src/HOL/Probability/Probability_Measure.thy
author hoelzl
Thu, 26 May 2011 17:40:01 +0200
changeset 42988 d8f3fc934ff6
parent 42981 fe7f5a26e4c6
child 42991 3fa22920bf86
permissions -rw-r--r--
add lemma indep_distribution_eq_measure
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Probability/Probability_Measure.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Armin Heller, TU München
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*)
66c8281349ec standardized headers
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header {*Probability measure*}
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theory Probability_Measure
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e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
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imports Lebesgue_Integration Radon_Nikodym Finite_Product_Measure Lebesgue_Measure
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begin
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    11
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    12
locale prob_space = measure_space +
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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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    13
  assumes measure_space_1: "measure M (space M) = 1"
38656
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    14
d5d342611edb Rewrite the Probability theory.
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    15
sublocale prob_space < finite_measure
d5d342611edb Rewrite the Probability theory.
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    16
proof
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    17
  from measure_space_1 show "\<mu> (space M) \<noteq> \<infinity>" by simp
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    18
qed
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    19
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
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    20
abbreviation (in prob_space) "events \<equiv> sets M"
41981
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abbreviation (in prob_space) "prob \<equiv> \<mu>'"
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    22
abbreviation (in prob_space) "random_variable M' X \<equiv> sigma_algebra M' \<and> X \<in> measurable M M'"
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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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abbreviation (in prob_space) "expectation \<equiv> integral\<^isup>L M"
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definition (in prob_space)
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  "distribution X A = \<mu>' (X -` A \<inter> space M)"
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abbreviation (in prob_space)
36624
25153c08655e Cleanup information theory
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  "joint_distribution X Y \<equiv> distribution (\<lambda>x. (X x, Y x))"
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41981
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declare (in finite_measure) positive_measure'[intro, simp]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    32
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943c7b348524 Moved lemmas to appropriate locations
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lemma (in prob_space) distribution_cong:
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    34
  assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = Y x"
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    35
  shows "distribution X = distribution Y"
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d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
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    36
  unfolding distribution_def fun_eq_iff
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parents: 41831
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    37
  using assms by (auto intro!: arg_cong[where f="\<mu>'"])
39097
943c7b348524 Moved lemmas to appropriate locations
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    38
943c7b348524 Moved lemmas to appropriate locations
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    39
lemma (in prob_space) joint_distribution_cong:
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    40
  assumes "\<And>x. x \<in> space M \<Longrightarrow> X x = X' x"
943c7b348524 Moved lemmas to appropriate locations
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    41
  assumes "\<And>x. x \<in> space M \<Longrightarrow> Y x = Y' x"
943c7b348524 Moved lemmas to appropriate locations
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parents: 39096
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    42
  shows "joint_distribution X Y = joint_distribution X' Y'"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
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    43
  unfolding distribution_def fun_eq_iff
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    44
  using assms by (auto intro!: arg_cong[where f="\<mu>'"])
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
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    45
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
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lemma (in prob_space) distribution_id[simp]:
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    47
  "N \<in> events \<Longrightarrow> distribution (\<lambda>x. x) N = prob N"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    48
  by (auto simp: distribution_def intro!: arg_cong[where f=prob])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
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parents: 39302
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    49
de0b30e6c2d2 Support product spaces on sigma finite measures.
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parents: 39302
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    50
lemma (in prob_space) prob_space: "prob (space M) = 1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    51
  using measure_space_1 unfolding \<mu>'_def by (simp add: one_extreal_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    52
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    53
lemma (in prob_space) prob_le_1[simp, intro]: "prob A \<le> 1"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    54
  using bounded_measure[of A] by (simp add: prob_space)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    55
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    56
lemma (in prob_space) distribution_positive[simp, intro]:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    57
  "0 \<le> distribution X A" unfolding distribution_def by auto
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    58
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    59
lemma (in prob_space) joint_distribution_remove[simp]:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    60
    "joint_distribution X X {(x, x)} = distribution X {x}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    61
  unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    62
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42902
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    63
lemma (in prob_space) measure_le_1: "X \<in> sets M \<Longrightarrow> \<mu> X \<le> 1"
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42902
diff changeset
    64
  unfolding measure_space_1[symmetric]
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42902
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    65
  using sets_into_space
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 42902
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    66
  by (intro measure_mono) auto
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
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    67
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    68
lemma (in prob_space) distribution_1:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    69
  "distribution X A \<le> 1"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    70
  unfolding distribution_def by simp
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parents:
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    71
40859
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    72
lemma (in prob_space) prob_compl:
41981
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    73
  assumes A: "A \<in> events"
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    74
  shows "prob (space M - A) = 1 - prob A"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
    75
  using finite_measure_compl[OF A] by (simp add: prob_space)
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parents:
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    76
40859
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    77
lemma (in prob_space) prob_space_increasing: "increasing M prob"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    78
  by (auto intro!: finite_measure_mono simp: increasing_def)
35582
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parents:
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    79
40859
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    80
lemma (in prob_space) prob_zero_union:
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    81
  assumes "s \<in> events" "t \<in> events" "prob t = 0"
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parents:
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    82
  shows "prob (s \<union> t) = prob s"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
    83
using assms
35582
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hoelzl
parents:
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    84
proof -
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
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    85
  have "prob (s \<union> t) \<le> prob s"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
    86
    using finite_measure_subadditive[of s t] assms by auto
35582
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hoelzl
parents:
diff changeset
    87
  moreover have "prob (s \<union> t) \<ge> prob s"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
    88
    using assms by (blast intro: finite_measure_mono)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
    89
  ultimately show ?thesis by simp
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
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    90
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
    91
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
    92
lemma (in prob_space) prob_eq_compl:
35582
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hoelzl
parents:
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    93
  assumes "s \<in> events" "t \<in> events"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
    94
  assumes "prob (space M - s) = prob (space M - t)"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
    95
  shows "prob s = prob t"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
    96
  using assms prob_compl by auto
35582
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hoelzl
parents:
diff changeset
    97
40859
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hoelzl
parents: 39302
diff changeset
    98
lemma (in prob_space) prob_one_inter:
35582
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parents:
diff changeset
    99
  assumes events:"s \<in> events" "t \<in> events"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   100
  assumes "prob t = 1"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   101
  shows "prob (s \<inter> t) = prob s"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   102
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   103
  have "prob ((space M - s) \<union> (space M - t)) = prob (space M - s)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   104
    using events assms  prob_compl[of "t"] by (auto intro!: prob_zero_union)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   105
  also have "(space M - s) \<union> (space M - t) = space M - (s \<inter> t)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   106
    by blast
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   107
  finally show "prob (s \<inter> t) = prob s"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   108
    using events by (auto intro!: prob_eq_compl[of "s \<inter> t" s])
35582
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hoelzl
parents:
diff changeset
   109
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   110
40859
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hoelzl
parents: 39302
diff changeset
   111
lemma (in prob_space) prob_eq_bigunion_image:
35582
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hoelzl
parents:
diff changeset
   112
  assumes "range f \<subseteq> events" "range g \<subseteq> events"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   113
  assumes "disjoint_family f" "disjoint_family g"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   114
  assumes "\<And> n :: nat. prob (f n) = prob (g n)"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   115
  shows "(prob (\<Union> i. f i) = prob (\<Union> i. g i))"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   116
using assms
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   117
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   118
  have a: "(\<lambda> i. prob (f i)) sums (prob (\<Union> i. f i))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   119
    by (rule finite_measure_UNION[OF assms(1,3)])
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   120
  have b: "(\<lambda> i. prob (g i)) sums (prob (\<Union> i. g i))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   121
    by (rule finite_measure_UNION[OF assms(2,4)])
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   122
  show ?thesis using sums_unique[OF b] sums_unique[OF a] assms by simp
35582
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hoelzl
parents:
diff changeset
   123
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   124
40859
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hoelzl
parents: 39302
diff changeset
   125
lemma (in prob_space) prob_countably_zero:
35582
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hoelzl
parents:
diff changeset
   126
  assumes "range c \<subseteq> events"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   127
  assumes "\<And> i. prob (c i) = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   128
  shows "prob (\<Union> i :: nat. c i) = 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   129
proof (rule antisym)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   130
  show "prob (\<Union> i :: nat. c i) \<le> 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   131
    using finite_measure_countably_subadditive[OF assms(1)]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   132
    by (simp add: assms(2) suminf_zero summable_zero)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   133
qed simp
35582
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hoelzl
parents:
diff changeset
   134
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   135
lemma (in prob_space) prob_equiprobable_finite_unions:
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   136
  assumes "s \<in> events"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   137
  assumes s_finite: "finite s" "\<And>x. x \<in> s \<Longrightarrow> {x} \<in> events"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   138
  assumes "\<And> x y. \<lbrakk>x \<in> s; y \<in> s\<rbrakk> \<Longrightarrow> (prob {x} = prob {y})"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   139
  shows "prob s = real (card s) * prob {SOME x. x \<in> s}"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   140
proof (cases "s = {}")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   141
  case False hence "\<exists> x. x \<in> s" by blast
