
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2061
  have PI_diff: "\<And>n. (\<integral>\<^sup>+ x. ereal (?diff n x) \<partial>M) =
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2062
    (\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) - (\<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2063
    using diff w diff_less_2w w_pos
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2064
    by (subst positive_integral_diff[symmetric])
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2065
       (auto simp: integrable_def intro!: positive_integral_cong_AE)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2066
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2067
  have "integrable M (\<lambda>x. 2 * w x)"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2068
    using w by auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2069
  hence I2w_fin: "(\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) \<noteq> \<infinity>" and
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  2070
    borel_2w: "(\<lambda>x. ereal (2 * w x)) \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2071
    unfolding integrable_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2072
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2073
  have "limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = 0" (is "limsup ?f = 0")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2074
  proof cases
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2075
    assume eq_0: "(\<integral>\<^sup>+ x. max 0 (ereal (2 * w x)) \<partial>M) = 0" (is "?wx = 0")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2076
    { fix n
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2077
      have "?f n \<le> ?wx" (is "integral\<^sup>P M ?f' \<le> _")
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2078
        using diff_less_2w unfolding positive_integral_max_0
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2079
        by (intro positive_integral_mono_AE) auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2080
      then have "?f n = 0"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2081
        using positive_integral_positive[of M ?f'] eq_0 by auto }
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2082
    then show ?thesis by (simp add: Limsup_const)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2083
  next
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2084
    assume neq_0: "(\<integral>\<^sup>+ x. max 0 (ereal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0")
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  2085
    have "0 = limsup (\<lambda>n. 0 :: ereal)" by (simp add: Limsup_const)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2086
    also have "\<dots> \<le> limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
51000
c9adb50f74ad use order topology for extended reals
hoelzl
parents: 50384
diff changeset
  2087
      by (simp add: Limsup_mono  positive_integral_positive)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2088
    finally have pos: "0 \<le> limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)" .
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2089
    have "?wx = (\<integral>\<^sup>+ x. liminf (\<lambda>n. max 0 (ereal (?diff n x))) \<partial>M)"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2090
      using u'
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2091
    proof (intro positive_integral_cong_AE, eventually_elim)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2092
      fix x assume u': "(\<lambda>i. u i x) ----> u' x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  2093
      show "max 0 (ereal (2 * w x)) = liminf (\<lambda>n. max 0 (ereal (?diff n x)))"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  2094
        unfolding ereal_max_0
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  2095
      proof (rule lim_imp_Liminf[symmetric], unfold lim_ereal)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2096
        have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2097
          using u' by (safe intro!: tendsto_intros)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2098
        then show "(\<lambda>i. max 0 (?diff i x)) ----> max 0 (2 * w x)"
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56571
diff changeset
  2099
          by (auto intro!: tendsto_max)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2100
      qed (rule trivial_limit_sequentially)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2101
    qed
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2102
    also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^sup>+ x. max 0 (ereal (?diff n x)) \<partial>M)"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2103
      using w u unfolding integrable_def
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2104
      by (intro positive_integral_lim_INF) (auto intro!: positive_integral_lim_INF)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2105
    also have "\<dots> = (\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) -
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2106
        limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2107
      unfolding PI_diff positive_integral_max_0
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2108
      using positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"]
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  2109
      by (subst liminf_ereal_cminus) auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2110
    finally show ?thesis
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2111
      using neq_0 I2w_fin positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"] pos
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2112
      unfolding positive_integral_max_0
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2113
      by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"])
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2114
         auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2115
  qed
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2116
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2117
  have "liminf ?f \<le> limsup ?f"
51000
c9adb50f74ad use order topology for extended reals
hoelzl
parents: 50384
diff changeset
  2118
    by (intro Liminf_le_Limsup trivial_limit_sequentially)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2119
  moreover
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  2120
  { have "0 = liminf (\<lambda>n. 0 :: ereal)" by (simp add: Liminf_const)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2121
    also have "\<dots> \<le> liminf ?f"
51000
c9adb50f74ad use order topology for extended reals
hoelzl
parents: 50384
diff changeset
  2122
      by (simp add: Liminf_mono positive_integral_positive)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2123
    finally have "0 \<le> liminf ?f" . }
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  2124
  ultimately have liminf_limsup_eq: "liminf ?f = ereal 0" "limsup ?f = ereal 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2125
    using `limsup ?f = 0` by auto
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2126
  have "\<And>n. (\<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = ereal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2127
    using diff positive_integral_positive[of M]
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2128
    by (subst integral_eq_positive_integral[of _ M]) (auto simp: ereal_real integrable_def)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2129
  then show ?lim_diff
51340
5e6296afe08d move Liminf / Limsup lemmas on complete_lattices to its own file
hoelzl
parents: 51000
diff changeset
  2130
    using Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq]
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2131
    by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2132
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2133
  show ?lim
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2134
  proof (rule LIMSEQ_I)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2135
    fix r :: real assume "0 < r"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2136
    from LIMSEQ_D[OF `?lim_diff` this]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2137
    obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M) < r"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2138
      using diff by (auto simp: integral_positive)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2139
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2140
    show "\<exists>N. \<forall>n\<ge>N. norm (integral\<^sup>L M (u n) - integral\<^sup>L M u') < r"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2141
    proof (safe intro!: exI[of _ N])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2142
      fix n assume "N \<le> n"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2143
      have "\<bar>integral\<^sup>L M (u n) - integral\<^sup>L M u'\<bar> = \<bar>(\<integral>x. u n x - u' x \<partial>M)\<bar>"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2144
        using u `integrable M u'` by auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2145
      also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)" using u `integrable M u'`
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2146
        by (rule_tac integral_triangle_inequality) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2147
      also note N[OF `N \<le> n`]
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2148
      finally show "norm (integral\<^sup>L M (u n) - integral\<^sup>L M u') < r" by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2149
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2150
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2151
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2152
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2153
lemma integral_sums:
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2154
  assumes integrable[measurable]: "\<And>i. integrable M (f i)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2155
  and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2156
  and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2157
  shows "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?S")
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2158
  and "(\<lambda>i. integral\<^sup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is ?integral)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2159
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2160
  have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2161
