src/HOL/Probability/Lebesgue_Integration.thy
author haftmann
Tue, 18 Mar 2014 22:11:46 +0100
changeset 56212 3253aaf73a01
parent 56193 c726ecfb22b6
child 56213 e5720d3c18f0
permissions -rw-r--r--
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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parents: 41981
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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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parents:
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header {*Lebesgue Integration*}
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parents:
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d5d342611edb Rewrite the Probability theory.
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theory Lebesgue_Integration
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  imports Measure_Space Borel_Space
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parents:
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    10
begin
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parents:
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    11
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    12
lemma tendsto_real_max:
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parents: 41831
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    13
  fixes x y :: real
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    14
  assumes "(X ---> x) net"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    15
  assumes "(Y ---> y) net"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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parents: 41831
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    16
  shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    17
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    18
  have *: "\<And>x y :: real. max x y = y + ((x - y) + norm (x - y)) / 2"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    19
    by (auto split: split_max simp: field_simps)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    20
  show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    21
    unfolding *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    22
    by (intro tendsto_add assms tendsto_divide tendsto_norm tendsto_diff) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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parents: 41831
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    23
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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parents: 41831
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    24
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lemma measurable_sets2[intro]:
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    26
  assumes "f \<in> measurable M M'" "g \<in> measurable M M''"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    27
  and "A \<in> sets M'" "B \<in> sets M''"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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parents: 41831
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    28
  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
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parents: 41831
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    29
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    30
  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
hoelzl
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    31
    by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    32
  then show ?thesis using assms by (auto intro: measurable_sets)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    33
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    34
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    35
section "Simple function"
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parents:
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text {*
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    38
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    39
Our simple functions are not restricted to positive real numbers. Instead
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they are just functions with a finite range and are measurable when singleton
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sets are measurable.
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*}
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    44
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definition "simple_function M g \<longleftrightarrow>
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    finite (g ` space M) \<and>
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    47
    (\<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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  assumes "simple_function M g"
40875
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hoelzl
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    51
  shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
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    52
proof -
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
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    53
  show "finite (g ` space M)"
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
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    54
    using assms unfolding simple_function_def by auto
40875
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
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    55
  have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
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    56
  also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
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    57
  finally show "g -` X \<inter> space M \<in> sets M" using assms
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hoelzl
parents: 50001
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    58
    by (auto simp del: UN_simps simp: simple_function_def)
40871
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hoelzl
parents: 40859
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    59
qed
36624
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    60
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parents: 46905
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    61
lemma simple_function_measurable2[intro]:
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parents: 41831
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    62
  assumes "simple_function M f" "simple_function M g"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
    63
  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
parents: 41831
diff changeset
    64
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
    65
  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
hoelzl
parents: 41831
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    66
    by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
    67
  then show ?thesis using assms[THEN simple_functionD(2)] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    68
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    69
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05663f75964c reworked Probability theory
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    70
lemma simple_function_indicator_representation:
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cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
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    71
  fixes f ::"'a \<Rightarrow> ereal"
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3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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parents: 41661
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    72
  assumes f: "simple_function M f" and x: "x \<in> space M"
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hoelzl
parents: 38642
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    73
  shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    74
  (is "?l = ?r")
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hoelzl
parents: 38642
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    75
proof -
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aaee86c0e237 moved generic lemmas in Probability to HOL
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parents: 38656
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    76
  have "?r = (\<Sum>y \<in> f ` space M.
38656
d5d342611edb Rewrite the Probability theory.
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    77
    (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
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hoelzl
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    78
    by (auto intro!: setsum_cong2)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    79
  also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    80
    using assms by (auto dest: simple_functionD simp: setsum_delta)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    81
  also have "... = f x" using x by (auto simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    82
  finally show ?thesis by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    83
qed
36624
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hoelzl
parents: 35977
diff changeset
    84
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hoelzl
parents: 46905
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    85
lemma simple_function_notspace:
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hoelzl
parents: 43339
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    86
  "simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h")
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
    87
proof -
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d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    88
  have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    89
  hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    90
  have "?h -` {0} \<inter> space M = space M" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    91
  thus ?thesis unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    92
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    93
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hoelzl
parents: 46905
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    94
lemma simple_function_cong:
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d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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    95
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
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
    96
  shows "simple_function M f \<longleftrightarrow> simple_function M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    97
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    98
  have "f ` space M = g ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    99
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   100
    using assms by (auto intro!: image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   101
  thus ?thesis unfolding simple_function_def using assms by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   102
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   103
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05663f75964c reworked Probability theory
hoelzl
parents: 46905
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   104
lemma simple_function_cong_algebra:
41689
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hoelzl
parents: 41661
diff changeset
   105
  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
   106
  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
   107
  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
   108
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
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   109
lemma borel_measurable_simple_function[measurable_dest]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
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   110
  assumes "simple_function M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   111
  shows "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   112
proof (rule borel_measurableI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   113
  fix S
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   114
  let ?I = "f ` (f -` S \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   115
  have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   116
  have "finite ?I"
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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
   117
    using assms unfolding simple_function_def
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
   118
    using finite_subset[of "f ` (f -` S \<inter> space M)" "f ` space M"] by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   119
  hence "?U \<in> sets M"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   120
    apply (rule sets.finite_UN)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   121
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   122
  thus "f -` S \<inter> space M \<in> sets M" unfolding * .