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   142
  from someI_ex[OF this] assms
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   143
  have prob_some: "\<And> x. x \<in> s \<Longrightarrow> prob {x} = prob {SOME y. y \<in> s}" by blast
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   144
  have "prob s = (\<Sum> x \<in> s. prob {x})"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   145
    using finite_measure_finite_singleton[OF s_finite] by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   146
  also have "\<dots> = (\<Sum> x \<in> s. prob {SOME y. y \<in> s})" using prob_some by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   147
  also have "\<dots> = real (card s) * prob {(SOME x. x \<in> s)}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   148
    using setsum_constant assms by (simp add: real_eq_of_nat)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   149
  finally show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   150
qed simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   151
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   152
lemma (in prob_space) prob_real_sum_image_fn:
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   153
  assumes "e \<in> events"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   154
  assumes "\<And> x. x \<in> s \<Longrightarrow> e \<inter> f x \<in> events"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   155
  assumes "finite s"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   156
  assumes disjoint: "\<And> x y. \<lbrakk>x \<in> s ; y \<in> s ; x \<noteq> y\<rbrakk> \<Longrightarrow> f x \<inter> f y = {}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   157
  assumes upper: "space M \<subseteq> (\<Union> i \<in> s. f i)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   158
  shows "prob e = (\<Sum> x \<in> s. prob (e \<inter> f x))"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   159
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   160
  have e: "e = (\<Union> i \<in> s. e \<inter> f i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   161
    using `e \<in> events` sets_into_space upper by blast
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   162
  hence "prob e = prob (\<Union> i \<in> s. e \<inter> f i)" by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   163
  also have "\<dots> = (\<Sum> x \<in> s. prob (e \<inter> f x))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   164
  proof (rule finite_measure_finite_Union)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   165
    show "finite s" by fact
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   166
    show "\<And>i. i \<in> s \<Longrightarrow> e \<inter> f i \<in> events" by fact
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   167
    show "disjoint_family_on (\<lambda>i. e \<inter> f i) s"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   168
      using disjoint by (auto simp: disjoint_family_on_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   169
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   170
  finally show ?thesis .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   171
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   172
42199
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   173
lemma (in prob_space) prob_space_vimage:
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   174
  assumes S: "sigma_algebra S"
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   175
  assumes T: "T \<in> measure_preserving M S"
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   176
  shows "prob_space S"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   177
proof -
42199
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   178
  interpret S: measure_space S
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   179
    using S and T by (rule measure_space_vimage)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   180
  show ?thesis
42199
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   181
  proof
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   182
    from T[THEN measure_preservingD2]
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   183
    have "T -` space S \<inter> space M = space M"
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   184
      by (auto simp: measurable_def)
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   185
    with T[THEN measure_preservingD, of "space S", symmetric]
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   186
    show  "measure S (space S) = 1"
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   187
      using measure_space_1 by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   188
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   189
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   190
42988
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   191
lemma prob_space_unique_Int_stable:
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   192
  fixes E :: "('a, 'b) algebra_scheme" and A :: "nat \<Rightarrow> 'a set"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   193
  assumes E: "Int_stable E" "space E \<in> sets E"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   194
  and M: "prob_space M" "space M = space E" "sets M = sets (sigma E)"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   195
  and N: "prob_space N" "space N = space E" "sets N = sets (sigma E)"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   196
  and eq: "\<And>X. X \<in> sets E \<Longrightarrow> finite_measure.\<mu>' M X = finite_measure.\<mu>' N X"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   197
  assumes "X \<in> sets (sigma E)"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   198
  shows "finite_measure.\<mu>' M X = finite_measure.\<mu>' N X"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   199
proof -
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   200
  interpret M!: prob_space M by fact
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   201
  interpret N!: prob_space N by fact
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   202
  have "measure M X = measure N X"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   203
  proof (rule measure_unique_Int_stable[OF `Int_stable E`])
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   204
    show "range (\<lambda>i. space M) \<subseteq> sets E" "incseq (\<lambda>i. space M)" "(\<Union>i. space M) = space E"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   205
      using E M N by auto
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   206
    show "\<And>i. M.\<mu> (space M) \<noteq> \<infinity>"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   207
      using M.measure_space_1 by simp
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   208
    show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = M.\<mu>\<rparr>"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   209
      using E M N by (auto intro!: M.measure_space_cong)
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   210
    show "measure_space \<lparr>space = space E, sets = sets (sigma E), measure_space.measure = N.\<mu>\<rparr>"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   211
      using E M N by (auto intro!: N.measure_space_cong)
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   212
    { fix X assume "X \<in> sets E"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   213
      then have "X \<in> sets (sigma E)"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   214
        by (auto simp: sets_sigma sigma_sets.Basic)
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   215
      with eq[OF `X \<in> sets E`] M N show "M.\<mu> X = N.\<mu> X"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   216
        by (simp add: M.finite_measure_eq N.finite_measure_eq) }
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   217
  qed fact
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   218
  with `X \<in> sets (sigma E)` M N show ?thesis
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   219
    by (simp add: M.finite_measure_eq N.finite_measure_eq)
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   220
qed
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   221
42199
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   222
lemma (in prob_space) distribution_prob_space:
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   223
  assumes X: "random_variable S X"
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   224
  shows "prob_space (S\<lparr>measure := extreal \<circ> distribution X\<rparr>)" (is "prob_space ?S")
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   225
proof (rule prob_space_vimage)
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   226
  show "X \<in> measure_preserving M ?S"
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   227
    using X
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   228
    unfolding measure_preserving_def distribution_def_raw
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   229
    by (auto simp: finite_measure_eq measurable_sets)
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   230
  show "sigma_algebra ?S" using X by simp
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   231
qed
aded34119213 add prob_space_vimage
hoelzl
parents: 42148
diff changeset
   232
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   233
lemma (in prob_space) AE_distribution:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   234
  assumes X: "random_variable MX X" and "AE x in MX\<lparr>measure := extreal \<circ> distribution X\<rparr>. Q x"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   235
  shows "AE x. Q (X x)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   236
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   237
  interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>" using X by (rule distribution_prob_space)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   238
  obtain N where N: "N \<in> sets MX" "distribution X N = 0" "{x\<in>space MX. \<not> Q x} \<subseteq> N"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   239
    using assms unfolding X.almost_everywhere_def by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   240
  from X[unfolded measurable_def] N show "AE x. Q (X x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   241
    by (intro AE_I'[where N="X -` N \<inter> space M"])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   242
       (auto simp: finite_measure_eq distribution_def measurable_sets)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   243
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   244
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   245
lemma (in prob_space) distribution_eq_integral:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   246
  "random_variable S X \<Longrightarrow> A \<in> sets S \<Longrightarrow> distribution X A = expectation (indicator (X -` A \<inter> space M))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   247
  using finite_measure_eq[of "X -` A \<inter> space M"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   248
  by (auto simp: measurable_sets distribution_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   249
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   250
lemma (in prob_space) distribution_eq_translated_integral:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   251
  assumes "random_variable S X" "A \<in> sets S"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   252
  shows "distribution X A = integral\<^isup>P (S\<lparr>measure := extreal \<circ> distribution X\<rparr>) (indicator A)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   253
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   254
  interpret S: prob_space "S\<lparr>measure := extreal \<circ> distribution X\<rparr>"
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
   255
    using assms(1) by (rule distribution_prob_space)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   256
  show ?thesis
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   257
    using S.positive_integral_indicator(1)[of A] assms by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   258
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   259
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   260
lemma (in prob_space) finite_expectation1:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   261
  assumes f: "finite (X`space M)" and rv: "random_variable borel X"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   262
  shows "expectation X = (\<Sum>r \<in> X ` space M. r * prob (X -` {r} \<inter> space M))" (is "_ = ?r")
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   263
proof (subst integral_on_finite)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   264
  show "X \<in> borel_measurable M" "finite (X`space M)" using assms by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   265
  show "(\<Sum> r \<in> X ` space M. r * real (\<mu> (X -` {r} \<inter> space M))) = ?r"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   266
    "\<And>x. \<mu> (X -` {x} \<inter> space M) \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   267
    using finite_measure_eq[OF borel_measurable_vimage, of X] rv by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   268
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   269
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   270
lemma (in prob_space) finite_expectation:
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
   271
  assumes "finite (X`space M)" "random_variable borel X"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   272
  shows "expectation X = (\<Sum> r \<in> X ` (space M). r * distribution X {r})"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   273
  using assms unfolding distribution_def using finite_expectation1 by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   274
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   275
lemma (in prob_space) prob_x_eq_1_imp_prob_y_eq_0:
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   276
  assumes "{x} \<in> events"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   277
  assumes "prob {x} = 1"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   278
  assumes "{y} \<in> events"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   279
  assumes "y \<noteq> x"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   280
  shows "prob {y} = 0"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   281
  using prob_one_inter[of "{y}" "{x}"] assms by auto
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   282
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   283
lemma (in prob_space) distribution_empty[simp]: "distribution X {} = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   284
  unfolding distribution_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   285
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   286
lemma (in prob_space) distribution_space[simp]: "distribution X (X ` space M) = 1"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   287
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   288
  have "X -` X ` space M \<inter> space M = space M" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   289
  thus ?thesis unfolding distribution_def by (simp add: prob_space)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   290
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   291
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   292
lemma (in prob_space) distribution_one:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   293
  assumes "random_variable M' X" and "A \<in> sets M'"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   294
  shows "distribution X A \<le> 1"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   295
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   296
  have "distribution X A \<le> \<mu>' (space M)" unfolding distribution_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   297