    using summable unfolding summable_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2162
  from bchoice[OF this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2163
  obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2164
  then have w_borel: "w \<in> borel_measurable M" unfolding sums_def
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2165
    by (rule borel_measurable_LIMSEQ) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2166
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  2167
  let ?w = "\<lambda>y. if y \<in> space M then w y else 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2168
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2169
  obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M)) sums x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2170
    using sums unfolding summable_def ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2171
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56166
diff changeset
  2172
  have 1: "\<And>n. integrable M (\<lambda>x. \<Sum>i<n. f i x)"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2173
    using integrable by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2174
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56166
diff changeset
  2175
  have 2: "\<And>j. AE x in M. \<bar>\<Sum>i<j. f i x\<bar> \<le> ?w x"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2176
    using AE_space
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2177
  proof eventually_elim
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2178
    fix j x assume [simp]: "x \<in> space M"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56166
diff changeset
  2179
    have "\<bar>\<Sum>i<j. f i x\<bar> \<le> (\<Sum>i<j. \<bar>f i x\<bar>)" by (rule setsum_abs)
56213
e5720d3c18f0 further renaming in Series
hoelzl
parents: 56212
diff changeset
  2180
    also have "\<dots> \<le> w x" using w[of x] setsum_le_suminf[of "\<lambda>i. \<bar>f i x\<bar>"] unfolding sums_iff by auto
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56166
diff changeset
  2181
    finally show "\<bar>\<Sum>i<j. f i x\<bar> \<le> ?w x" by simp
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2182
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2183
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2184
  have 3: "integrable M ?w"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2185
  proof (rule integral_monotone_convergence(1))
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56166
diff changeset
  2186
    let ?F = "\<lambda>n y. (\<Sum>i<n. \<bar>f i y\<bar>)"
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  2187
    let ?w' = "\<lambda>n y. if y \<in> space M then ?F n y else 0"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2188
    have "\<And>n. integrable M (?F n)"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2189
      using integrable by (auto intro!: integrable_abs)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2190
    thus "\<And>n. integrable M (?w' n)" by (simp cong: integrable_cong)
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2191
    show "AE x in M. mono (\<lambda>n. ?w' n x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2192
      by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2193
    show "AE x in M. (\<lambda>n. ?w' n x) ----> ?w x"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2194
        using w by (simp_all add: tendsto_const sums_def)
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56166
diff changeset
  2195
    have *: "\<And>n. integral\<^sup>L M (?w' n) = (\<Sum>i< n. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2196
      using integrable by (simp add: integrable_abs cong: integral_cong)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2197
    from abs_sum
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2198
    show "(\<lambda>i. integral\<^sup>L M (?w' i)) ----> x" unfolding * sums_def .
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2199
  qed (simp add: w_borel measurable_If_set)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2200
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2201
  from summable[THEN summable_rabs_cancel]
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56166
diff changeset
  2202
  have 4: "AE x in M. (\<lambda>n. \<Sum>i<n. f i x) ----> (\<Sum>i. f i x)"
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 56166
diff changeset
  2203
    by (auto intro: summable_LIMSEQ)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2204
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2205
  note int = integral_dominated_convergence(1,3)[OF 1 2 3 4
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2206
    borel_measurable_suminf[OF integrableD(1)[OF integrable]]]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2207
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2208
  from int show "integrable M ?S" by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2209
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2210
  show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF integrable]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2211
    using int(2) by simp
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  2212
qed
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  2213
50384
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2214
lemma integrable_mult_indicator:
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2215
  "A \<in> sets M \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. f x * indicator A x)"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2216
  by (rule integrable_bound[where f="\<lambda>x. \<bar>f x\<bar>"])
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2217
     (auto intro: integrable_abs split: split_indicator)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2218
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2219
lemma tendsto_integral_at_top:
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2220
  fixes M :: "real measure"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2221
  assumes M: "sets M = sets borel" and f[measurable]: "integrable M f"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2222
  shows "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2223
proof -
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2224
  have M_measure[simp]: "borel_measurable M = borel_measurable borel"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2225
    using M by (simp add: sets_eq_imp_space_eq measurable_def)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2226
  { fix f assume f: "integrable M f" "\<And>x. 0 \<le> f x"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2227
    then have [measurable]: "f \<in> borel_measurable borel"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2228
      by (simp add: integrable_def)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2229
    have "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2230
    proof (rule tendsto_at_topI_sequentially)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2231
      have "\<And>j. AE x in M. \<bar>f x * indicator {.. j} x\<bar> \<le> f x"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2232
        using f(2) by (intro AE_I2) (auto split: split_indicator)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2233
      have int: "\<And>n. integrable M (\<lambda>x. f x * indicator {.. n} x)"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2234
        by (rule integrable_mult_indicator) (auto simp: M f)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2235
      show "(\<lambda>n. \<integral> x. f x * indicator {..real n} x \<partial>M) ----> integral\<^sup>L M f"
50384
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2236
      proof (rule integral_dominated_convergence)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2237
        { fix x have "eventually (\<lambda>n. f x * indicator {..real n} x = f x) sequentially"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2238
            by (rule eventually_sequentiallyI[of "natceiling x"])
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2239
               (auto split: split_indicator simp: natceiling_le_eq) }
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2240
        from filterlim_cong[OF refl refl this]
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2241
        show "AE x in M. (\<lambda>n. f x * indicator {..real n} x) ----> f x"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2242
          by (simp add: tendsto_const)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2243
      qed (fact+, simp)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2244
      show "mono (\<lambda>y. \<integral> x. f x * indicator {..y} x \<partial>M)"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2245
        by (intro monoI integral_mono int) (auto split: split_indicator intro: f)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2246
    qed }
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2247
  note nonneg = this
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2248
  let ?P = "\<lambda>y. \<integral> x. max 0 (f x) * indicator {..y} x \<partial>M"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2249
  let ?N = "\<lambda>y. \<integral> x. max 0 (- f x) * indicator {..y} x \<partial>M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2250
  let ?p = "integral\<^sup>L M (\<lambda>x. max 0 (f x))"
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2251
  let ?n = "integral\<^sup>L M (\<lambda>x. max 0 (- f x))"
50384
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2252
  have "(?P ---> ?p) at_top" "(?N ---> ?n) at_top"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2253
    by (auto intro!: nonneg integrable_max f)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2254
  note tendsto_diff[OF this]
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2255
  also have "(\<lambda>y. ?P y - ?N y) = (\<lambda>y. \<integral> x. f x * indicator {..y} x \<partial>M)"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2256
    by (subst integral_diff(2)[symmetric])
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2257
       (auto intro!: integrable_mult_indicator integrable_max f integral_cong ext
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2258
             simp: M split: split_max)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2259
  also have "?p - ?n = integral\<^sup>L M f"
50384
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2260
    by (subst integral_diff(2)[symmetric])
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2261
       (auto intro!: integrable_max f integral_cong ext simp: M split: split_max)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2262
  finally show ?thesis .