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   123
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   124
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   125
lemma simple_function_borel_measurable:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   126
  fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   127
  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
   128
  shows "simple_function M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   129
  using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   130
  by (auto intro: borel_measurable_vimage)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   131
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   132
lemma simple_function_eq_borel_measurable:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   133
  fixes f :: "'a \<Rightarrow> ereal"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   134
  shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   135
  using simple_function_borel_measurable[of f] borel_measurable_simple_function[of M f]
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44666
diff changeset
   136
  by (fastforce simp: simple_function_def)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   137
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   138
lemma simple_function_const[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
   139
  "simple_function M (\<lambda>x. c)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   140
  by (auto intro: finite_subset simp: simple_function_def)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   141
lemma simple_function_compose[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
   142
  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
   143
  shows "simple_function M (g \<circ> f)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   144
  unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   145
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   146
  show "finite ((g \<circ> f) ` space M)"
56154
f0a927235162 more complete set of lemmas wrt. image and composition
haftmann
parents: 54611
diff changeset
   147
    using assms unfolding simple_function_def by (auto simp: image_comp [symmetric])
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   148
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   149
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   150
  let ?G = "g -` {g (f x)} \<inter> (f`space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   151
  have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   152
    (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   153
  show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   154
    using assms unfolding simple_function_def *
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   155
    by (rule_tac sets.finite_UN) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   156
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   157
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   158
lemma simple_function_indicator[intro, simp]:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   159
  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
   160
  shows "simple_function M (indicator A)"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   161
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   162
  have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   163
    by (auto simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   164
  hence "finite ?S" by (rule finite_subset) simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   165
  moreover have "- A \<inter> space M = space M - A" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   166
  ultimately show ?thesis unfolding simple_function_def
46905
6b1c0a80a57a prefer abs_def over def_raw;
wenzelm
parents: 46884
diff changeset
   167
    using assms by (auto simp: indicator_def [abs_def])
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   168
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   169
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   170
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
   171
  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
   172
  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
   173
  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
   174
  unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   175
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   176
  show "finite (?p ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   177
    using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   178
    by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   179
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   180
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   181
  have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   182
      (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   183
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   184
  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
   185
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   186
qed
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   187
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   188
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
   189
  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
   190
  shows "simple_function M (\<lambda>x. g (f x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   191
  using simple_function_compose[OF assms, of g]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   192
  by (simp add: comp_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   193
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   194
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
   195
  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
   196
  shows "simple_function M (\<lambda>x. h (f x) (g x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   197
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
   198
  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
   199
    using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   200
  thus ?thesis by (simp_all add: comp_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   201
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   202
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   203
lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   204
  and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   205
  and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   206
  and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   207
  and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   208
  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
   209
  and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   210
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   211
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
   212
  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
   213
  shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   214
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   215
  assume "finite P" from this assms show ?thesis by induct auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   216
qed auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   217
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   218
lemma
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   219
  fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   220
  shows simple_function_ereal[intro, simp]: "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
   221
  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
   222
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   223
lemma
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   224
  fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   225
  shows simple_function_real_of_nat[intro, simp]: "simple_function M (\<lambda>x. real (f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   226
  by (auto intro!: simple_function_compose1[OF sf])
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   227
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   228
lemma borel_measurable_implies_simple_function_sequence:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   229
  fixes u :: "'a \<Rightarrow> ereal"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   230
  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
   231
  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
   232
             (\<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
   233
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   234
  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
   235
  { 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
   236
    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
   237
      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
   238
      then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   239
         by (cases "u x") (auto intro!: natfloor_mono simp: mult_nonneg_nonneg)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   240
      moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   241
        by (intro real_natfloor_le) (auto simp: mult_nonneg_nonneg)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   242
      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
   243
        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
   244
    qed auto }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   245
  note f_upper = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   246
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   247
  have real_f:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   248
    "\<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
   249
    unfolding f_def by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   250
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   251
  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
   252
  show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   253
  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
   254
    fix i
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   255
    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
   256
    proof (intro simple_function_borel_measurable)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   257
      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
   258
        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
   259
      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
   260
        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
   261
      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
   262
        by (rule finite_subset) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   263
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   264
    then show "simple_function M (?g i)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   265
      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
   266
  next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   267