    using assms[unfolded measurable_def] by (auto intro!: finite_measure_mono)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   298
  thus ?thesis by (simp add: prob_space)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   299
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   300
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   301
lemma (in prob_space) distribution_x_eq_1_imp_distribution_y_eq_0:
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   302
  assumes X: "random_variable \<lparr>space = X ` (space M), sets = Pow (X ` (space M))\<rparr> X"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   303
    (is "random_variable ?S X")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   304
  assumes "distribution X {x} = 1"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   305
  assumes "y \<noteq> x"
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   306
  shows "distribution X {y} = 0"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   307
proof cases
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
   308
  { fix x have "X -` {x} \<inter> space M \<in> sets 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
   309
    proof cases
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
   310
      assume "x \<in> X`space M" with X show ?thesis
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
   311
        by (auto simp: measurable_def image_iff)
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
   312
    next
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
   313
      assume "x \<notin> X`space M" then have "X -` {x} \<inter> space M = {}" by auto
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
   314
      then show ?thesis by auto
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
   315
    qed } note single = this
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
   316
  have "X -` {x} \<inter> space M - X -` {y} \<inter> space M = X -` {x} \<inter> 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
   317
    "X -` {y} \<inter> space M \<inter> (X -` {x} \<inter> 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
   318
    using `y \<noteq> x` by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   319
  with finite_measure_inter_full_set[OF single single, of x y] assms(2)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   320
  show ?thesis by (auto simp: distribution_def prob_space)
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
   321
next
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
   322
  assume "{y} \<notin> sets ?S"
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
   323
  then have "X -` {y} \<inter> space M = {}" by auto
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
   324
  thus "distribution X {y} = 0" unfolding distribution_def by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   325
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   326
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   327
lemma (in prob_space) joint_distribution_Times_le_fst:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   328
  assumes X: "random_variable MX X" and Y: "random_variable MY Y"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   329
    and A: "A \<in> sets MX" and B: "B \<in> sets MY"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   330
  shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   331
  unfolding distribution_def
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   332
proof (intro finite_measure_mono)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   333
  show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   334
  show "X -` A \<inter> space M \<in> events"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   335
    using X A unfolding measurable_def by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   336
  have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   337
    (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   338
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   339
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   340
lemma (in prob_space) joint_distribution_commute:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   341
  "joint_distribution X Y x = joint_distribution Y X ((\<lambda>(x,y). (y,x))`x)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   342
  unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   343
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   344
lemma (in prob_space) joint_distribution_Times_le_snd:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   345
  assumes X: "random_variable MX X" and Y: "random_variable MY Y"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   346
    and A: "A \<in> sets MX" and B: "B \<in> sets MY"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   347
  shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   348
  using assms
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   349
  by (subst joint_distribution_commute)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   350
     (simp add: swap_product joint_distribution_Times_le_fst)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   351
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   352
lemma (in prob_space) random_variable_pairI:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   353
  assumes "random_variable MX X"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   354
  assumes "random_variable MY Y"
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
   355
  shows "random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   356
proof
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   357
  interpret MX: sigma_algebra MX using assms by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   358
  interpret MY: sigma_algebra MY using assms by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   359
  interpret P: pair_sigma_algebra MX MY by default
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
   360
  show "sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   361
  have sa: "sigma_algebra M" by default
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
   362
  show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
41095
c335d880ff82 cleanup bijectivity btw. product spaces and pairs
hoelzl
parents: 41023
diff changeset
   363
    unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   364
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   365
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   366
lemma (in prob_space) joint_distribution_commute_singleton:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   367
  "joint_distribution X Y {(x, y)} = joint_distribution Y X {(y, x)}"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   368
  unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   369
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   370
lemma (in prob_space) joint_distribution_assoc_singleton:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   371
  "joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)} =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   372
   joint_distribution (\<lambda>x. (X x, Y x)) Z {((x, y), z)}"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   373
  unfolding distribution_def by (auto intro!: arg_cong[where f=\<mu>'])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   374
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
   375
locale pair_prob_space = M1: prob_space M1 + M2: prob_space M2 for M1 M2
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   376
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
   377
sublocale pair_prob_space \<subseteq> pair_sigma_finite M1 M2 by default
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
   378
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
   379
sublocale pair_prob_space \<subseteq> P: prob_space P
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
   380
by default (simp add: pair_measure_times M1.measure_space_1 M2.measure_space_1 space_pair_measure)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   381
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   382
lemma countably_additiveI[case_names countably]:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   383
  assumes "\<And>A. \<lbrakk> range A \<subseteq> sets M ; disjoint_family A ; (\<Union>i. A i) \<in> sets M\<rbrakk> \<Longrightarrow>
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   384
    (\<Sum>n. \<mu> (A n)) = \<mu> (\<Union>i. A i)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   385
  shows "countably_additive M \<mu>"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   386
  using assms unfolding countably_additive_def by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   387
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   388
lemma (in prob_space) joint_distribution_prob_space:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   389
  assumes "random_variable MX X" "random_variable MY Y"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   390
  shows "prob_space ((MX \<Otimes>\<^isub>M MY) \<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)"
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
   391
  using random_variable_pairI[OF assms] by (rule distribution_prob_space)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   392
42988
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   393
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   394
locale finite_product_prob_space =
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   395
  fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   396
    and I :: "'i set"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   397
  assumes prob_space: "\<And>i. prob_space (M i)" and finite_index: "finite I"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   398
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   399
sublocale finite_product_prob_space \<subseteq> M: prob_space "M i" for i
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   400
  by (rule prob_space)
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   401
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   402
sublocale finite_product_prob_space \<subseteq> finite_product_sigma_finite M I
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   403
  by default (rule finite_index)
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   404
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   405
sublocale finite_product_prob_space \<subseteq> prob_space "\<Pi>\<^isub>M i\<in>I. M i"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   406
  proof qed (simp add: measure_times M.measure_space_1 setprod_1)
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   407
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   408
lemma (in finite_product_prob_space) prob_times:
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   409
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets (M i)"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   410
  shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   411
proof -
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   412
  have "extreal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   413
    using X by (intro finite_measure_eq[symmetric] in_P) auto
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   414
  also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   415
    using measure_times X by simp
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   416
  also have "\<dots> = extreal (\<Prod>i\<in>I. M.\<mu>' i (X i))"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   417
    using X by (simp add: M.finite_measure_eq setprod_extreal)
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   418
  finally show ?thesis by simp
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   419
qed
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   420
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   421
lemma (in prob_space) random_variable_restrict:
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   422
  assumes I: "finite I"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   423
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> random_variable (N i) (X i)"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   424
  shows "random_variable (Pi\<^isub>M I N) (\<lambda>x. \<lambda>i\<in>I. X i x)"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   425
proof
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   426
  { fix i assume "i \<in> I"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   427
    with X interpret N: sigma_algebra "N i" by simp
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   428
    have "sets (N i) \<subseteq> Pow (space (N i))" by (rule N.space_closed) }
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   429
  note N_closed = this
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   430
  then show "sigma_algebra (Pi\<^isub>M I N)"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   431
    by (simp add: product_algebra_def)
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   432
       (intro sigma_algebra_sigma product_algebra_generator_sets_into_space)
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   433
  show "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable M (Pi\<^isub>M I N)"
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   434
    using X by (intro measurable_restrict[OF I N_closed]) auto
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   435
qed
d8f3fc934ff6 add lemma indep_distribution_eq_measure
hoelzl
parents: 42981
diff changeset
   436
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   437
section "Probability spaces on finite sets"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   438
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35929
diff changeset
   439
locale finite_prob_space = prob_space + finite_measure_space
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35929
diff changeset
   440
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   441
abbreviation (in prob_space) "finite_random_variable M' X \<equiv> finite_sigma_algebra M' \<and> X \<in> measurable M M'"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   442
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   443
lemma (in prob_space) finite_random_variableD:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   444
  assumes "finite_random_variable M' X" shows "random_variable M' X"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   445
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   446
  interpret M': finite_sigma_algebra M' using assms by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   447
  then show "random_variable M' X" using assms by simp default
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   448
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   449
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   450
lemma (in prob_space) distribution_finite_prob_space:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   451
  assumes "finite_random_variable MX X"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   452
  shows "finite_prob_space (MX\<lparr>measure := extreal \<circ> distribution X\<rparr>)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   453
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   454
  interpret X: prob_space "MX\<lparr>measure := extreal \<circ> distribution X\<rparr>"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   455
    using assms[THEN finite_random_variableD] by (rule distribution_prob_space)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   456
  interpret MX: finite_sigma_algebra MX
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
   457
    using assms by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   458
  show ?thesis by default (simp_all add: MX.finite_space)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   459
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   460
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   461