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2263
qed
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2264
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2265
lemma integral_monotone_convergence_at_top:
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2266
  fixes M :: "real measure"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2267
  assumes M: "sets M = sets borel"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2268
  assumes nonneg: "AE x in M. 0 \<le> f x"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2269
  assumes borel: "f \<in> borel_measurable borel"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2270
  assumes int: "\<And>y. integrable M (\<lambda>x. f x * indicator {.. y} x)"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2271
  assumes conv: "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> x) at_top"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2272
  shows "integrable M f" "integral\<^sup>L M f = x"
50384
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2273
proof -
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2274
  from nonneg have "AE x in M. mono (\<lambda>n::nat. f x * indicator {..real n} x)"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2275
    by (auto split: split_indicator intro!: monoI)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2276
  { fix x have "eventually (\<lambda>n. f x * indicator {..real n} x = f x) sequentially"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2277
      by (rule eventually_sequentiallyI[of "natceiling x"])
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2278
         (auto split: split_indicator simp: natceiling_le_eq) }
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2279
  from filterlim_cong[OF refl refl this]
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2280
  have "AE x in M. (\<lambda>i. f x * indicator {..real i} x) ----> f x"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2281
    by (simp add: tendsto_const)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2282
  have "(\<lambda>i. \<integral> x. f x * indicator {..real i} x \<partial>M) ----> x"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2283
    using conv filterlim_real_sequentially by (rule filterlim_compose)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2284
  have M_measure[simp]: "borel_measurable M = borel_measurable borel"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2285
    using M by (simp add: sets_eq_imp_space_eq measurable_def)
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2286
  have "f \<in> borel_measurable M"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2287
    using borel by simp
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2288
  show "integrable M f"
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2289
    by (rule integral_monotone_convergence) fact+
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2290
  show "integral\<^sup>L M f = x"
50384
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2291
    by (rule integral_monotone_convergence) fact+
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2292
qed
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2293
b9b967da28e9 rules for improper Lebesgue integrals (using tendsto at_top)
hoelzl
parents: 50252
diff changeset
  2294
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2295
section "Lebesgue integration on countable spaces"
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2296
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2297
lemma integral_on_countable:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2298
  assumes f: "f \<in> borel_measurable M"
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2299
  and bij: "bij_betw enum S (f ` space M)"
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2300
  and enum_zero: "enum ` (-S) \<subseteq> {0}"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2301
  and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> (emeasure M) (f -` {x} \<inter> space M) \<noteq> \<infinity>"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2302
  and abs_summable: "summable (\<lambda>r. \<bar>enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))\<bar>)"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2303
  shows "integrable M f"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2304
  and "(\<lambda>r. enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))) sums integral\<^sup>L M f" (is ?sums)
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2305
proof -
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  2306
  let ?A = "\<lambda>r. f -` {enum r} \<inter> space M"
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  2307
  let ?F = "\<lambda>r x. enum r * indicator (?A r) x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2308
  have enum_eq: "\<And>r. enum r * real ((emeasure M) (?A r)) = integral\<^sup>L M (?F r)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2309
    using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2310
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2311
  { fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2312
    hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2313
    then obtain i where "i\<in>S" "enum i = f x" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2314
    have F: "\<And>j. ?F j x = (if j = i then f x else 0)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2315
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2316
      fix j assume "j = i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2317
      thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2318
    next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2319
      fix j assume "j \<noteq> i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2320
      show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2321
        by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2322
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2323
    hence F_abs: "\<And>j. \<bar>if j = i then f x else 0\<bar> = (if j = i then \<bar>f x\<bar> else 0)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2324
    have "(\<lambda>i. ?F i x) sums f x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2325
         "(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2326
      by (auto intro!: sums_single simp: F F_abs) }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2327
  note F_sums_f = this(1) and F_abs_sums_f = this(2)
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2328
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2329
  have int_f: "integral\<^sup>L M f = (\<integral>x. (\<Sum>r. ?F r x) \<partial>M)" "integrable M f = integrable M (\<lambda>x. \<Sum>r. ?F r x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2330
    using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2331
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2332
  { fix r
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2333
    have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2334
      by (auto simp: indicator_def intro!: integral_cong)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2335
    also have "\<dots> = \<bar>enum r\<bar> * real ((emeasure M) (?A r))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2336
      using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2337
    finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real ((emeasure M) (?A r))\<bar>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  2338
      using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_ereal_pos measurable_sets) }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2339
  note int_abs_F = this
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2340
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2341
  have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2342
    using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2343
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2344
  have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2345
    using F_abs_sums_f unfolding sums_iff by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2346
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2347
  from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2348
  show ?sums unfolding enum_eq int_f by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2349
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2350
  from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2351
  show "integrable M f" unfolding int_f by simp
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2352
qed
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2353
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2354
section {* Distributions *}
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2355
49797
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2356
lemma positive_integral_distr':
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2357
  assumes T: "T \<in> measurable M M'"
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2358
  and f: "f \<in> borel_measurable (distr M M' T)" "\<And>x. 0 \<le> f x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2359
  shows "integral\<^sup>P (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
49797
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2360
  using f 
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2361
proof induct
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2362
  case (cong f g)
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2363
  with T show ?case
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2364
    apply (subst positive_integral_cong[of _ f g])
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2365
    apply simp
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2366
    apply (subst positive_integral_cong[of _ "\<lambda>x. f (T x)" "\<lambda>x. g (T x)"])
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2367
    apply (simp add: measurable_def Pi_iff)
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2368
    apply simp
49797
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2369
    done
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2370
next
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2371
  case (set A)
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2372
  then have eq: "\<And>x. x \<in> space M \<Longrightarrow> indicator A (T x) = indicator (T -` A \<inter> space M) x"
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2373
    by (auto simp: indicator_def)
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2374
  from set T show ?case
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2375
    by (subst positive_integral_cong[OF eq])
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
  2376
       (auto simp add: emeasure_distr intro!: positive_integral_indicator[symmetric] measurable_sets)