    show "incseq ?g"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   268
    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
   269
      fix x and i :: nat
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   270
      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
   271
      proof ((split split_if)+, intro conjI impI)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   272
        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
   273
        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
   274
          by (cases "u x") (auto intro!: le_natfloor)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   275
      next
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   276
        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
   277
        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
   278
          by (cases "u x") auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   279
      next
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   280
        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
   281
        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
   282
          by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   283
        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
   284
        proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   285
          assume "0 \<le> u x" then show ?thesis
46671
3a40ea076230 removing unnecessary assumptions in RComplete;
bulwahn
parents: 45342
diff changeset
   286
            by (intro le_mult_natfloor) 
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   287
        next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   288
          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
   289
            by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   290
        qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   291
        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
   292
          by (simp add: ac_simps)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   293
        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
   294
      qed simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   295
      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
   296
        by (auto simp: field_simps)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   297
    qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   298
  next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   299
    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
   300
    proof (rule SUP_eqI)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   301
      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
   302
        by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   303
                                     mult_nonpos_nonneg mult_nonneg_nonneg)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   304
    next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   305
      fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   306
      have "\<And>i. 0 \<le> ?g i x" by (auto simp: divide_nonneg_pos)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   307
      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
   308
      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
   309
      proof (cases y)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   310
        case (real r)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   311
        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
   312
        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
   313
        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
   314
        then guess p .. note ux = this
44666
8670a39d4420 remove more duplicate lemmas
huffman
parents: 44568
diff changeset
   315
        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
   316
        have "p \<le> r"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   317
        proof (rule ccontr)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   318
          assume "\<not> p \<le> r"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   319
          with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   320
          obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: inverse_eq_divide field_simps)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   321
          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
   322
          moreover
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   323
          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
   324
            using *[of "max N m"] m unfolding real_f using ux
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   325
            by (cases "0 \<le> u x") (simp_all add: max_def mult_nonneg_nonneg split: split_if_asm)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   326
          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
   327
            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
   328
          ultimately show False by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   329
        qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   330
        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
   331
      qed (insert `0 \<le> y`, auto)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   332
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   333
  qed (auto simp: divide_nonneg_pos)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   334
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   335
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   336
lemma borel_measurable_implies_simple_function_sequence':
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   337
  fixes u :: "'a \<Rightarrow> ereal"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   338
  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
   339
  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
   340
    "\<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
   341
  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
   342
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   343
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
   344
  fixes u :: "'a \<Rightarrow> ereal"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   345
  assumes u: "simple_function M u"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   346
  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
   347
  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
   348
  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
   349
  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
   350
  shows "P u"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   351
proof (rule cong)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   352
  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
   353
  proof eventually_elim
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   354
    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
   355
    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
   356
    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
   357
  qed
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   358
next
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   359
  from u have "finite (u ` space M)"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   360
    unfolding simple_function_def by auto
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   361
  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
   362
  proof induct
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   363
    case empty show ?case
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   364
      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
   365
  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
   366
next
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   367
  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
   368
    apply (subst simple_function_cong)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   369
    apply (rule simple_function_indicator_representation[symmetric])
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   370
    apply (auto intro: u)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   371
    done
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   372
qed fact
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   373
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   374
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
   375
  fixes u :: "'a \<Rightarrow> ereal"
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   376
  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
   377
  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
   378
  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
   379
  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
   380
  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
   381
  shows "P u"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   382
proof -
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   383
  show ?thesis
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   384
  proof (rule cong)
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   385
    fix x assume x: "x \<in> space M"
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   386
    from simple_function_indicator_representation[OF u x]
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   387
    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
   388
  next
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   389
    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
   390
      apply (subst simple_function_cong)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   391
      apply (rule simple_function_indicator_representation[symmetric])
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   392
      apply (auto intro: u)
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   393
      done
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   394
  next
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   395
    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
   396
      unfolding simple_function_def by auto
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   397
    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
   398
    proof induct
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   399
      case empty show ?case
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   400
        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
   401
    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
   402
       simple_function_setsum)
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   403
  qed fact
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   404
qed
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   405
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   406
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
   407
  fixes u :: "'a \<Rightarrow> ereal"
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   408