lemma (in prob_space) simple_function_imp_finite_random_variable[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   462
  assumes "simple_function M X"
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
   463
  shows "finite_random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = x \<rparr> X"
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
   464
    (is "finite_random_variable ?X _")
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   465
proof (intro conjI)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   466
  have [simp]: "finite (X ` space M)" using assms unfolding simple_function_def by 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
   467
  interpret X: sigma_algebra ?X by (rule sigma_algebra_Pow)
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
  show "finite_sigma_algebra ?X"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   469
    by default auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   470
  show "X \<in> measurable M ?X"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   471
  proof (unfold measurable_def, clarsimp)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   472
    fix A assume A: "A \<subseteq> X`space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   473
    then have "finite A" by (rule finite_subset) simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   474
    then have "X -` (\<Union>a\<in>A. {a}) \<inter> space M \<in> events"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   475
      unfolding vimage_UN UN_extend_simps
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   476
      apply (rule finite_UN)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   477
      using A assms unfolding simple_function_def by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   478
    then show "X -` A \<inter> space M \<in> events" by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   479
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   480
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   481
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   482
lemma (in prob_space) simple_function_imp_random_variable[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   483
  assumes "simple_function M X"
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
  shows "random_variable \<lparr> space = X`space M, sets = Pow (X`space M), \<dots> = ext \<rparr> X"
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
   485
  using simple_function_imp_finite_random_variable[OF assms, of ext]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   486
  by (auto dest!: finite_random_variableD)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   487
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   488
lemma (in prob_space) sum_over_space_real_distribution:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   489
  "simple_function M X \<Longrightarrow> (\<Sum>x\<in>X`space M. distribution X {x}) = 1"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   490
  unfolding distribution_def prob_space[symmetric]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   491
  by (subst finite_measure_finite_Union[symmetric])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   492
     (auto simp add: disjoint_family_on_def simple_function_def
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   493
           intro!: arg_cong[where f=prob])
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   494
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   495
lemma (in prob_space) finite_random_variable_pairI:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   496
  assumes "finite_random_variable MX X"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   497
  assumes "finite_random_variable MY Y"
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
   498
  shows "finite_random_variable (MX \<Otimes>\<^isub>M MY) (\<lambda>x. (X x, Y x))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   499
proof
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   500
  interpret MX: finite_sigma_algebra MX using assms by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   501
  interpret MY: finite_sigma_algebra MY using assms by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   502
  interpret P: pair_finite_sigma_algebra MX MY by default
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
   503
  show "finite_sigma_algebra (MX \<Otimes>\<^isub>M MY)" by default
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   504
  have sa: "sigma_algebra M" by default
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
   505
  show "(\<lambda>x. (X x, Y x)) \<in> measurable M (MX \<Otimes>\<^isub>M MY)"
41095
c335d880ff82 cleanup bijectivity btw. product spaces and pairs
hoelzl
parents: 41023
diff changeset
   506
    unfolding P.measurable_pair_iff[OF sa] using assms by (simp add: comp_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   507
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   508
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   509
lemma (in prob_space) finite_random_variable_imp_sets:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   510
  "finite_random_variable MX X \<Longrightarrow> x \<in> space MX \<Longrightarrow> {x} \<in> sets MX"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   511
  unfolding finite_sigma_algebra_def finite_sigma_algebra_axioms_def by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   512
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   513
lemma (in prob_space) finite_random_variable_measurable:
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   514
  assumes X: "finite_random_variable MX X" shows "X -` A \<inter> space M \<in> events"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   515
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   516
  interpret X: finite_sigma_algebra MX using X by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   517
  from X have vimage: "\<And>A. A \<subseteq> space MX \<Longrightarrow> X -` A \<inter> space M \<in> events" and
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   518
    "X \<in> space M \<rightarrow> space MX"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   519
    by (auto simp: measurable_def)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   520
  then have *: "X -` A \<inter> space M = X -` (A \<inter> space MX) \<inter> space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   521
    by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   522
  show "X -` A \<inter> space M \<in> events"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   523
    unfolding * by (intro vimage) auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   524
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   525
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   526
lemma (in prob_space) joint_distribution_finite_Times_le_fst:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   527
  assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   528
  shows "joint_distribution X Y (A \<times> B) \<le> distribution X A"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   529
  unfolding distribution_def
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   530
proof (intro finite_measure_mono)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   531
  show "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M \<subseteq> X -` A \<inter> space M" by force
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   532
  show "X -` A \<inter> space M \<in> events"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   533
    using finite_random_variable_measurable[OF X] .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   534
  have *: "(\<lambda>x. (X x, Y x)) -` (A \<times> B) \<inter> space M =
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   535
    (X -` A \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   536
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   537
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   538
lemma (in prob_space) joint_distribution_finite_Times_le_snd:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   539
  assumes X: "finite_random_variable MX X" and Y: "finite_random_variable MY Y"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   540
  shows "joint_distribution X Y (A \<times> B) \<le> distribution Y B"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   541
  using assms
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   542
  by (subst joint_distribution_commute)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   543
     (simp add: swap_product joint_distribution_finite_Times_le_fst)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   544
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   545
lemma (in prob_space) finite_distribution_order:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   546
  fixes MX :: "('c, 'd) measure_space_scheme" and MY :: "('e, 'f) measure_space_scheme"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   547
  assumes "finite_random_variable MX X" "finite_random_variable MY Y"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   548
  shows "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   549
    and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   550
    and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   551
    and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   552
    and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   553
    and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   554
  using joint_distribution_finite_Times_le_snd[OF assms, of "{x}" "{y}"]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   555
  using joint_distribution_finite_Times_le_fst[OF assms, of "{x}" "{y}"]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   556
  by (auto intro: antisym)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   557
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   558
lemma (in prob_space) setsum_joint_distribution:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   559
  assumes X: "finite_random_variable MX X"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   560
  assumes Y: "random_variable MY Y" "B \<in> sets MY"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   561
  shows "(\<Sum>a\<in>space MX. joint_distribution X Y ({a} \<times> B)) = distribution Y B"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   562
  unfolding distribution_def
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   563
proof (subst finite_measure_finite_Union[symmetric])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   564
  interpret MX: finite_sigma_algebra MX using X by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   565
  show "finite (space MX)" using MX.finite_space .
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   566
  let "?d i" = "(\<lambda>x. (X x, Y x)) -` ({i} \<times> B) \<inter> space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   567
  { fix i assume "i \<in> space MX"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   568
    moreover have "?d i = (X -` {i} \<inter> space M) \<inter> (Y -` B \<inter> space M)" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   569
    ultimately show "?d i \<in> events"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   570
      using measurable_sets[of X M MX] measurable_sets[of Y M MY B] X Y
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   571
      using MX.sets_eq_Pow by auto }
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   572
  show "disjoint_family_on ?d (space MX)" by (auto simp: disjoint_family_on_def)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   573
  show "\<mu>' (\<Union>i\<in>space MX. ?d i) = \<mu>' (Y -` B \<inter> space M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   574
    using X[unfolded measurable_def] by (auto intro!: arg_cong[where f=\<mu>'])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   575
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   576
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   577
lemma (in prob_space) setsum_joint_distribution_singleton:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   578
  assumes X: "finite_random_variable MX X"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   579
  assumes Y: "finite_random_variable MY Y" "b \<in> space MY"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   580
  shows "(\<Sum>a\<in>space MX. joint_distribution X Y {(a, b)}) = distribution Y {b}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   581
  using setsum_joint_distribution[OF X
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   582
    finite_random_variableD[OF Y(1)]
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   583
    finite_random_variable_imp_sets[OF Y]] by simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   584
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
   585
locale pair_finite_prob_space = M1: finite_prob_space M1 + M2: finite_prob_space M2 for M1 M2
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   586
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
   587
sublocale pair_finite_prob_space \<subseteq> pair_prob_space M1 M2 by default
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
   588
sublocale pair_finite_prob_space \<subseteq> pair_finite_space M1 M2  by default
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
   589
sublocale pair_finite_prob_space \<subseteq> finite_prob_space P by default
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   590
42859
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   591
locale product_finite_prob_space =
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   592
  fixes M :: "'i \<Rightarrow> ('a,'b) measure_space_scheme"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   593
    and I :: "'i set"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   594
  assumes finite_space: "\<And>i. finite_prob_space (M i)" and finite_index: "finite I"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   595
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   596
sublocale product_finite_prob_space \<subseteq> M!: finite_prob_space "M i" using finite_space .
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   597
sublocale product_finite_prob_space \<subseteq> finite_product_sigma_finite M I by default (rule finite_index)
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   598
sublocale product_finite_prob_space \<subseteq> prob_space "Pi\<^isub>M I M"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   599
proof
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   600
  show "\<mu> (space P) = 1"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   601
    using measure_times[OF M.top] M.measure_space_1
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   602
    by (simp add: setprod_1 space_product_algebra)
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   603
qed
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   604
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   605
lemma funset_eq_UN_fun_upd_I:
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   606
  assumes *: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   607
  and **: "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a:=d))"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   608
  and ***: "\<And>f x. \<lbrakk> f \<in> F A ; x \<in> G f \<rbrakk> \<Longrightarrow> f(a:=x) \<in> F (insert a A)"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   609
  shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` (G f))"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   610
proof safe
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   611
  fix f assume f: "f \<in> F (insert a A)"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   612
  show "f \<in> (\<Union>f\<in>F A. fun_upd f a ` G f)"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   613
  proof (rule UN_I[of "f(a := d)"])
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   614
    show "f(a := d) \<in> F A" using *[OF f] .