49798
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2377
qed (simp_all add: measurable_compose[OF T] T positive_integral_cmult positive_integral_add
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2378
                   positive_integral_monotone_convergence_SUP le_fun_def incseq_def)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2379
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2380
lemma positive_integral_distr:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2381
  "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^sup>P (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2382
  by (subst (1 2) positive_integral_max_0[symmetric])
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2383
     (simp add: positive_integral_distr')
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  2384
49800
a6678da5692c induction prove for positive_integral_fst
hoelzl
parents: 49799
diff changeset
  2385
lemma integral_distr:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2386
  "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^sup>L (distr M M' T) f = (\<integral> x. f (T x) \<partial>M)"
49800
a6678da5692c induction prove for positive_integral_fst
hoelzl
parents: 49799
diff changeset
  2387
  unfolding lebesgue_integral_def
a6678da5692c induction prove for positive_integral_fst
hoelzl
parents: 49799
diff changeset
  2388
  by (subst (1 2) positive_integral_distr) auto
a6678da5692c induction prove for positive_integral_fst
hoelzl
parents: 49799
diff changeset
  2389
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2390
lemma integrable_distr_eq:
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2391
  "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integrable (distr M M' T) f \<longleftrightarrow> integrable M (\<lambda>x. f (T x))"
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2392
  unfolding integrable_def 
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2393
  by (subst (1 2) positive_integral_distr) (auto simp: comp_def)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2394
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2395
lemma integrable_distr:
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2396
  "T \<in> measurable M M' \<Longrightarrow> integrable (distr M M' T) f \<Longrightarrow> integrable M (\<lambda>x. f (T x))"
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2397
  by (subst integrable_distr_eq[symmetric]) auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2398
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2399
section {* Lebesgue integration on @{const count_space} *}
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2400
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2401
lemma simple_function_count_space[simp]:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2402
  "simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2403
  unfolding simple_function_def by simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2404
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2405
lemma positive_integral_count_space:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2406
  assumes A: "finite {a\<in>A. 0 < f a}"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2407
  shows "integral\<^sup>P (count_space A) f = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2408
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2409
  have *: "(\<integral>\<^sup>+x. max 0 (f x) \<partial>count_space A) =
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2410
    (\<integral>\<^sup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2411
    by (auto intro!: positive_integral_cong
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2412
             simp add: indicator_def if_distrib setsum_cases[OF A] max_def le_less)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2413
  also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. \<integral>\<^sup>+ x. f a * indicator {a} x \<partial>count_space A)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2414
    by (subst positive_integral_setsum)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2415
       (simp_all add: AE_count_space ereal_zero_le_0_iff less_imp_le)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2416
  also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2417
    by (auto intro!: setsum_cong simp: positive_integral_cmult_indicator one_ereal_def[symmetric])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2418
  finally show ?thesis by (simp add: positive_integral_max_0)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2419
qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2420
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2421
lemma integrable_count_space:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2422
  "finite X \<Longrightarrow> integrable (count_space X) f"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2423
  by (auto simp: positive_integral_count_space integrable_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2424
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2425
lemma positive_integral_count_space_finite:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2426
    "finite A \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. max 0 (f a))"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2427
  by (subst positive_integral_max_0[symmetric])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2428
     (auto intro!: setsum_mono_zero_left simp: positive_integral_count_space less_le)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2429
49775
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2430
lemma lebesgue_integral_count_space_finite_support:
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2431
  assumes f: "finite {a\<in>A. f a \<noteq> 0}" shows "(\<integral>x. f x \<partial>count_space A) = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. f a)"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2432
proof -
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2433
  have *: "\<And>r::real. 0 < max 0 r \<longleftrightarrow> 0 < r" "\<And>x. max 0 (ereal x) = ereal (max 0 x)"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2434
    "\<And>a. a \<in> A \<and> 0 < f a \<Longrightarrow> max 0 (f a) = f a"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2435
    "\<And>a. a \<in> A \<and> f a < 0 \<Longrightarrow> max 0 (- f a) = - f a"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2436
    "{a \<in> A. f a \<noteq> 0} = {a \<in> A. 0 < f a} \<union> {a \<in> A. f a < 0}"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2437
    "({a \<in> A. 0 < f a} \<inter> {a \<in> A. f a < 0}) = {}"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2438
    by (auto split: split_max)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2439
  have "finite {a \<in> A. 0 < f a}" "finite {a \<in> A. f a < 0}"
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2440
    by (auto intro: finite_subset[OF _ f])
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2441
  then show ?thesis
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2442
    unfolding lebesgue_integral_def
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2443
    apply (subst (1 2) positive_integral_max_0[symmetric])
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2444
    apply (subst (1 2) positive_integral_count_space)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2445
    apply (auto simp add: * setsum_negf setsum_Un
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2446
                simp del: ereal_max)
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2447
    done
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2448
qed
970964690b3d remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents: 47761
diff changeset
  2449
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2450
lemma lebesgue_integral_count_space_finite:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2451
    "finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2452
  apply (auto intro!: setsum_mono_zero_left
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2453
              simp: positive_integral_count_space_finite lebesgue_integral_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2454
  apply (subst (1 2)  setsum_real_of_ereal[symmetric])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2455
  apply (auto simp: max_def setsum_subtractf[symmetric] intro!: setsum_cong)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2456
  done
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2457
54418
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2458
lemma emeasure_UN_countable:
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2459
  assumes sets: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M" and I: "countable I" 
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2460
  assumes disj: "disjoint_family_on X I"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2461
  shows "emeasure M (UNION I X) = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2462
proof cases
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2463
  assume "finite I" with sets disj show ?thesis
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2464
    by (subst setsum_emeasure[symmetric])
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2465
       (auto intro!: setsum_cong simp add: max_def subset_eq positive_integral_count_space_finite emeasure_nonneg)
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2466
next
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2467
  assume f: "\<not> finite I"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2468
  then have [intro]: "I \<noteq> {}" by auto
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2469
  from from_nat_into_inj_infinite[OF I f] from_nat_into[OF this] disj
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2470
  have disj2: "disjoint_family (\<lambda>i. X (from_nat_into I i))"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2471
    unfolding disjoint_family_on_def by metis
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2472
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2473
  from f have "bij_betw (from_nat_into I) UNIV I"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2474
    using bij_betw_from_nat_into[OF I] by simp
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2475
  then have "(\<Union>i\<in>I. X i) = (\<Union>i. (X \<circ> from_nat_into I) i)"
56154
f0a927235162 more complete set of lemmas wrt. image and composition
haftmann
parents: 54611
diff changeset
  2476
    unfolding SUP_def image_comp [symmetric] by (simp add: bij_betw_def)
54418
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2477
  then have "emeasure M (UNION I X) = emeasure M (\<Union>i. X (from_nat_into I i))"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2478