  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
   409
  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
   410
  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
   411
  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
   412
  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
   413
  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
   414
  shows "P u"
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   415
  using u
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   416
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
   417
  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
   418
    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
   419
  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
   420
    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
   421
  
49797
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   422
  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
   423
    by (simp add: borel_measurable_simple_function)
28066863284c add induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49796
diff changeset
   424
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   425
  show "P u"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   426
    unfolding u_eq
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   427
  proof (rule seq)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   428
    fix i show "P (U i)"
49799
15ea98537c76 strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents: 49798
diff changeset
   429
      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
   430
      by (induct rule: simple_function_induct_nn)
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   431
         (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
   432
  qed fact+
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   433
qed
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   434
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   435
lemma simple_function_If_set:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   436
  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
   437
  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
   438
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   439
  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
   440
  show ?thesis unfolding simple_function_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   441
  proof safe
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   442
    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
   443
    from finite_subset[OF this] assms
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   444
    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
   445
  next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   446
    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
   447
    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
   448
      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
   449
      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
   450
      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
   451
    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
   452
      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
   453
    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
   454
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   455
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   456
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   457
lemma simple_function_If:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   458
  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
   459
  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
   460
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   461
  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
   462
  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
   463
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   464
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   465
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
   466
  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
   467
  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
   468
  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
   469
  using assms unfolding simple_function_def by auto
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   470
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   471
lemma simple_function_comp:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   472
  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
   473
    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
   474
  shows "simple_function M (\<lambda>x. f (T x))"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   475
proof (intro simple_function_def[THEN iffD2] conjI ballI)
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   476
  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
   477
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   478
  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
   479
    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
   480
  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
   481
  then have "i \<in> f ` space M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   482
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   483
  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
   484
    using f unfolding simple_function_def by auto
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   485
  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
   486
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   487
  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
   488
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   489
  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
   490
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   491
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   492
section "Simple integral"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   493
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   494
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
   495
  "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
   496
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
   497
syntax
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   498
  "_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
   499
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
   500
translations
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   501
  "\<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
   502
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   503
lemma simple_integral_cong:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   504
  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
   505
  shows "integral\<^sup>S M f = integral\<^sup>S M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   506
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   507
  have "f ` space M = g ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   508
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   509
    using assms by (auto intro!: image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   510
  thus ?thesis unfolding simple_integral_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   511
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   512
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   513
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
   514
  "(\<integral>\<^sup>Sx. c \<partial>M) = c * (emeasure M) (space M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   515
proof (cases "space M = {}")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   516
  case True thus ?thesis unfolding simple_integral_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   517
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   518
  case False hence "(\<lambda>x. c) ` space M = {c}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   519
  thus ?thesis unfolding simple_integral_def by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   520
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   521
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   522
lemma simple_function_partition:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   523
  assumes f: "simple_function M f" and g: "simple_function M g"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   524
  shows "integral\<^sup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * (emeasure M) A)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   525
    (is "_ = setsum _ (?p ` space M)")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   526
proof-
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   527
  let ?sub = "\<lambda>x. ?p ` (f -` {x} \<inter> space M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   528
  let ?SIGMA = "Sigma (f`space M) ?sub"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   529
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   530
  have [intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   531
    "finite (f ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   532
    "finite (g ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   533
    using assms unfolding simple_function_def by simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   534
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   535
  { fix A
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   536
    have "?p ` (A \<inter> space M) \<subseteq>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   537
      (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   538
      by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   539
    hence "finite (?p ` (A \<inter> space M))"
40786
0a54cfc9add3 gave more standard finite set rules simp and intro attribute
nipkow
parents: 39910
diff changeset
   540
      by (rule finite_subset) auto }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   541
  note this[intro, simp]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   542
  note sets = simple_function_measurable2[OF f g]
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   543
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   544
  { fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   545
    have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   546
    with sets have "(emeasure M) (f -` {f x} \<inter> space M) = setsum (emeasure M) (?sub (f x))"
47761
dfe747e72fa8 moved lemmas to appropriate places
hoelzl
parents: 47694
diff changeset
   547
      by (subst setsum_emeasure) (auto simp: disjoint_family_on_def) }
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   548
  hence "integral\<^sup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * (emeasure M) A)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   549
    unfolding simple_integral_def using f sets
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   550
    by (subst setsum_Sigma[symmetric])
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   551