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   615
    show "f \<in> fun_upd (f(a:=d)) a ` G (f(a:=d))"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   616
    proof (rule image_eqI[of _ _ "f a"])
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   617
      show "f a \<in> G (f(a := d))" using **[OF f] .
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   618
    qed simp
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   619
  qed
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   620
next
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   621
  fix f x assume "f \<in> F A" "x \<in> G f"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   622
  from ***[OF this] show "f(a := x) \<in> F (insert a A)" .
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   623
qed
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   624
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   625
lemma extensional_funcset_insert_eq[simp]:
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   626
  assumes "a \<notin> A"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   627
  shows "extensional (insert a A) \<inter> (insert a A \<rightarrow> B) = (\<Union>f \<in> extensional A \<inter> (A \<rightarrow> B). (\<lambda>b. f(a := b)) ` B)"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   628
  apply (rule funset_eq_UN_fun_upd_I)
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   629
  using assms
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   630
  by (auto intro!: inj_onI dest: inj_onD split: split_if_asm simp: extensional_def)
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   631
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   632
lemma finite_extensional_funcset[simp, intro]:
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   633
  assumes "finite A" "finite B"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   634
  shows "finite (extensional A \<inter> (A \<rightarrow> B))"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   635
  using assms by induct (auto simp: extensional_funcset_insert_eq)
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   636
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   637
lemma finite_PiE[simp, intro]:
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   638
  assumes fin: "finite A" "\<And>i. i \<in> A \<Longrightarrow> finite (B i)"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   639
  shows "finite (Pi\<^isub>E A B)"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   640
proof -
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   641
  have *: "(Pi\<^isub>E A B) \<subseteq> extensional A \<inter> (A \<rightarrow> (\<Union>i\<in>A. B i))" by auto
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   642
  show ?thesis
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   643
    using fin by (intro finite_subset[OF *] finite_extensional_funcset) auto
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   644
qed
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   645
42892
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   646
lemma (in product_finite_prob_space) singleton_eq_product:
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   647
  assumes x: "x \<in> space P" shows "{x} = (\<Pi>\<^isub>E i\<in>I. {x i})"
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   648
proof (safe intro!: ext[of _ x])
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   649
  fix y i assume *: "y \<in> (\<Pi> i\<in>I. {x i})" "y \<in> extensional I"
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   650
  with x show "y i = x i"
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   651
    by (cases "i \<in> I") (auto simp: extensional_def)
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   652
qed (insert x, auto)
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   653
42859
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   654
sublocale product_finite_prob_space \<subseteq> finite_prob_space "Pi\<^isub>M I M"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   655
proof
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   656
  show "finite (space P)"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   657
    using finite_index M.finite_space by auto
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   658
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   659
  { fix x assume "x \<in> space P"
42892
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   660
    with this[THEN singleton_eq_product]
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   661
    have "{x} \<in> sets P"
42859
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   662
      by (auto intro!: in_P) }
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   663
  note x_in_P = this
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   664
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   665
  have "Pow (space P) \<subseteq> sets P"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   666
  proof
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   667
    fix X assume "X \<in> Pow (space P)"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   668
    moreover then have "finite X"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   669
      using `finite (space P)` by (blast intro: finite_subset)
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   670
    ultimately have "(\<Union>x\<in>X. {x}) \<in> sets P"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   671
      by (intro finite_UN x_in_P) auto
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   672
    then show "X \<in> sets P" by simp
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   673
  qed
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   674
  with space_closed show [simp]: "sets P = Pow (space P)" ..
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   675
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   676
  { fix x assume "x \<in> space P"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   677
    from this top have "\<mu> {x} \<le> \<mu> (space P)" by (intro measure_mono) auto
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   678
    then show "\<mu> {x} \<noteq> \<infinity>"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   679
      using measure_space_1 by auto }
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   680
qed
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   681
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   682
lemma (in product_finite_prob_space) measure_finite_times:
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   683
  "(\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)) \<Longrightarrow> \<mu> (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.\<mu> i (X i))"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   684
  by (rule measure_times) simp
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   685
42892
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   686
lemma (in product_finite_prob_space) measure_singleton_times:
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   687
  assumes x: "x \<in> space P" shows "\<mu> {x} = (\<Prod>i\<in>I. M.\<mu> i {x i})"
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   688
  unfolding singleton_eq_product[OF x] using x
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   689
  by (intro measure_finite_times) auto
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   690
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   691
lemma (in product_finite_prob_space) prob_finite_times:
42859
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   692
  assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<subseteq> space (M i)"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   693
  shows "prob (\<Pi>\<^isub>E i\<in>I. X i) = (\<Prod>i\<in>I. M.prob i (X i))"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   694
proof -
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   695
  have "extreal (\<mu>' (\<Pi>\<^isub>E i\<in>I. X i)) = \<mu> (\<Pi>\<^isub>E i\<in>I. X i)"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   696
    using X by (intro finite_measure_eq[symmetric] in_P) auto
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   697
  also have "\<dots> = (\<Prod>i\<in>I. M.\<mu> i (X i))"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   698
    using measure_finite_times X by simp
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   699
  also have "\<dots> = extreal (\<Prod>i\<in>I. M.\<mu>' i (X i))"
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   700
    using X by (simp add: M.finite_measure_eq setprod_extreal)
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   701
  finally show ?thesis by simp
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   702
qed
d9dfc733f25c add product of probability spaces with finite cardinality
hoelzl
parents: 42858
diff changeset
   703
42892
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   704
lemma (in product_finite_prob_space) prob_singleton_times:
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   705
  assumes x: "x \<in> space P"
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   706
  shows "prob {x} = (\<Prod>i\<in>I. M.prob i {x i})"
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   707
  unfolding singleton_eq_product[OF x] using x
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   708
  by (intro prob_finite_times) auto
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   709
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   710
lemma (in product_finite_prob_space) prob_finite_product:
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   711
  "A \<subseteq> space P \<Longrightarrow> prob A = (\<Sum>x\<in>A. \<Prod>i\<in>I. M.prob i {x i})"
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   712
  by (auto simp add: finite_measure_singleton prob_singleton_times
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   713
           simp del: space_product_algebra
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   714
           intro!: setsum_cong prob_singleton_times)
a61e30bfd0bc add lemma prob_finite_product
hoelzl
parents: 42860
diff changeset
   715
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   716
lemma (in prob_space) joint_distribution_finite_prob_space:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   717
  assumes X: "finite_random_variable MX X"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   718
  assumes Y: "finite_random_variable MY Y"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   719
  shows "finite_prob_space ((MX \<Otimes>\<^isub>M MY)\<lparr> measure := extreal \<circ> joint_distribution X Y\<rparr>)"
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
   720
  by (intro distribution_finite_prob_space finite_random_variable_pairI X Y)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   721
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   722
lemma finite_prob_space_eq:
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
   723
  "finite_prob_space M \<longleftrightarrow> finite_measure_space M \<and> measure M (space M) = 1"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   724
  unfolding finite_prob_space_def finite_measure_space_def prob_space_def prob_space_axioms_def
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   725
  by auto
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   726
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   727
lemma (in prob_space) not_empty: "space M \<noteq> {}"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   728
  using prob_space empty_measure' by auto
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   729
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   730
lemma (in finite_prob_space) sum_over_space_eq_1: "(\<Sum>x\<in>space M. \<mu> {x}) = 1"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 36624
diff changeset
   731
  using measure_space_1 sum_over_space by simp
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   732
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   733
lemma (in finite_prob_space) joint_distribution_restriction_fst:
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   734
  "joint_distribution X Y A \<le> distribution X (fst ` A)"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   735
  unfolding distribution_def
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   736
proof (safe intro!: finite_measure_mono)
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   737
  fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   738
  show "x \<in> X -` fst ` A"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   739
    by (auto intro!: image_eqI[OF _ *])
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   740
qed (simp_all add: sets_eq_Pow)
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   741
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   742
lemma (in finite_prob_space) joint_distribution_restriction_snd:
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   743
  "joint_distribution X Y A \<le> distribution Y (snd ` A)"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   744
  unfolding distribution_def
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   745
proof (safe intro!: finite_measure_mono)
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   746
  fix x assume "x \<in> space M" and *: "(X x, Y x) \<in> A"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   747
  show "x \<in> Y -` snd ` A"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   748
    by (auto intro!: image_eqI[OF _ *])
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   749
qed (simp_all add: sets_eq_Pow)
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   750
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   751
lemma (in finite_prob_space) distribution_order:
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   752
  shows "0 \<le> distribution X x'"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   753
  and "(distribution X x' \<noteq> 0) \<longleftrightarrow> (0 < distribution X x')"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   754
  and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution X {x}"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   755
  and "r \<le> joint_distribution X Y {(x, y)} \<Longrightarrow> r \<le> distribution Y {y}"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   756
  and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution X {x}"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   757