    by simp
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2479
  also have "\<dots> = (\<Sum>i. emeasure M (X (from_nat_into I i)))"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2480
    by (intro suminf_emeasure[symmetric] disj disj2) (auto intro!: sets from_nat_into[OF `I \<noteq> {}`])
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2481
  also have "\<dots> = (\<Sum>n. \<integral>\<^sup>+i. emeasure M (X i) * indicator {from_nat_into I n} i \<partial>count_space I)"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2482
  proof (intro arg_cong[where f=suminf] ext)
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2483
    fix i
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2484
    have eq: "{a \<in> I. 0 < emeasure M (X a) * indicator {from_nat_into I i} a}
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2485
     = (if 0 < emeasure M (X (from_nat_into I i)) then {from_nat_into I i} else {})"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2486
     using ereal_0_less_1
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2487
     by (auto simp: ereal_zero_less_0_iff indicator_def from_nat_into `I \<noteq> {}` simp del: ereal_0_less_1)
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2488
    have "(\<integral>\<^sup>+ ia. emeasure M (X ia) * indicator {from_nat_into I i} ia \<partial>count_space I) =
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2489
      (if 0 < emeasure M (X (from_nat_into I i)) then emeasure M (X (from_nat_into I i)) else 0)"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2490
      by (subst positive_integral_count_space) (simp_all add: eq)
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2491
    also have "\<dots> = emeasure M (X (from_nat_into I i))"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2492
      by (simp add: less_le emeasure_nonneg)
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2493
    finally show "emeasure M (X (from_nat_into I i)) =
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2494
         \<integral>\<^sup>+ ia. emeasure M (X ia) * indicator {from_nat_into I i} ia \<partial>count_space I" ..
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2495
  qed
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2496
  also have "\<dots> = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2497
    apply (subst positive_integral_suminf[symmetric])
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2498
    apply (auto simp: emeasure_nonneg intro!: positive_integral_cong)
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2499
  proof -
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2500
    fix x assume "x \<in> I"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2501
    then have "(\<Sum>i. emeasure M (X x) * indicator {from_nat_into I i} x) = (\<Sum>i\<in>{to_nat_on I x}. emeasure M (X x) * indicator {from_nat_into I i} x)"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2502
      by (intro suminf_finite) (auto simp: indicator_def I f)
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2503
    also have "\<dots> = emeasure M (X x)"
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2504
      by (simp add: I f `x\<in>I`)
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2505
    finally show "(\<Sum>i. emeasure M (X x) * indicator {from_nat_into I i} x) = emeasure M (X x)" .
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2506
  qed
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2507
  finally show ?thesis .
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2508
qed
3b8e33d1a39a measure of a countable union
hoelzl
parents: 54417
diff changeset
  2509
54417
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2510
section {* Measures with Restricted Space *}
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2511
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2512
lemma positive_integral_restrict_space:
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2513
  assumes \<Omega>: "\<Omega> \<in> sets M" and f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "\<And>x. x \<in> space M - \<Omega> \<Longrightarrow> f x = 0"
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2514
  shows "positive_integral (restrict_space M \<Omega>) f = positive_integral M f"
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2515
using f proof (induct rule: borel_measurable_induct)
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2516
  case (cong f g) then show ?case
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2517
    using positive_integral_cong[of M f g] positive_integral_cong[of "restrict_space M \<Omega>" f g]
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2518
      sets.sets_into_space[OF `\<Omega> \<in> sets M`]
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2519
    by (simp add: subset_eq space_restrict_space)
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2520
next
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2521
  case (set A)
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2522
  then have "A \<subseteq> \<Omega>"
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2523
    unfolding indicator_eq_0_iff by (auto dest: sets.sets_into_space)
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2524
  with set `\<Omega> \<in> sets M` sets.sets_into_space[OF `\<Omega> \<in> sets M`] show ?case
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2525
    by (subst positive_integral_indicator')
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2526
       (auto simp add: sets_restrict_space_iff space_restrict_space
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2527
                  emeasure_restrict_space Int_absorb2
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2528
                dest: sets.sets_into_space)
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2529
next
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2530
  case (mult f c) then show ?case
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2531
    by (cases "c = 0") (simp_all add: measurable_restrict_space1 \<Omega> positive_integral_cmult)
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2532
next
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2533
  case (add f g) then show ?case
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2534
    by (simp add: measurable_restrict_space1 \<Omega> positive_integral_add ereal_add_nonneg_eq_0_iff)
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2535
next
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2536
  case (seq F) then show ?case
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2537
    by (auto simp add: SUP_eq_iff measurable_restrict_space1 \<Omega> positive_integral_monotone_convergence_SUP)
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2538
qed
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 54230
diff changeset
  2539
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2540
section {* Measure spaces with an associated density *}
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2541
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2542
definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2543
  "density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2544
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2545
lemma 
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2546
  shows sets_density[simp]: "sets (density M f) = sets M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2547
    and space_density[simp]: "space (density M f) = space M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2548
  by (auto simp: density_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2549
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2550
(* FIXME: add conversion to simplify space, sets and measurable *)
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2551
lemma space_density_imp[measurable_dest]:
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2552
  "\<And>x M f. x \<in> space (density M f) \<Longrightarrow> x \<in> space M" by auto
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2553
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2554
lemma 
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2555
  shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2556
    and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2557
    and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2558
  unfolding measurable_def simple_function_def by simp_all
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2559
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2560
lemma density_cong: "f \<in> borel_measurable M \<Longrightarrow> f' \<in> borel_measurable M \<Longrightarrow>
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2561
  (AE x in M. f x = f' x) \<Longrightarrow> density M f = density M f'"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
  2562
  unfolding density_def by (auto intro!: measure_of_eq positive_integral_cong_AE sets.space_closed)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2563
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2564
lemma density_max_0: "density M f = density M (\<lambda>x. max 0 (f x))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2565
proof -
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2566
  have "\<And>x A. max 0 (f x) * indicator A x = max 0 (f x * indicator A x)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2567
    by (auto simp: indicator_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2568
  then show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2569
    unfolding density_def by (simp add: positive_integral_max_0)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2570
qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2571
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2572
lemma density_ereal_max_0: "density M (\<lambda>x. ereal (f x)) = density M (\<lambda>x. ereal (max 0 (f x)))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2573
  by (subst density_max_0) (auto intro!: arg_cong[where f="density M"] split: split_max)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2574
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2575
lemma emeasure_density:
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  2576
  assumes f[measurable]: "f \<in> borel_measurable M" and A[measurable]: "A \<in> sets M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2577
  shows "emeasure (density M f) A = (\<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2578
    (is "_ = ?\<mu> A")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2579
  unfolding density_def
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2580
proof (rule emeasure_measure_of_sigma)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2581
  show "sigma_algebra (space M) (sets M)" ..