       (auto intro!: setsum_cong setsum_ereal_right_distrib)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   552
  also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * (emeasure M) A)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   553
  proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   554
    have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   555
    have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   556
      = (\<lambda>x. (f x, ?p x)) ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   557
    proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   558
      fix x assume "x \<in> space M"
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   559
      thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   560
        by (auto intro!: image_eqI[of _ _ "?p x"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   561
    qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   562
    thus ?thesis
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   563
      apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   564
      apply (rule_tac x="xa" in image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   565
      by simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   566
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   567
  finally show ?thesis .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   568
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   569
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   570
lemma simple_integral_add[simp]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   571
  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
   572
  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
   573
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   574
  { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   575
    assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   576
    hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   577
        "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   578
      by auto }
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   579
  with assms show ?thesis
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   580
    unfolding
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   581
      simple_function_partition[OF simple_function_add[OF f g] simple_function_Pair[OF f g]]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   582
      simple_function_partition[OF f g]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   583
      simple_function_partition[OF g f]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   584
    by (subst (3) Int_commute)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   585
       (auto simp add: ereal_left_distrib setsum_addf[symmetric] intro!: setsum_cong)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   586
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   587
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   588
lemma simple_integral_setsum[simp]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   589
  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
   590
  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
   591
  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
   592
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   593
  assume "finite P"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   594
  from this assms show ?thesis
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   595
    by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   596
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   597
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   598
lemma simple_integral_mult[simp]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   599
  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
   600
  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
   601
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   602
  note mult = simple_function_mult[OF simple_function_const[of _ c] f(1)]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   603
  { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   604
    assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   605
    hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   606
      by auto }
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   607
  with assms show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   608
    unfolding simple_function_partition[OF mult f(1)]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   609
              simple_function_partition[OF f(1) mult]
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   610
    by (subst setsum_ereal_right_distrib)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   611
       (auto intro!: ereal_0_le_mult setsum_cong simp: mult_assoc)
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   612
qed
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   613
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   614
lemma simple_integral_mono_AE:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   615
  assumes f: "simple_function M f" and g: "simple_function M g"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   616
  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
   617
  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
   618
proof -
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   619
  let ?S = "\<lambda>x. (g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   620
  have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   621
    "\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   622
  show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   623
    unfolding *
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   624
      simple_function_partition[OF f g]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   625
      simple_function_partition[OF g f]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   626
  proof (safe intro!: setsum_mono)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   627
    fix x assume "x \<in> space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   628
    then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   629
    show "the_elem (f`?S x) * (emeasure M) (?S x) \<le> the_elem (g`?S x) * (emeasure M) (?S x)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   630
    proof (cases "f x \<le> g x")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   631
      case True then show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   632
        using * assms(1,2)[THEN simple_functionD(2)]
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   633
        by (auto intro!: ereal_mult_right_mono)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   634
    next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   635
      case False
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   636
      obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "(emeasure M) N = 0"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   637
        using mono by (auto elim!: AE_E)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   638
      have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   639
      moreover have "?S x \<in> sets M" using assms
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   640
        by (rule_tac sets.Int) (auto intro!: simple_functionD)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   641
      ultimately have "(emeasure M) (?S x) \<le> (emeasure M) N"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   642
        using `N \<in> sets M` by (auto intro!: emeasure_mono)
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   643
      moreover have "0 \<le> (emeasure M) (?S x)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   644
        using assms(1,2)[THEN simple_functionD(2)] by auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   645
      ultimately have "(emeasure M) (?S x) = 0" using `(emeasure M) N = 0` by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   646
      then show ?thesis by simp
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   647
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   648
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   649
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   650
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   651
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
   652
  assumes "simple_function M f" and "simple_function M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   653
  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
   654
  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
   655
  using assms by (intro simple_integral_mono_AE) auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   656
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   657
lemma simple_integral_cong_AE:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   658
  assumes "simple_function M f" and "simple_function M g"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   659
  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
   660
  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
   661
  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
   662
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   663
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
   664
  assumes sf: "simple_function M f" "simple_function M g"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   665
  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
   666
  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
   667
proof (intro simple_integral_cong_AE sf AE_I)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   668
  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
   669
  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
   670
    using sf[THEN borel_measurable_simple_function] by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   671
qed simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   672
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   673
lemma simple_integral_indicator:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   674
  assumes "A \<in> sets M"
49796
182fa22e7ee8 introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents: 49795
diff changeset
   675
  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
   676
  shows "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   677
    (\<Sum>x \<in> f ` space M. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))"
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
   678
proof (cases "A = space M")
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
   679
  case True
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
   680
  then have "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) = integral\<^sup>S M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   681
    by (auto intro!: simple_integral_cong)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
   682
  with True show ?thesis by (simp add: simple_integral_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   683