  and "r < joint_distribution X Y {(x, y)} \<Longrightarrow> r < distribution Y {y}"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   758
  and "distribution X {x} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   759
  and "distribution Y {y} = 0 \<Longrightarrow> joint_distribution X Y {(x, y)} = 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   760
  using
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   761
    joint_distribution_restriction_fst[of X Y "{(x, y)}"]
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   762
    joint_distribution_restriction_snd[of X Y "{(x, y)}"]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   763
  by (auto intro: antisym)
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   764
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   765
lemma (in finite_prob_space) distribution_mono:
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   766
  assumes "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   767
  shows "distribution X x \<le> distribution Y y"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   768
  unfolding distribution_def
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   769
  using assms by (auto simp: sets_eq_Pow intro!: finite_measure_mono)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   770
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   771
lemma (in finite_prob_space) distribution_mono_gt_0:
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   772
  assumes gt_0: "0 < distribution X x"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   773
  assumes *: "\<And>t. \<lbrakk> t \<in> space M ; X t \<in> x \<rbrakk> \<Longrightarrow> Y t \<in> y"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   774
  shows "0 < distribution Y y"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   775
  by (rule less_le_trans[OF gt_0 distribution_mono]) (rule *)
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   776
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   777
lemma (in finite_prob_space) sum_over_space_distrib:
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   778
  "(\<Sum>x\<in>X`space M. distribution X {x}) = 1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   779
  unfolding distribution_def prob_space[symmetric] using finite_space
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   780
  by (subst finite_measure_finite_Union[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   781
     (auto simp add: disjoint_family_on_def sets_eq_Pow
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   782
           intro!: arg_cong[where f=\<mu>'])
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   783
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   784
lemma (in finite_prob_space) finite_sum_over_space_eq_1:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   785
  "(\<Sum>x\<in>space M. prob {x}) = 1"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   786
  using prob_space finite_space
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   787
  by (subst (asm) finite_measure_finite_singleton) auto
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   788
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   789
lemma (in prob_space) distribution_remove_const:
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   790
  shows "joint_distribution X (\<lambda>x. ()) {(x, ())} = distribution X {x}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   791
  and "joint_distribution (\<lambda>x. ()) X {((), x)} = distribution X {x}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   792
  and "joint_distribution X (\<lambda>x. (Y x, ())) {(x, y, ())} = joint_distribution X Y {(x, y)}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   793
  and "joint_distribution X (\<lambda>x. ((), Y x)) {(x, (), y)} = joint_distribution X Y {(x, y)}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   794
  and "distribution (\<lambda>x. ()) {()} = 1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   795
  by (auto intro!: arg_cong[where f=\<mu>'] simp: distribution_def prob_space[symmetric])
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35929
diff changeset
   796
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   797
lemma (in finite_prob_space) setsum_distribution_gen:
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   798
  assumes "Z -` {c} \<inter> space M = (\<Union>x \<in> X`space M. Y -` {f x}) \<inter> space M"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   799
  and "inj_on f (X`space M)"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   800
  shows "(\<Sum>x \<in> X`space M. distribution Y {f x}) = distribution Z {c}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   801
  unfolding distribution_def assms
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   802
  using finite_space assms
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   803
  by (subst finite_measure_finite_Union[symmetric])
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   804
     (auto simp add: disjoint_family_on_def sets_eq_Pow inj_on_def
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   805
      intro!: arg_cong[where f=prob])
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   806
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   807
lemma (in finite_prob_space) setsum_distribution:
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   808
  "(\<Sum>x \<in> X`space M. joint_distribution X Y {(x, y)}) = distribution Y {y}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   809
  "(\<Sum>y \<in> Y`space M. joint_distribution X Y {(x, y)}) = distribution X {x}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   810
  "(\<Sum>x \<in> X`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution Y Z {(y, z)}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   811
  "(\<Sum>y \<in> Y`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Z {(x, z)}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   812
  "(\<Sum>z \<in> Z`space M. joint_distribution X (\<lambda>x. (Y x, Z x)) {(x, y, z)}) = joint_distribution X Y {(x, y)}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   813
  by (auto intro!: inj_onI setsum_distribution_gen)
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   814
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   815
lemma (in finite_prob_space) uniform_prob:
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   816
  assumes "x \<in> space M"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   817
  assumes "\<And> x y. \<lbrakk>x \<in> space M ; y \<in> space M\<rbrakk> \<Longrightarrow> prob {x} = prob {y}"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   818
  shows "prob {x} = 1 / card (space M)"
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   819
proof -
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   820
  have prob_x: "\<And> y. y \<in> space M \<Longrightarrow> prob {y} = prob {x}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   821
    using assms(2)[OF _ `x \<in> space M`] by blast
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   822
  have "1 = prob (space M)"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   823
    using prob_space by auto
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   824
  also have "\<dots> = (\<Sum> x \<in> space M. prob {x})"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   825
    using finite_measure_finite_Union[of "space M" "\<lambda> x. {x}", simplified]
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   826
      sets_eq_Pow inj_singleton[unfolded inj_on_def, rule_format]
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   827
      finite_space unfolding disjoint_family_on_def  prob_space[symmetric]
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   828
    by (auto simp add:setsum_restrict_set)
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   829
  also have "\<dots> = (\<Sum> y \<in> space M. prob {x})"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   830
    using prob_x by auto
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   831
  also have "\<dots> = real_of_nat (card (space M)) * prob {x}" by simp
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   832
  finally have one: "1 = real (card (space M)) * prob {x}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   833
    using real_eq_of_nat by auto
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   834
  hence two: "real (card (space M)) \<noteq> 0" by fastsimp 
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   835
  from one have three: "prob {x} \<noteq> 0" by fastsimp
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   836
  thus ?thesis using one two three divide_cancel_right
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   837
    by (auto simp:field_simps)
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   838
qed
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35929
diff changeset
   839
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   840
lemma (in prob_space) prob_space_subalgebra:
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   841
  assumes "sigma_algebra N" "sets N \<subseteq> sets M" "space N = 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
   842
    and "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   843
  shows "prob_space N"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   844
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
   845
  interpret N: measure_space N
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
   846
    by (rule measure_space_subalgebra[OF assms])
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   847
  show ?thesis
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
   848
  proof qed (insert assms(4)[OF N.top], simp add: assms measure_space_1)
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35929
diff changeset
   849
qed
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35929
diff changeset
   850
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   851
lemma (in prob_space) prob_space_of_restricted_space:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   852
  assumes "\<mu> A \<noteq> 0" "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
   853
  shows "prob_space (restricted_space A \<lparr>measure := \<lambda>S. \<mu> S / \<mu> A\<rparr>)"
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
   854
    (is "prob_space ?P")
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
   855
proof -
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
   856
  interpret A: measure_space "restricted_space A"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   857
    using `A \<in> sets M` by (rule restricted_measure_space)
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
   858
  interpret A': sigma_algebra ?P
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
   859
    by (rule A.sigma_algebra_cong) auto
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
   860
  show "prob_space ?P"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   861
  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
   862
    show "measure ?P (space ?P) = 1"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   863
      using real_measure[OF `A \<in> events`] `\<mu> A \<noteq> 0` by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   864
    show "positive ?P (measure ?P)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   865
    proof (simp add: positive_def, safe)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   866
      show "0 / \<mu> A = 0" using `\<mu> A \<noteq> 0` by (cases "\<mu> A") (auto simp: zero_extreal_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   867
      fix B assume "B \<in> events"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   868
      with real_measure[of "A \<inter> B"] real_measure[OF `A \<in> events`] `A \<in> sets M`
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   869
      show "0 \<le> \<mu> (A \<inter> B) / \<mu> A" by (auto simp: Int)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   870
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   871
    show "countably_additive ?P (measure ?P)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   872
    proof (simp add: countably_additive_def, safe)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   873
      fix B and F :: "nat \<Rightarrow> 'a set"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   874
      assume F: "range F \<subseteq> op \<inter> A ` events" "disjoint_family F"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   875
      { fix i
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   876
        from F have "F i \<in> op \<inter> A ` events" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   877
        with `A \<in> events` have "F i \<in> events" by auto }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   878
      moreover then have "range F \<subseteq> events" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   879
      moreover have "\<And>S. \<mu> S / \<mu> A = inverse (\<mu> A) * \<mu> S"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   880
        by (simp add: mult_commute divide_extreal_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   881
      moreover have "0 \<le> inverse (\<mu> A)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   882
        using real_measure[OF `A \<in> events`] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   883
      ultimately show "(\<Sum>i. \<mu> (F i) / \<mu> A) = \<mu> (\<Union>i. F i) / \<mu> A"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   884
        using measure_countably_additive[of F] F
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   885
        by (auto simp: suminf_cmult_extreal)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   886
    qed
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   887
  qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   888
qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   889
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   890
lemma finite_prob_spaceI:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   891
  assumes "finite (space M)" "sets M = Pow(space M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   892
    and "measure M (space M) = 1" "measure M {} = 0" "\<And>A. A \<subseteq> space M \<Longrightarrow> 0 \<le> measure M A"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   893
    and "\<And>A B. A\<subseteq>space M \<Longrightarrow> B\<subseteq>space M \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> measure M (A \<union> B) = measure M A + measure M B"
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