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2582
  show "positive (sets M) ?\<mu>"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2583
    using f by (auto simp: positive_def intro!: positive_integral_positive)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2584
  have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^sup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'")
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2585
    apply (subst positive_integral_max_0[symmetric])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2586
    apply (intro ext positive_integral_cong_AE AE_I2)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2587
    apply (auto simp: indicator_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2588
    done
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2589
  show "countably_additive (sets M) ?\<mu>"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2590
    unfolding \<mu>_eq
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2591
  proof (intro countably_additiveI)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2592
    fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  2593
    then have "\<And>i. A i \<in> sets M" by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2594
    then have *: "\<And>i. (\<lambda>x. max 0 (f x) * indicator (A i) x) \<in> borel_measurable M"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  2595
      by (auto simp: set_eq_iff)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2596
    assume disj: "disjoint_family A"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2597
    have "(\<Sum>n. ?\<mu>' (A n)) = (\<integral>\<^sup>+ x. (\<Sum>n. max 0 (f x) * indicator (A n) x) \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2598
      using f * by (simp add: positive_integral_suminf)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2599
    also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * (\<Sum>n. indicator (A n) x) \<partial>M)" using f
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2600
      by (auto intro!: suminf_cmult_ereal positive_integral_cong_AE)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2601
    also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * indicator (\<Union>n. A n) x \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2602
      unfolding suminf_indicator[OF disj] ..
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2603
    finally show "(\<Sum>n. ?\<mu>' (A n)) = ?\<mu>' (\<Union>x. A x)" by simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2604
  qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2605
qed fact
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2606
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2607
lemma null_sets_density_iff:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2608
  assumes f: "f \<in> borel_measurable M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2609
  shows "A \<in> null_sets (density M f) \<longleftrightarrow> A \<in> sets M \<and> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2610
proof -
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2611
  { assume "A \<in> sets M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2612
    have eq: "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. max 0 (f x) * indicator A x \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2613
      apply (subst positive_integral_max_0[symmetric])
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2614
      apply (intro positive_integral_cong)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2615
      apply (auto simp: indicator_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2616
      done
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2617
    have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> 
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2618
      emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2619
      unfolding eq
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2620
      using f `A \<in> sets M`
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2621
      by (intro positive_integral_0_iff) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2622
    also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2623
      using f `A \<in> sets M`
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  2624
      by (intro AE_iff_measurable[OF _ refl, symmetric]) auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2625
    also have "(AE x in M. max 0 (f x) * indicator A x = 0) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2626
      by (auto simp add: indicator_def max_def split: split_if_asm)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2627
    finally have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)" . }
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2628
  with f show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2629
    by (simp add: null_sets_def emeasure_density cong: conj_cong)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2630
qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2631
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2632
lemma AE_density:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2633
  assumes f: "f \<in> borel_measurable M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2634
  shows "(AE x in density M f. P x) \<longleftrightarrow> (AE x in M. 0 < f x \<longrightarrow> P x)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2635
proof
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2636
  assume "AE x in density M f. P x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2637
  with f obtain N where "{x \<in> space M. \<not> P x} \<subseteq> N" "N \<in> sets M" and ae: "AE x in M. x \<in> N \<longrightarrow> f x \<le> 0"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2638
    by (auto simp: eventually_ae_filter null_sets_density_iff)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2639
  then have "AE x in M. x \<notin> N \<longrightarrow> P x" by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2640
  with ae show "AE x in M. 0 < f x \<longrightarrow> P x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2641
    by (rule eventually_elim2) auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2642
next
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2643
  fix N assume ae: "AE x in M. 0 < f x \<longrightarrow> P x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2644
  then obtain N where "{x \<in> space M. \<not> (0 < f x \<longrightarrow> P x)} \<subseteq> N" "N \<in> null_sets M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2645
    by (auto simp: eventually_ae_filter)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2646
  then have *: "{x \<in> space (density M f). \<not> P x} \<subseteq> N \<union> {x\<in>space M. \<not> 0 < f x}"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2647
    "N \<union> {x\<in>space M. \<not> 0 < f x} \<in> sets M" and ae2: "AE x in M. x \<notin> N"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
  2648
    using f by (auto simp: subset_eq intro!: sets.sets_Collect_neg AE_not_in)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2649
  show "AE x in density M f. P x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2650
    using ae2
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2651
    unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f]
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2652
    by (intro exI[of _ "N \<union> {x\<in>space M. \<not> 0 < f x}"] conjI *)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2653
       (auto elim: eventually_elim2)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2654
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2655
49798
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2656
lemma positive_integral_density':
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2657
  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2658
  assumes g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2659
  shows "integral\<^sup>P (density M f) g = (\<integral>\<^sup>+ x. f x * g x \<partial>M)"
49798
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2660
using g proof induct
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2661
  case (cong u v)
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2662
  then show ?case
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2663
    apply (subst positive_integral_cong[OF cong(3)])
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
  2664
    apply (simp_all cong: positive_integral_cong)
49798
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2665
    done
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2666
next
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2667
  case (set A) then show ?case
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2668
    by (simp add: emeasure_density f)
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2669
next
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2670
  case (mult u c)
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2671
  moreover have "\<And>x. f x * (c * u x) = c * (f x * u x)" by (simp add: field_simps)
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2672
  ultimately show ?case
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
  2673
    using f by (simp add: positive_integral_cmult)
49798
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2674
next
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2675
  case (add u v)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
  2676
  then have "\<And>x. f x * (v x + u x) = f x * v x + f x * u x"
49798
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2677
    by (simp add: ereal_right_distrib)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
  2678
  with add f show ?case
49798
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2679
    by (auto simp add: positive_integral_add ereal_zero_le_0_iff intro!: positive_integral_add[symmetric])
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2680
next
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2681
  case (seq U)
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2682
  from f(2) have eq: "AE x in M. f x * (SUP i. U i x) = (SUP i. f x * U i x)"
56212
3253aaf73a01 consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents: 56193
diff changeset
  2683