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   684
  assume "A \<noteq> space M"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   685
  then obtain x where x: "x \<in> space M" "x \<notin> A" using sets.sets_into_space[OF assms(1)] by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   686
  have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   687
  proof safe
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   688
    fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   689
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   690
    fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   691
      using sets.sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   692
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   693
    show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   694
  qed
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   695
  have *: "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   696
    (\<Sum>x \<in> f ` space M \<union> {0}. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   697
    unfolding simple_integral_def I
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   698
  proof (rule setsum_mono_zero_cong_left)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   699
    show "finite (f ` space M \<union> {0})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   700
      using assms(2) unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   701
    show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   702
      using sets.sets_into_space[OF assms(1)] by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   703
    have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   704
      by (auto simp: image_iff)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   705
    thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   706
      i * (emeasure M) (f -` {i} \<inter> space M \<inter> A) = 0" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   707
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   708
    fix x assume "x \<in> f`A \<union> {0}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   709
    hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   710
      by (auto simp: indicator_def split: split_if_asm)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   711
    thus "x * (emeasure M) (?I -` {x} \<inter> space M) =
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   712
      x * (emeasure M) (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   713
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   714
  show ?thesis unfolding *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   715
    using assms(2) unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   716
    by (auto intro!: setsum_mono_zero_cong_right)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   717
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   718
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   719
lemma simple_integral_indicator_only[simp]:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   720
  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
   721
  shows "integral\<^sup>S M (indicator A) = emeasure M A"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   722
proof cases
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   723
  assume "space M = {}" hence "A = {}" using sets.sets_into_space[OF assms] by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   724
  thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   725
next
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   726
  assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::ereal}" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   727
  thus ?thesis
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   728
    using simple_integral_indicator[OF assms simple_function_const[of _ 1]]
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   729
    using sets.sets_into_space[OF assms]
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   730
    by (auto intro!: arg_cong[where f="(emeasure M)"])
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   731
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   732
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   733
lemma simple_integral_null_set:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   734
  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
   735
  shows "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   736
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   737
  have "AE x in M. indicator N x = (0 :: ereal)"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   738
    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
   739
  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
   740
    using assms apply (intro simple_integral_cong_AE) by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   741
  then show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   742
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   743
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   744
lemma simple_integral_cong_AE_mult_indicator:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   745
  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
   746
  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
   747
  using assms by (intro simple_integral_cong_AE) auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   748
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   749
lemma simple_integral_cmult_indicator:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   750
  assumes A: "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
   751
  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
   752
  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
   753
  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
   754
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   755
lemma simple_integral_positive:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   756
  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
   757
  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
   758
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   759
  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
   760
    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
   761
  then show ?thesis by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   762
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   763
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
   764
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
   765
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   766
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
   767
  "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
   768
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
   769
syntax
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   770
  "_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
   771
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
   772
translations
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   773
  "\<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
   774
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   775
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
   776
  "0 \<le> integral\<^sup>P M f"
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   777
  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
   778
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   779
lemma positive_integral_not_MInfty[simp]: "integral\<^sup>P M f \<noteq> -\<infinity>"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   780
  using positive_integral_positive[of M f] by auto
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   781
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   782
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
   783
  "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)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   784
    (is "_ = SUPR ?A ?f")
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   785
  unfolding positive_integral_def
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   786
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
   787
  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
   788
  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
   789
  note gM = g(1)[THEN borel_measurable_simple_function]
50252
4aa34bd43228 eliminated slightly odd identifiers;
wenzelm
parents: 50244
diff changeset
   790
  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
   791
  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
   792
  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
   793
    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
   794
    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
   795
    done
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   796
  show "integral\<^sup>S M g \<le> SUPR ?A ?f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   797
  proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   798
    have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   799
    assume "(emeasure M) ?G = 0"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   800
    with gM have "AE x in M. x \<notin> ?G"
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   801
      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
   802
    with gM g show ?thesis
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   803
      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
   804
         (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
   805
  next
50252
4aa34bd43228 eliminated slightly odd identifiers;
wenzelm
parents: 50244
diff changeset
   806
    assume \<mu>_G: "(emeasure M) ?G \<noteq> 0"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   807
    have "SUPR ?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
   808
    proof (intro SUP_PInfty)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   809
      fix n :: nat
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   810
      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
   811
      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
   812
      then have "?g ?y \<in> ?A" by (rule g_in_A)
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   813
      have "real n \<le> ?y * (emeasure M) ?G"
50252
4aa34bd43228 eliminated slightly odd identifiers;
wenzelm
parents: 50244
diff changeset
   814
        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
   815
      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
   816
        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
   817
        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
   818
      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
   819
        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
   820
      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
   821
        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
   822
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   823