   894
  shows "finite_prob_space M"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   895
  unfolding finite_prob_space_eq
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   896
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
   897
  show "finite_measure_space M" using assms
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   898
    by (auto intro!: finite_measure_spaceI)
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
   899
  show "measure M (space M) = 1" by fact
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   900
qed
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   901
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   902
lemma (in finite_prob_space) finite_measure_space:
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   903
  fixes X :: "'a \<Rightarrow> 'x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   904
  shows "finite_measure_space \<lparr>space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X\<rparr>"
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
   905
    (is "finite_measure_space ?S")
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   906
proof (rule finite_measure_spaceI, simp_all)
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   907
  show "finite (X ` space M)" using finite_space by simp
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   908
next
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   909
  fix A B :: "'x set" assume "A \<inter> B = {}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   910
  then show "distribution X (A \<union> B) = distribution X A + distribution X B"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   911
    unfolding distribution_def
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   912
    by (subst finite_measure_Union[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   913
       (auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow)
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   914
qed
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   915
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   916
lemma (in finite_prob_space) finite_prob_space_of_images:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   917
  "finite_prob_space \<lparr> space = X ` space M, sets = Pow (X ` space M), measure = extreal \<circ> distribution X \<rparr>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   918
  by (simp add: finite_prob_space_eq finite_measure_space measure_space_1 one_extreal_def)
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   919
39096
hoelzl
parents: 39092
diff changeset
   920
lemma (in finite_prob_space) finite_product_measure_space:
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   921
  fixes X :: "'a \<Rightarrow> 'x" and Y :: "'a \<Rightarrow> 'y"
39096
hoelzl
parents: 39092
diff changeset
   922
  assumes "finite s1" "finite s2"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   923
  shows "finite_measure_space \<lparr> space = s1 \<times> s2, sets = Pow (s1 \<times> s2), measure = extreal \<circ> joint_distribution X Y\<rparr>"
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
   924
    (is "finite_measure_space ?M")
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   925
proof (rule finite_measure_spaceI, simp_all)
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   926
  show "finite (s1 \<times> s2)"
39096
hoelzl
parents: 39092
diff changeset
   927
    using assms by auto
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   928
next
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   929
  fix A B :: "('x*'y) set" assume "A \<inter> B = {}"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   930
  then show "joint_distribution X Y (A \<union> B) = joint_distribution X Y A + joint_distribution X Y B"
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   931
    unfolding distribution_def
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   932
    by (subst finite_measure_Union[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   933
       (auto intro!: arg_cong[where f=\<mu>'] simp: sets_eq_Pow)
39096
hoelzl
parents: 39092
diff changeset
   934
qed
hoelzl
parents: 39092
diff changeset
   935
39097
943c7b348524 Moved lemmas to appropriate locations
hoelzl
parents: 39096
diff changeset
   936
lemma (in finite_prob_space) finite_product_measure_space_of_images:
39096
hoelzl
parents: 39092
diff changeset
   937
  shows "finite_measure_space \<lparr> space = X ` space M \<times> Y ` 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
   938
                                sets = Pow (X ` space M \<times> Y ` space M),
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   939
                                measure = extreal \<circ> joint_distribution X Y \<rparr>"
39096
hoelzl
parents: 39092
diff changeset
   940
  using finite_space by (auto intro!: finite_product_measure_space)
hoelzl
parents: 39092
diff changeset
   941
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   942
lemma (in finite_prob_space) finite_product_prob_space_of_images:
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
   943
  "finite_prob_space \<lparr> space = X ` space M \<times> Y ` space M, sets = Pow (X ` space M \<times> Y ` space M),
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   944
                       measure = extreal \<circ> joint_distribution X Y \<rparr>"
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
   945
  (is "finite_prob_space ?S")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   946
proof (simp add: finite_prob_space_eq finite_product_measure_space_of_images one_extreal_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   947
  have "X -` X ` space M \<inter> Y -` Y ` space M \<inter> space M = space M" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   948
  thus "joint_distribution X Y (X ` space M \<times> Y ` space M) = 1"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   949
    by (simp add: distribution_def prob_space vimage_Times comp_def measure_space_1)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   950
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 39302
diff changeset
   951
39085
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
   952
section "Conditional Expectation and Probability"
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
   953
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
   954
lemma (in prob_space) conditional_expectation_exists:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   955
  fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   956
  assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X 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
   957
  and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   958
  shows "\<exists>Y\<in>borel_measurable N. (\<forall>x. 0 \<le> Y x) \<and> (\<forall>C\<in>sets N.
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   959
      (\<integral>\<^isup>+x. Y x * indicator C x \<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x \<partial>M))"
39083
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   960
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
   961
  note N(4)[simp]
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
   962
  interpret P: prob_space N
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   963
    using prob_space_subalgebra[OF N] .
39083
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   964
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   965
  let "?f A" = "\<lambda>x. X x * indicator A 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
   966
  let "?Q A" = "integral\<^isup>P M (?f A)"
39083
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   967
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   968
  from measure_space_density[OF borel]
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
   969
  have Q: "measure_space (N\<lparr> measure := ?Q \<rparr>)"
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
   970
    apply (rule measure_space.measure_space_subalgebra[of "M\<lparr> measure := ?Q \<rparr>"])
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
   971
    using N by (auto intro!: P.sigma_algebra_cong)
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
   972
  then interpret Q: measure_space "N\<lparr> measure := ?Q \<rparr>" .
39083
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   973
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   974
  have "P.absolutely_continuous ?Q"
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   975
    unfolding P.absolutely_continuous_def
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   976
  proof safe
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
   977
    fix A assume "A \<in> sets N" "P.\<mu> A = 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   978
    then have f_borel: "?f A \<in> borel_measurable M" "AE x. x \<notin> A"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   979
      using borel N by (auto intro!: borel_measurable_indicator AE_not_in)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   980
    then show "?Q A = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   981
      by (auto simp add: positive_integral_0_iff_AE)
39083
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   982
  qed
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   983
  from P.Radon_Nikodym[OF Q this]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   984
  obtain Y where Y: "Y \<in> borel_measurable N" "\<And>x. 0 \<le> Y 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
   985
    "\<And>A. A \<in> sets N \<Longrightarrow> ?Q A =(\<integral>\<^isup>+x. Y x * indicator A x \<partial>N)"
39083
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   986
    by blast
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   987
  with N(2) show ?thesis
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   988
    by (auto intro!: bexI[OF _ Y(1)] simp: positive_integral_subalgebra[OF _ _ N(2,3,4,1)])
39083
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   989
qed
e46acc0ea1fe introduced integration on subalgebras
hoelzl
parents: 38656
diff changeset
   990
39085
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
   991
definition (in prob_space)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   992
  "conditional_expectation N X = (SOME Y. Y\<in>borel_measurable N \<and> (\<forall>x. 0 \<le> Y 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
   993
    \<and> (\<forall>C\<in>sets N. (\<integral>\<^isup>+x. Y x * indicator C x\<partial>M) = (\<integral>\<^isup>+x. X x * indicator C x\<partial>M)))"
39085
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
   994
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
   995
abbreviation (in prob_space)
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 39091
diff changeset
   996
  "conditional_prob N A \<equiv> conditional_expectation N (indicator A)"
39085
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
   997
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
   998
lemma (in prob_space)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   999
  fixes X :: "'a \<Rightarrow> extreal" and N :: "('a, 'b) measure_space_scheme"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1000
  assumes borel: "X \<in> borel_measurable M" "AE x. 0 \<le> X 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
  1001
  and N: "sigma_algebra N" "sets N \<subseteq> sets M" "space N = space M" "\<And>A. measure N A = \<mu> A"
39085
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
  1002
  shows borel_measurable_conditional_expectation:
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1003
    "conditional_expectation N X \<in> borel_measurable N"
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1004
  and conditional_expectation: "\<And>C. C \<in> sets N \<Longrightarrow>
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
  1005
      (\<integral>\<^isup>+x. conditional_expectation N X x * indicator C x \<partial>M) =
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1006
      (\<integral>\<^isup>+x. X x * indicator C x \<partial>M)"
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1007
   (is "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C")
39085
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
  1008
proof -
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
  1009
  note CE = conditional_expectation_exists[OF assms, unfolded Bex_def]
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1010
  then show "conditional_expectation N X \<in> borel_measurable N"
39085
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
  1011
    unfolding conditional_expectation_def by (rule someI2_ex) blast
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
  1012
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1013
  from CE show "\<And>C. C \<in> sets N \<Longrightarrow> ?eq C"
39085
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
  1014
    unfolding conditional_expectation_def by (rule someI2_ex) blast
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
  1015
qed
8b7c009da23c added definition of conditional expectation
hoelzl
parents: 39084
diff changeset
  1016
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1017
lemma (in sigma_algebra) factorize_measurable_function_pos:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1018
  fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c"
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1019
  assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1020
  assumes Z: "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1021
  shows "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. max 0 (Z x) = g (Y x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1022
proof -
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1023
  interpret M': sigma_algebra M' by fact
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1024
  have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1025
  from M'.sigma_algebra_vimage[OF this]
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1026
  interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1027
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1028
  from va.borel_measurable_implies_simple_function_sequence'[OF Z] guess f . note f = this
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1029
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
  1030
  have "\<forall>i. \<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1031
  proof
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1032
    fix i
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1033