    by eventually_elim (simp add: SUP_ereal_cmult seq)
49798
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2684
  from seq f show ?case
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2685
    apply (simp add: positive_integral_monotone_convergence_SUP)
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2686
    apply (subst positive_integral_cong_AE[OF eq])
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2687
    apply (subst positive_integral_monotone_convergence_SUP_AE)
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2688
    apply (auto simp: incseq_def le_fun_def intro!: ereal_mult_left_mono)
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2689
    done
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2690
qed
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2691
49798
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2692
lemma positive_integral_density:
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2693
  "f \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> g' \<in> borel_measurable M \<Longrightarrow> 
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2694
    integral\<^sup>P (density M f) g' = (\<integral>\<^sup>+ x. f x * g' x \<partial>M)"
49798
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2695
  by (subst (1 2) positive_integral_max_0[symmetric])
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2696
     (auto intro!: positive_integral_cong_AE
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2697
           simp: measurable_If max_def ereal_zero_le_0_iff positive_integral_density')
8d5668f73c42 induction prove for positive_integral_density
hoelzl
parents: 49797
diff changeset
  2698
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2699
lemma integral_density:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2700
  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2701
    and g: "g \<in> borel_measurable M"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2702
  shows "integral\<^sup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2703
    and "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2704
  unfolding lebesgue_integral_def integrable_def using f g
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2705
  by (auto simp: positive_integral_density)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2706
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2707
lemma emeasure_restricted:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2708
  assumes S: "S \<in> sets M" and X: "X \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2709
  shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)"
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2710
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2711
  have "emeasure (density M (indicator S)) X = (\<integral>\<^sup>+x. indicator S x * indicator X x \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2712
    using S X by (simp add: emeasure_density)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2713
  also have "\<dots> = (\<integral>\<^sup>+x. indicator (S \<inter> X) x \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2714
    by (auto intro!: positive_integral_cong simp: indicator_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2715
  also have "\<dots> = emeasure M (S \<inter> X)"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
  2716
    using S X by (simp add: sets.Int)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2717
  finally show ?thesis .
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2718
qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2719
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2720
lemma measure_restricted:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2721
  "S \<in> sets M \<Longrightarrow> X \<in> sets M \<Longrightarrow> measure (density M (indicator S)) X = measure M (S \<inter> X)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2722
  by (simp add: emeasure_restricted measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2723
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2724
lemma (in finite_measure) finite_measure_restricted:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2725
  "S \<in> sets M \<Longrightarrow> finite_measure (density M (indicator S))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2726
  by default (simp add: emeasure_restricted)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2727
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2728
lemma emeasure_density_const:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2729
  "A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> emeasure (density M (\<lambda>_. c)) A = c * emeasure M A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2730
  by (auto simp: positive_integral_cmult_indicator emeasure_density)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2731
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2732
lemma measure_density_const:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2733
  "A \<in> sets M \<Longrightarrow> 0 < c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real c * measure M A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2734
  by (auto simp: emeasure_density_const measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2735
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2736
lemma density_density_eq:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2737
   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow>
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2738
   density (density M f) g = density M (\<lambda>x. f x * g x)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2739
  by (auto intro!: measure_eqI simp: emeasure_density positive_integral_density ac_simps)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2740
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2741
lemma distr_density_distr:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2742
  assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2743
    and inv: "\<forall>x\<in>space M. T' (T x) = x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2744
  assumes f: "f \<in> borel_measurable M'"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2745
  shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)" (is "?R = ?L")
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2746
proof (rule measure_eqI)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2747
  fix A assume A: "A \<in> sets ?R"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2748
  { fix x assume "x \<in> space M"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
  2749
    with sets.sets_into_space[OF A]
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2750
    have "indicator (T' -` A \<inter> space M') (T x) = (indicator A x :: ereal)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2751
      using T inv by (auto simp: indicator_def measurable_space) }
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2752
  with A T T' f show "emeasure ?R A = emeasure ?L A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2753
    by (simp add: measurable_comp emeasure_density emeasure_distr
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2754
                  positive_integral_distr measurable_sets cong: positive_integral_cong)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2755
qed simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2756
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2757
lemma density_density_divide:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2758
  fixes f g :: "'a \<Rightarrow> real"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2759
  assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2760
  assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2761
  assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2762
  shows "density (density M f) (\<lambda>x. g x / f x) = density M g"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2763
proof -
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2764
  have "density M g = density M (\<lambda>x. f x * (g x / f x))"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2765
    using f g ac by (auto intro!: density_cong measurable_If)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2766
  then show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2767
    using f g by (subst density_density_eq) auto
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2768
qed
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2769
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2770
section {* Point measure *}
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2771
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2772
definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2773
  "point_measure A f = density (count_space A) f"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2774
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2775
lemma
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2776
  shows space_point_measure: "space (point_measure A f) = A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2777
    and sets_point_measure: "sets (point_measure A f) = Pow A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2778
  by (auto simp: point_measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2779
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2780
lemma measurable_point_measure_eq1[simp]:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2781
  "g \<in> measurable (point_measure A f) M \<longleftrightarrow> g \<in> A \<rightarrow> space M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2782
  unfolding point_measure_def by simp
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2783
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2784
lemma measurable_point_measure_eq2_finite[simp]:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2785
  "finite A \<Longrightarrow>
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2786
   g \<in> measurable M (point_measure A f) \<longleftrightarrow>
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2787
    (g \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. g -` {a} \<inter> space M \<in> sets M))"
50002
ce0d316b5b44 add measurability prover; add support for Borel sets
hoelzl
parents: 50001
diff changeset
  2788
  unfolding point_measure_def by (simp add: measurable_count_space_eq2)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2789