    then show ?thesis by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   824
  qed
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   825
qed (auto intro: SUP_upper)
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   826
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   827
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
   828
  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
   829
  unfolding positive_integral_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   830
proof (safe intro!: SUP_mono)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   831
  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
   832
  from ae[THEN AE_E] guess N . note N = this
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   833
  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
   834
  let ?n = "\<lambda>x. n x * indicator (space M - N) x"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   835
  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
   836
    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
   837
  moreover
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   838
  { 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
   839
    proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   840
      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
   841
      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
   842
      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
   843
        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
   844
    qed simp }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   845
  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
   846
  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
   847
    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
   848
  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
   849
    by force
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   850
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   851
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   852
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
   853
  "(\<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
   854
  by (auto intro: positive_integral_mono_AE)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   855
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   856
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
   857
  "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
   858
  by (auto simp: eq_iff intro!: positive_integral_mono_AE)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   859
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   860
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
   861
  "(\<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
   862
  by (auto intro: positive_integral_cong_AE)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   863
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   864
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
   865
  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
   866
proof -
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   867
  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
   868
  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
   869
  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
   870
    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
   871
  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
   872
    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
   873
  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
   874
    unfolding positive_integral_def
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44890
diff changeset
   875
    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
   876
  ultimately show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   877
    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
   878
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   879
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   880
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
   881
  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
   882
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   883
  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
   884
  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
   885
    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
   886
             add: positive_integral_eq_simple_integral)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   887
  with assms show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   888
    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
   889
qed
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   890
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   891
lemma positive_integral_SUP_approx:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   892
  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
   893
  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
   894
  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
   895
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
   896
  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
   897
    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
   898
  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
   899
  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
   900
    using u(3) by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   901
  fix a :: ereal assume "0 < a" "a < 1"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   902
  hence "a \<noteq> 0" by auto
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   903
  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
   904
  have B: "\<And>i. ?B i \<in> sets M"
50003
8c213922ed49 use measurability prover
hoelzl
parents: 50002
diff changeset
   905
    using f `simple_function M u` by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   906
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   907
  let ?uB = "\<lambda>i x. u x * indicator (?B i) x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   908
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   909
  { fix i have "?B i \<subseteq> ?B (Suc i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   910
    proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   911
      fix i x assume "a * u x \<le> f i x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   912
      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
   913
        using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   914
      finally show "a * u x \<le> f (Suc i) x" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   915
    qed }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   916
  note B_mono = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   917
50244
de72bbe42190 qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents: 50104
diff changeset
   918
  note B_u = sets.Int[OF u(1)[THEN simple_functionD(2)] B]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   919
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
   920
  let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   921
  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
   922
  proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   923
    fix i
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   924
    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
   925
    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
   926
    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
   927
    proof safe
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   928
      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
   929
      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
   930
      proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   931
        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
   932
      next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   933
        assume "u x \<noteq> 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   934
        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
   935
        have "a * u x < 1 * u x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
   936
          by (intro ereal_mult_strict_right_mono) (auto simp: image_iff)
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46731
diff changeset
   937
        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
   938
        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
   939
          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
   940
        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
   941
        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
   942
      qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   943
    qed
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   944
    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
   945
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   946
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   947
  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
   948
    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
   949
  proof (subst SUP_ereal_setsum, safe)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   950
    fix x n assume "x \<in> space M"
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   951
    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
   952
      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
   953
  next
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   954
    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
   955
      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
   956
      by (auto intro!: setsum_cong SUP_ereal_cmult [symmetric])
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   957
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   958
  moreover
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   959
  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
   960
    apply (subst SUP_ereal_cmult [symmetric])
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
   961
  proof (safe intro!: SUP_mono bexI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   962
    fix i
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   963
    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
   964
      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
   965
      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
   966
    also have "\<dots> \<le> integral\<^sup>P M (f i)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   967
    proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   968
      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
   969
      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
   970
        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
   971
           (auto intro!: positive_integral_mono split: split_indicator)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
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
    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
   974