    from f(1)[of i] have "finite (f i`space M)" and B_ex:
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1034
      "\<forall>z\<in>(f i)`space M. \<exists>B. B \<in> sets M' \<and> (f i) -` {z} \<inter> space M = Y -` B \<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
  1035
      unfolding simple_function_def by auto
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1036
    from B_ex[THEN bchoice] guess B .. note B = this
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1037
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1038
    let ?g = "\<lambda>x. \<Sum>z\<in>f i`space M. z * indicator (B z) x"
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1039
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
  1040
    show "\<exists>g. simple_function M' g \<and> (\<forall>x\<in>space M. f i x = g (Y x))"
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1041
    proof (intro exI[of _ ?g] conjI ballI)
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
  1042
      show "simple_function M' ?g" using B by auto
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1043
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1044
      fix x assume "x \<in> space M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1045
      then have "\<And>z. z \<in> f i`space M \<Longrightarrow> indicator (B z) (Y x) = (indicator (f i -` {z} \<inter> space M) x::extreal)"
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1046
        unfolding indicator_def using B by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1047
      then show "f i x = ?g (Y x)" using `x \<in> space M` f(1)[of i]
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1048
        by (subst va.simple_function_indicator_representation) auto
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1049
    qed
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1050
  qed
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1051
  from choice[OF this] guess g .. note g = this
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1052
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1053
  show ?thesis
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1054
  proof (intro ballI bexI)
41097
a1abfa4e2b44 use SUPR_ and INFI_apply instead of SUPR_, INFI_fun_expand
hoelzl
parents: 41095
diff changeset
  1055
    show "(\<lambda>x. SUP i. g i x) \<in> borel_measurable M'"
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1056
      using g by (auto intro: M'.borel_measurable_simple_function)
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1057
    fix x assume "x \<in> space M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1058
    have "max 0 (Z x) = (SUP i. f i x)" using f by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1059
    also have "\<dots> = (SUP i. g i (Y x))"
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1060
      using g `x \<in> space M` by simp
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1061
    finally show "max 0 (Z x) = (SUP i. g i (Y x))" .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1062
  qed
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
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1065
lemma (in sigma_algebra) factorize_measurable_function:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1066
  fixes Z :: "'a \<Rightarrow> extreal" and Y :: "'a \<Rightarrow> 'c"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1067
  assumes "sigma_algebra M'" and "Y \<in> measurable M M'" "Z \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1068
  shows "Z \<in> borel_measurable (sigma_algebra.vimage_algebra M' (space M) Y)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1069
    \<longleftrightarrow> (\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1070
proof safe
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1071
  interpret M': sigma_algebra M' by fact
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1072
  have Y: "Y \<in> space M \<rightarrow> space M'" using assms unfolding measurable_def by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1073
  from M'.sigma_algebra_vimage[OF this]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1074
  interpret va: sigma_algebra "M'.vimage_algebra (space M) Y" .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1075
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1076
  { fix g :: "'c \<Rightarrow> extreal" assume "g \<in> borel_measurable M'"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1077
    with M'.measurable_vimage_algebra[OF Y]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1078
    have "g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1079
      by (rule measurable_comp)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1080
    moreover assume "\<forall>x\<in>space M. Z x = g (Y x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1081
    then have "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y) \<longleftrightarrow>
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1082
       g \<circ> Y \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1083
       by (auto intro!: measurable_cong)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1084
    ultimately show "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1085
      by simp }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1086
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1087
  assume Z: "Z \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1088
  with assms have "(\<lambda>x. - Z x) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1089
    "(\<lambda>x. - Z x) \<in> borel_measurable (M'.vimage_algebra (space M) Y)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1090
    by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1091
  from factorize_measurable_function_pos[OF assms(1,2) this] guess n .. note n = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1092
  from factorize_measurable_function_pos[OF assms Z] guess p .. note p = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1093
  let "?g x" = "p x - n x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1094
  show "\<exists>g\<in>borel_measurable M'. \<forall>x\<in>space M. Z x = g (Y x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1095
  proof (intro bexI ballI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1096
    show "?g \<in> borel_measurable M'" using p n by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1097
    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
  1098
    then have "p (Y x) = max 0 (Z x)" "n (Y x) = max 0 (- Z x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1099
      using p n by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1100
    then show "Z x = ?g (Y x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1101
      by (auto split: split_max)
39091
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1102
  qed
11314c196e11 factorizable measurable functions
hoelzl
parents: 39090
diff changeset
  1103
qed
39090
a2d38b8b693e Introduced sigma algebra generated by function preimages.
hoelzl
parents: 39089
diff changeset
  1104
42902
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1105
subsection "Borel Measure on {0 .. 1}"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1106
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1107
definition pborel :: "real measure_space" where
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1108
  "pborel = lborel.restricted_space {0 .. 1}"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1109
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1110
lemma space_pborel[simp]:
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1111
  "space pborel = {0 .. 1}"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1112
  unfolding pborel_def by auto
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1113
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1114
lemma sets_pborel:
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1115
  "A \<in> sets pborel \<longleftrightarrow> A \<in> sets borel \<and> A \<subseteq> { 0 .. 1}"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1116
  unfolding pborel_def by auto
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1117
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1118
lemma in_pborel[intro, simp]:
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1119
  "A \<subseteq> {0 .. 1} \<Longrightarrow> A \<in> sets borel \<Longrightarrow> A \<in> sets pborel"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1120
  unfolding pborel_def by auto
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1121
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1122
interpretation pborel: measure_space pborel
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1123
  using lborel.restricted_measure_space[of "{0 .. 1}"]
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1124
  by (simp add: pborel_def)
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1125
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1126
interpretation pborel: prob_space pborel
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1127
  by default (simp add: one_extreal_def pborel_def)
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1128
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1129
lemma pborel_prob: "pborel.prob A = (if A \<in> sets borel \<and> A \<subseteq> {0 .. 1} then real (lborel.\<mu> A) else 0)"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1130
  unfolding pborel.\<mu>'_def by (auto simp: pborel_def)
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1131
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1132
lemma pborel_singelton[simp]: "pborel.prob {a} = 0"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1133
  by (auto simp: pborel_prob)
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1134
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1135
lemma
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1136
  shows pborel_atLeastAtMost[simp]: "pborel.\<mu>' {a .. b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1137
    and pborel_atLeastLessThan[simp]: "pborel.\<mu>' {a ..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1138
    and pborel_greaterThanAtMost[simp]: "pborel.\<mu>' {a <.. b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1139
    and pborel_greaterThanLessThan[simp]: "pborel.\<mu>' {a <..< b} = (if 0 \<le> a \<and> a \<le> b \<and> b \<le> 1 then b - a else 0)"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1140
  unfolding pborel_prob by (auto simp: atLeastLessThan_subseteq_atLeastAtMost_iff
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1141
    greaterThanAtMost_subseteq_atLeastAtMost_iff greaterThanLessThan_subseteq_atLeastAtMost_iff)
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1142
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1143
lemma pborel_lebesgue_measure:
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1144
  "A \<in> sets pborel \<Longrightarrow> pborel.prob A = real (measure lebesgue A)"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1145
  by (simp add: sets_pborel pborel_prob)
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1146
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1147
lemma pborel_alt:
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1148
  "pborel = sigma \<lparr>
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1149
    space = {0..1},
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1150
    sets = range (\<lambda>(x,y). {x..y} \<inter> {0..1}),
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1151
    measure = measure lborel \<rparr>" (is "_ = ?R")
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1152
proof -
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1153
  have *: "{0..1::real} \<in> sets borel" by auto
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1154
  have **: "op \<inter> {0..1::real} ` range (\<lambda>(x, y). {x..y}) = range (\<lambda>(x,y). {x..y} \<inter> {0..1})"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1155
    unfolding image_image by (intro arg_cong[where f=range]) auto
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1156
  have "pborel = algebra.restricted_space (sigma \<lparr>space=UNIV, sets=range (\<lambda>(a, b). {a .. b :: real}),
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1157
    measure = measure pborel\<rparr>) {0 .. 1}"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1158
    by (simp add: sigma_def lebesgue_def pborel_def borel_eq_atLeastAtMost lborel_def)
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1159
  also have "\<dots> = ?R"
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1160
    by (subst restricted_sigma)
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1161
       (simp_all add: sets_sigma sigma_sets.Basic ** pborel_def image_eqI[of _ _ "(0,1)"])
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1162
  finally show ?thesis .
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1163
qed
e8dbf90a2f3b Add restricted borel measure to {0 .. 1}
hoelzl
parents: 42892
diff changeset
  1164
42860
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1165
subsection "Bernoulli space"
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1166
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1167
definition "bernoulli_space p = \<lparr> space = UNIV, sets = UNIV,
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1168
  measure = extreal \<circ> setsum (\<lambda>b. if b then min 1 (max 0 p) else 1 - min 1 (max 0 p)) \<rparr>"
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1169
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1170
interpretation bernoulli: finite_prob_space "bernoulli_space p" for p
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1171
  by (rule finite_prob_spaceI)
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1172
     (auto simp: bernoulli_space_def UNIV_bool one_extreal_def setsum_Un_disjoint intro!: setsum_nonneg)
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1173
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1174
lemma bernoulli_measure:
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1175
  "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p B = (\<Sum>b\<in>B. if b then p else 1 - p)"
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1176
  unfolding bernoulli.\<mu>'_def unfolding bernoulli_space_def by (auto intro!: setsum_cong)
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1177
b02349e70d5a add Bernoulli space
hoelzl
parents: 42859
diff changeset
  1178
lemma bernoulli_measure_True: "0 \<le> p \<Longrightarrow> p \<le> 1 \<Longrightarrow> bernoulli.prob p {True} = p"