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2790
lemma simple_function_point_measure[simp]:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2791
  "simple_function (point_measure A f) g \<longleftrightarrow> finite (g ` A)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2792
  by (simp add: point_measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2793
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2794
lemma emeasure_point_measure:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2795
  assumes A: "finite {a\<in>X. 0 < f a}" "X \<subseteq> A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2796
  shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)"
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35833
diff changeset
  2797
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2798
  have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2799
    using `X \<subseteq> A` by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2800
  with A show ?thesis
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2801
    by (simp add: emeasure_density positive_integral_count_space ereal_zero_le_0_iff
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2802
                  point_measure_def indicator_def)
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35833
diff changeset
  2803
qed
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35833
diff changeset
  2804
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2805
lemma emeasure_point_measure_finite:
49795
9f2fb9b25a77 joint distribution of independent variables
hoelzl
parents: 49775
diff changeset
  2806
  "finite A \<Longrightarrow> (\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2807
  by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2808
49795
9f2fb9b25a77 joint distribution of independent variables
hoelzl
parents: 49775
diff changeset
  2809
lemma emeasure_point_measure_finite2:
9f2fb9b25a77 joint distribution of independent variables
hoelzl
parents: 49775
diff changeset
  2810
  "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> (\<And>i. i \<in> X \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
9f2fb9b25a77 joint distribution of independent variables
hoelzl
parents: 49775
diff changeset
  2811
  by (subst emeasure_point_measure)
9f2fb9b25a77 joint distribution of independent variables
hoelzl
parents: 49775
diff changeset
  2812
     (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
9f2fb9b25a77 joint distribution of independent variables
hoelzl
parents: 49775
diff changeset
  2813
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2814
lemma null_sets_point_measure_iff:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2815
  "X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x \<le> 0)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2816
 by (auto simp: AE_count_space null_sets_density_iff point_measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2817
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2818
lemma AE_point_measure:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2819
  "(AE x in point_measure A f. P x) \<longleftrightarrow> (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2820
  unfolding point_measure_def
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2821
  by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2822
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2823
lemma positive_integral_point_measure:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2824
  "finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow>
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2825
    integral\<^sup>P (point_measure A f) g = (\<Sum>a|a\<in>A \<and> 0 < f a \<and> 0 < g a. f a * g a)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2826
  unfolding point_measure_def
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2827
  apply (subst density_max_0)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2828
  apply (subst positive_integral_density)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2829
  apply (simp_all add: AE_count_space positive_integral_density)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2830
  apply (subst positive_integral_count_space )
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2831
  apply (auto intro!: setsum_cong simp: max_def ereal_zero_less_0_iff)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2832
  apply (rule finite_subset)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2833
  prefer 2
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2834
  apply assumption
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2835
  apply auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2836
  done
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2837
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2838
lemma positive_integral_point_measure_finite:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2839
  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> g a) \<Longrightarrow>
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2840
    integral\<^sup>P (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2841
  by (subst positive_integral_point_measure) (auto intro!: setsum_mono_zero_left simp: less_le)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2842
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2843
lemma lebesgue_integral_point_measure_finite:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2844
  "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> integral\<^sup>L (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2845
  by (simp add: lebesgue_integral_count_space_finite AE_count_space integral_density point_measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2846
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2847
lemma integrable_point_measure_finite:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2848
  "finite A \<Longrightarrow> integrable (point_measure A (\<lambda>x. ereal (f x))) g"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2849
  unfolding point_measure_def
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2850
  apply (subst density_ereal_max_0)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2851
  apply (subst integral_density)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2852
  apply (auto simp: AE_count_space integrable_count_space)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2853
  done
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2854
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2855
section {* Uniform measure *}
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2856
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2857
definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2858
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2859
lemma
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2860
  shows sets_uniform_measure[simp]: "sets (uniform_measure M A) = sets M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2861
    and space_uniform_measure[simp]: "space (uniform_measure M A) = space M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2862
  by (auto simp: uniform_measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2863
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2864
lemma emeasure_uniform_measure[simp]:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2865
  assumes A: "A \<in> sets M" and B: "B \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2866
  shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2867
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  2868
  from A B have "emeasure (uniform_measure M A) B = (\<integral>\<^sup>+x. (1 / emeasure M A) * indicator (A \<inter> B) x \<partial>M)"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2869
    by (auto simp add: uniform_measure_def emeasure_density split: split_indicator
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2870
             intro!: positive_integral_cong)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2871
  also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2872
    using A B
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
  2873
    by (subst positive_integral_cmult_indicator) (simp_all add: sets.Int emeasure_nonneg)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2874
  finally show ?thesis .
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2875
qed
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2876
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2877
lemma measure_uniform_measure[simp]:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2878
  assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>" and B: "B \<in> sets M"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2879
  shows "measure (uniform_measure M A) B = measure M (A \<inter> B) / measure M A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2880
  using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2881
  by (cases "emeasure M A" "emeasure M (A \<inter> B)" rule: ereal2_cases) (simp_all add: measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2882
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2883
section {* Uniform count measure *}
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2884
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2885
definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2886
 
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2887
lemma 
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2888
  shows space_uniform_count_measure: "space (uniform_count_measure A) = A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2889
    and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2890
    unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2891
 
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2892
lemma emeasure_uniform_count_measure:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2893
  "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2894
  by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2895
 
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2896
lemma measure_uniform_count_measure:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2897
  "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2898
  by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  2899
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2900
end