      by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   975
  next
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   976
    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
   977
      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
   978
  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
   979
  ultimately show "a * integral\<^sup>S M u \<le> ?S" by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   980
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   981
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   982
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
   983
  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
   984
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   985
  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
   986
    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
   987
  then show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   988
    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
   989
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   990
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   991
text {* Beppo-Levi monotone convergence theorem *}
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
   992
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
   993
  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
   994
  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
   995
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
   996
  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
   997
    by (auto intro!: SUP_least SUP_upper positive_integral_mono)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   998
next
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
   999
  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
  1000
    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
  1001
  proof (safe intro!: SUP_least)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1002
    fix g assume g: "simple_function M g"
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
  1003
      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
  1004
    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
  1005
      using f by (auto intro!: SUP_upper2)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53015
diff changeset
  1006
    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
  1007
      by (intro  positive_integral_SUP_approx[OF f g _ g'])
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46731
diff changeset
  1008
         (auto simp: le_fun_def max_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1009
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1010
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1011
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1012
lemma positive_integral_monotone_convergence_SUP_AE:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1013
  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
  1014
  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
  1015
proof -
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1016
  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
  1017
    by (simp add: AE_all_countable)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1018
  from this[THEN AE_E] guess N . note N = this
46731
5302e932d1e5 avoid undeclared variables in let bindings;
wenzelm
parents: 46671
diff changeset
  1019
  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
  1020
  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
  1021
  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
  1022
    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
  1023
  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
  1024
  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
  1025
    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
  1026
    { 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
  1027
        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
  1028
      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
  1029
        using N(1) by auto }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1030
  qed
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1031
  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
  1032
    using f_eq by (force intro!: arg_cong[where f="SUPR UNIV"] positive_integral_cong_AE ext)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1033
  finally show ?thesis .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1034
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1035
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1036
lemma positive_integral_monotone_convergence_SUP_AE_incseq:
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1037
  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
  1038
  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
  1039
  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
  1040
  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
  1041
     auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1042
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1043
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
  1044
  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
  1045
  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
  1046
  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
  1047
    f(3)[THEN borel_measurable_simple_function] f(2)]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1048
  by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPR UNIV"] ext)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1049
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1050
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
  1051
  "(\<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
  1052
  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
  1053
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1054
lemma positive_integral_cong_pos:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1055
  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
  1056
  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
  1057
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1058
  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
  1059
  proof (intro positive_integral_cong)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1060
    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
  1061
    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
  1062
      by (auto split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1063
  qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1064
  then show ?thesis by (simp add: positive_integral_max_0)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1065
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1066
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1067
lemma SUP_simple_integral_sequences:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1068
  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
  1069
  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
  1070
  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
  1071
  shows "(SUP i. integral\<^sup>S M (f i)) = (SUP i. integral\<^sup>S M (g i))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1072
    (is "SUPR _ ?F = SUPR _ ?G")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1073
proof -
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1074
  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
  1075
    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
  1076
  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
  1077
    unfolding eq[THEN positive_integral_cong_AE] ..
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1078
  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
  1079
    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
  1080
  finally show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1081
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1082
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1083
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
  1084
  "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
  1085
  by (subst positive_integral_eq_simple_integral) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1086
47694
05663f75964c reworked Probability theory
hoelzl
parents: 46905
diff changeset
  1087
lemma positive_integral_linear:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1088
  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
  1089
  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
  1090
  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
  1091
    (is "integral\<^sup>P M ?L = _")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1092
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1093
  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
  1094
  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
  1095
  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
  1096
  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
  1097
  let ?L' = "\<lambda>i x. a * u i x + v i x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1098
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1099
  have "?L \<in> borel_measurable M" using assms by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1100
  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
  1101
  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
  1102
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1103
  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
  1104
    using u v `0 \<le> a`
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1105
    by (auto simp: incseq_Suc_iff le_fun_def
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1106
             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
  1107
  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
  1108
    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
  1109
  { 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
  1110
      by (auto split: split_if_asm) }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1111
  note not_MInf = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1112
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 51340
diff changeset
  1113
  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
  1114
  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
  1115
    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
  1116
      using u v  `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 43339
diff changeset
  1117
      by (auto intro!: add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1118
    { fix x
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1119
      { 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
  1120
          by auto }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1121
      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
  1122
        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
  1123
        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
  1124
           (auto intro!: SUP_ereal_add
43920
cedb5cb948fd Rename extreal => ereal
hoelzl</